{"id":1863,"date":"2022-08-01T11:49:40","date_gmt":"2022-08-01T11:49:40","guid":{"rendered":"https:\/\/blog.praxilabs.com\/?p=1863"},"modified":"2025-10-19T18:09:19","modified_gmt":"2025-10-19T18:09:19","slug":"all-you-need-about-pascals-law","status":"publish","type":"post","link":"https:\/\/praxilabs.com\/en\/blog\/2022\/08\/01\/all-you-need-about-pascals-law\/","title":{"rendered":"Pascal\u2019s Law: All You Need to Know about this Powerful Law!"},"content":{"rendered":"<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Pressure is an intrinsic property of matter. You can exert pressure on a solid, a liquid, or on a gas, and so can solids, liquids, and gasses according to the surrounding conditions. A famous example in this context is the cautionary signs on your perfume bottle, saying \u201cthe bottle should not be left in the sun, or in high temperatures, to avoid explosions.\u201d<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Liquids and gasses\u2014known as fluids\u2014are the two types of matter that go through continuous and noticeable changes caused by pressure. Many laws of physics discuss and explain the behavior and mechanisms of fluids. And Pascal\u2019s law is on the top of this list. The law is also known as <i>the principle of transmission of fluid-pressure,<\/i> or <i>Pascal&#8217;s principle.<\/i><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Discover everything about Pascal\u2019s law definition, its formula,\u00a0 and applications in this article. Besides, we are going to see how Pascal\u2019s law is used in hydraulic lift and how to calculate Pascal\u2019s law. Let\u2019s start figuring this law out\u2026<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><a class=\"maxbutton-3 maxbutton\" href=\"https:\/\/praxilabs.com\/en\/virtual-physics-lab\"><span class='mb-text'>Discover Praxilabs Virtual Physics Lab<\/span><\/a><\/strong><\/span><\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\r\n<div class=\"ez-toc-title-container\">\r\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\r\n<span class=\"ez-toc-title-toggle\"><\/span><\/div>\r\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/08\/01\/all-you-need-about-pascals-law\/#Pascals_Law_of_Hydrostatics\" >Pascal&#8217;s Law of Hydrostatics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/08\/01\/all-you-need-about-pascals-law\/#Intermolecular_Forces_and_Pascals_Law\" >Intermolecular Forces and Pascal\u2019s Law<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/08\/01\/all-you-need-about-pascals-law\/#Pascals_Law_Formula\" >Pascal\u2019s Law Formula<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/08\/01\/all-you-need-about-pascals-law\/#Check_Your_Understanding\" >Check Your Understanding!<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/08\/01\/all-you-need-about-pascals-law\/#State_and_Prove_Pascals_Law\" >State and Prove Pascal\u2019s Law<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/08\/01\/all-you-need-about-pascals-law\/#How_to_Calculate_Pascals_Law\" >How to Calculate Pascal\u2019s Law?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/praxilabs.com\/en\/blog\/2022\/08\/01\/all-you-need-about-pascals-law\/#Applications_of_Pascals_Law\" >Applications of Pascal\u2019s Law<\/a><\/li><\/ul><\/nav><\/div>\r\n<h2><span class=\"ez-toc-section\" id=\"Pascals_Law_of_Hydrostatics\"><\/span><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><b>Pascal&#8217;s Law of Hydrostatics<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The French mathematician, philosopher, and physicist Blaise Pascal was interested in developing the concept of pressure (1623-1662). He discovered his famous Pascal\u2019s law while conducting an interesting experiment. In addition to his interest in pressure, Pascal was the scientist who developed the modern theory of probability.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The experiment consisted of a 10 m vertically inserted long pole into a barrel. The barrel was filled with water, and a water flow was running through that long pole. As the water was poured, he noticed that the barrel burst due to the water\u2019s pressure. He concluded that the pressure caused by the water\u2014or any fluid in general\u2014was transmitted through the water in all directions.<\/span><\/p>\n<p><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5016 aligncenter\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-2-129x300.webp\" alt=\"Pascal's law and Pascal's barrel experiment\" width=\"129\" height=\"300\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-2-129x300.webp 129w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-2.webp 265w\" sizes=\"auto, (max-width: 129px) 100vw, 129px\" \/><\/span><\/p>\n<p><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\">An illustration showing the setup of Pascal&#8217;s barrel (Source: <span style=\"text-decoration: underline;\"><a href=\"https:\/\/www.ck12.org\/c\/physics\/fluid-pressure\/rwa\/Pascals-Principle\/\" target=\"_blank\" rel=\"noopener\">Ck-12<\/a><\/span>)<\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\"><br \/>\nPascal realized that, for a <\/span>static fluid, the pressure at a point will be the same across all planes passing through that specific point of the fluid. For example, if you are using static oil instead of water, and you are conducting the same experiment as Pascal did, the final result will be the same. Such that:<br \/>\n<span style=\"font-weight: 400;\">The oil going through the tube will cause the barrel to burst. Hence, the pressure caused by it will be transmitted to the oil in the barrel and in all directions.<\/span><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">In itself, the term fluid statics\u2014also known as hydrostatics\u2014is a branch of fluid mechanics that is concerned with studying the equilibrium condition of a floating body and a submerged body. In other words, it studies fluids at hydrostatic equilibrium and the pressure <i>in<\/i> a fluid, or <i>exerted by<\/i> a fluid, on any immersed body.<\/span><\/p>\n<p><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5017 size-full\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-3.webp\" alt=\"Pascal's law and hydrostatic fluids\" width=\"768\" height=\"469\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-3.webp 768w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-3-300x183.webp 300w\" sizes=\"auto, (max-width: 768px) 100vw, 768px\" \/><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Fluid statics is very contrary to fluid dynamics, since the first studies the conditions under which fluid are at rest and are in a stable equilibrium, whereas the latter refers to studying fluids that are in motion. Nevertheless, hydrostatics is a subcategory of fluid statics.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Hydrostatics is fundamental to hydraulics. Hydrostatics is relevant to both astrophysics (in understanding the anomalies of earth\u2019s gravitational field) and geophysics (in understanding plate tectonics), while it is also important to many other fields, such as meteorology and medicine (in the context of blood pressure).<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Why do oil and wood float on the water\u2019s surface?<br \/>\n<\/span><span style=\"font-weight: 400;\">Why does atmospheric pressure change with altitude?<br \/>\n<\/span><span style=\"font-weight: 400;\">Why is still water\u2019s surface always level according to the earth\u2019s curvature?<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The answer to these three previous questions is hydrostatics as many phenomena in our daily life can be elaborately explained using this term, and how it functions and works. Thus, Pascal\u2019s law of hydrostatics helps us justify and understand many of the questions around us.<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Intermolecular_Forces_and_Pascals_Law\"><\/span><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><b>Intermolecular Forces and Pascal\u2019s Law<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">One thing to note is that Pascal\u2019s law is not applicable on gasses. And it is only used with liquids\u2026 Can you guess the reason behind this?<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">All types of matter are composed of particles at the microscopic level, where these particles are in random motion all the time. In the solid state of matter, particles are strongly bound together, where their motion is restricted to only jiggle about virtual and fixed equilibrium positions relative to each other. Because of this, we consider solids to be rigid.<\/span><\/p>\n<figure id=\"attachment_5018\" aria-describedby=\"caption-attachment-5018\" style=\"width: 695px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5018 size-full\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-4.webp\" alt=\"Pascal's law &amp; intermolecular forces\" width=\"695\" height=\"275\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-4.webp 695w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-4-300x119.webp 300w\" sizes=\"auto, (max-width: 695px) 100vw, 695px\" \/><figcaption id=\"caption-attachment-5018\" class=\"wp-caption-text\"><\/span> <span style=\"font-family: tahoma, arial, helvetica, sans-serif;\">Intermolecular forces, Pascal&#8217;s law and gases<\/span><\/figcaption><\/figure>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><br \/>\nOn the other hand, gas particles interact briefly under certain circumstances, and they cause intense collisions with other gas\u2019 particles or with the container\u2019s walls. Liquids stand alone in the middle between solids and gasses, where the motion of particles in a liquid is random but not as chaotic as that of the gas.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The average attractive forces between the particles of a liquid are sufficient to make the liquid cohesive yet are weak enough to allow the liquid to flow. Liquid particles pass each other smoothly while flowing, without causing the liquid\u2019s mass to undergo any macroscopic separation.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><a class=\"maxbutton-3 maxbutton\" href=\"https:\/\/praxilabs.com\/en\/sign-up\"><span class='mb-text'>Create an account to try our virtual experiment for FREE<\/span><\/a><\/strong><\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Pascals_Law_Formula\"><\/span><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><b>Pascal\u2019s Law Formula<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Now, can you deduce Pascal\u2019s law formula based on what we discussed previously?<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Pascal\u2019s law formula tackles three quantities of physics, which are:<\/span><\/p>\n<ol>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Pressure, denoted as <\/span><b>P<\/b>.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Force, denoted by <\/span><b>F<\/b>.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Area (or surface area), and it is denoted by <\/span><b>A<\/b>.<\/span><\/li>\n<\/ol>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">And the law is stated as follows:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">F = p*A<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">whereas the formula of <\/span><b>hydrostatic pressure<\/b><span style=\"font-weight: 400;\"> is given as<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">\u0394p = g(\u0394h)<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">where<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b><span style=\"font-weight: 400;\">\u0394<\/span>p:<\/b><span style=\"font-weight: 400;\"> the hydrostatic pressure exerted by the liquid, measured in (N.m<\/span><sup><span style=\"font-weight: 400;\">-2<\/span><\/sup><span style=\"font-weight: 400;\">, or Pa),<br \/>\n<\/span><b>\u03c1:<\/b><span style=\"font-weight: 400;\"> the density of the liquid, measured in (kg.m<\/span><sup><span style=\"font-weight: 400;\">-3<\/span><\/sup><span style=\"font-weight: 400;\">),<br \/>\n<\/span><b>g:<\/b><span style=\"font-weight: 400;\"> the acceleration due to gravity, put as <\/span><b>9.81 <\/b><span style=\"font-weight: 400;\">m.s<\/span><sup><span style=\"font-weight: 400;\">-2<\/span><\/sup><span style=\"font-weight: 400;\">, and<br \/>\n<\/span><b><span style=\"font-weight: 400;\">\u0394<\/span>h:<\/b><span style=\"font-weight: 400;\"> the difference in height of the fluid column, given in (m).<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">According to Pascal\u2019s law, the change in pressure of a fluid <\/span><i><span style=\"font-weight: 400;\">at rest<\/span><\/i><span style=\"font-weight: 400;\"> and in a <\/span><i><span style=\"font-weight: 400;\">closed<\/span><\/i><span style=\"font-weight: 400;\"> container is transmitted <\/span><i><span style=\"font-weight: 400;\">undiminished<\/span><\/i><span style=\"font-weight: 400;\"> and without any loss to every part of the fluid, as well as to the walls of the container.<\/span><\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Check_Your_Understanding\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Check Your Understanding!<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Check the figure below and ask yourself: which pressure will give the <b>highest<\/b> value of the three readings?<br \/>\n<\/span><\/p>\n<figure id=\"attachment_5019\" aria-describedby=\"caption-attachment-5019\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5019 size-medium\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-5-300x149.webp\" alt=\"An example of Pascal's law\" width=\"300\" height=\"149\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-5-300x149.webp 300w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-5.webp 635w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-5019\" class=\"wp-caption-text\"><\/span> <span style=\"font-family: tahoma, arial, helvetica, sans-serif;\">According to Pascal&#8217;s law, determine which pressure is the highest! (Source: <a href=\"https:\/\/www.grc.nasa.gov\/www\/k-12\/WindTunnel\/Activities\/Pascals_principle.html\" target=\"_blank\" rel=\"noopener\">NASA<\/a>)<\/span><\/figcaption><\/figure>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\"><br \/>\nThe right answer to this question is <\/span><b>P3. <\/b><span style=\"font-weight: 400;\">The reason that P3 has the highest value between the other two readings is because it has the <\/span><i><span style=\"font-weight: 400;\">highest level of fluid <\/span><\/i><span style=\"font-weight: 400;\">above it.<\/span><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">If the container undergoes an increase in overall pressure, this pressure will affect all the gauges <\/span><i><span style=\"font-weight: 400;\">equally<\/span><\/i><span style=\"font-weight: 400;\">. To comprehend this example more, suppose that our initial readings were 1, 3, and 5 units of pressure for P1, P2, and P3, respectively. We will add 5 more units of pressure to the system; thus the new readings will be 6, 8, and 10 for P1, P2, and P3, respectively.<\/span><\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"State_and_Prove_Pascals_Law\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>State and Prove Pascal\u2019s Law<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Pascal\u2019s principle, or Pascal\u2019s law, can be defined and stated in three different statements, which are:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><i><span style=\"font-weight: 400;\">\u201cA change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid\u201d<\/span><\/i><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">or<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><i><span style=\"font-weight: 400;\">\u201cPressure exerted on a fluid in an enclosed container is transmitted equally and undiminished to all parts of the container and acts at right angle to the enclosing walls\u201d<\/span><\/i><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">and<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><i><span style=\"font-weight: 400;\">\u201cThe pressure applied to any part of the enclosed liquid will be transmitted equally in all directions through the liquid.\u201d<\/span><\/i><\/span><\/p>\n<figure id=\"attachment_5020\" aria-describedby=\"caption-attachment-5020\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5020 size-medium\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-6-300x169.webp\" alt=\"The syringe as an example pf Pascal's law\" width=\"300\" height=\"169\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-6-300x169.webp 300w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-6.webp 768w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-5020\" class=\"wp-caption-text\"><\/span> <span style=\"font-family: tahoma, arial, helvetica, sans-serif;\">Syringe injection as a form of Pascal&#8217;s law applications (Source: <a href=\"https:\/\/www.science.org\/content\/article\/us-wants-end-most-payouts-leading-vaccination-related-injury\" target=\"_blank\" rel=\"noopener\">Science.org<\/a>)<\/span><\/figcaption><\/figure>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><br \/>\nSince Pascal was a mathematician, his law can be mathematically proved without any doubts. If you are a student, the derivation of Pascal\u2019s law is a very common question that you will encounter during your physics or engineering study.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">You might be wondering what is the point of deriving or proving physics laws mathematically, right?<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">The thing is mathematical derivations and proofs will help you understand further details regarding the law you are tackling or working with. In addition to its significant help in memorizing the laws more clearly, it also develops your skillset over time, especially when it comes to solving more complicated problems and questions. Thus, in this section, <\/span><b>Pascal\u2019s law will be proved mathematically<\/b><span style=\"font-weight: 400;\">\u2026 Let\u2019s start!<\/span><\/span><\/p>\n<figure id=\"attachment_5021\" aria-describedby=\"caption-attachment-5021\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5021 size-medium\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-7-300x154.webp\" alt=\"derivation and prove of Pascal's law\" width=\"300\" height=\"154\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-7-300x154.webp 300w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-7.webp 750w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-5021\" class=\"wp-caption-text\"><\/span> <span style=\"font-family: tahoma, arial, helvetica, sans-serif;\">An illustration of the pressure&#8217;s triangle to prove Pascal&#8217;s law (Source: <a href=\"https:\/\/byjus.com\/physics\/pascals-law-and-its-application\/\" target=\"_blank\" rel=\"noopener\">Byju&#8217;s<\/a>)<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-size: 14pt;\"><span style=\"font-weight: 400;\"><br \/>\nConsider an arbitrary <\/span><i><span style=\"font-weight: 400;\">prismatic right-angled <\/span><\/i><span style=\"font-weight: 400;\">triangle, put in a liquid of de<\/span><\/span><span style=\"font-size: 14pt;\"><span style=\"font-weight: 400;\">nsity <\/span><i><span style=\"font-weight: 400;\">rho <\/span><\/i><span style=\"font-weight: 400;\">(<\/span><span style=\"font-weight: 400;\">). The prismatic triangle is very small. Thus we consider every point of it to be at the same depth as that of the liquid\u2019s surface. As a consequence, <\/span><i><span style=\"font-weight: 400;\">temperature<\/span><\/i><span style=\"font-weight: 400;\"> (T) is the same at all these points, as well as <\/span><i><span style=\"font-weight: 400;\">the effect of gravity<\/span><\/i><span style=\"font-weight: 400;\">.<\/span><\/span><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">In the triangle:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Let ad, bd, and cd be the area of the faces ABFE, ABDC, and CDFE, respectively.<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Let P1, P2, and P3 be the pressure on the faces ABFE, ABDC, and CDFE, respectively.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400;\">Since pressure exerts a <\/span><b>normal<\/b><span style=\"font-weight: 400;\"> force to the surface, let P1 exert force F1 on the surface ABFE, P2 exert force F2 on the surface ABDC, and P3 exert force F3 on the surface CDFE.<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The forces F1, F2, and F3 are hence given by<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">F1 <span style=\"font-weight: 400;\">= P1 \u00d7 area of ABFE = (P1)*(ad)<br \/>\n<\/span>F2<span style=\"font-weight: 400;\"> = P2 \u00d7 area of ABDC = (P2)*(bd)<br \/>\n<\/span>F3<span style=\"font-weight: 400;\"> = P3 \u00d7 area of CDFE = (P3)*(cd)<\/span><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Note: <\/b><span style=\"font-weight: 400;\">the net force acting on the prism will be equal to zero, as long as the prism is in <\/span><b>equilibrium.<\/b><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Thus,<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">F1 sin(\u03b8) = F2, and<br \/>\nF2 cos(\u03b8) = F3<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Consequently,<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">(P1)*(ad) = (P2)*(bd)<br \/>\n(P1)*(ad) = (P3)*(cd)<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">From #1 and #2, we realize that<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">P1 = P2 and P1 = P3<br \/>\n\u2234 P1 = P2 = P3<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">This last equation shows that all the definitions of Pascal\u2019s law are proved right and they follow a well-constructed mathematical logic.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><a class=\"maxbutton-3 maxbutton\" href=\"https:\/\/praxilabs.com\/en\/request-free-demo\"><span class='mb-text'>Request a Demo To Start The Virtual Labs Experience<\/span><\/a><\/strong><\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"How_to_Calculate_Pascals_Law\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>How to Calculate Pascal\u2019s Law?<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">In the previous pages, you were introduced to all the needed information and tools you need to know about Pascal\u2019s law; from the law\u2019s history and how it works to the mathematical derivation and the different formulas of it. At this point, you are now ready to know how to really calculate and account for Pascal\u2019s law.<\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">To not waste your time, we will straightforwardly show two problems and walk through their solution.<\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Calculate the pressure acting on the water at a depth of 1 m at 4\u00b0C?<\/span><\/li>\n<\/ul>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Solution<\/b><\/span><\/p>\n<p><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><span style=\"font-weight: 400; font-size: 14pt;\">The depth of the water column = 1 m<br \/>\n<\/span><span style=\"font-size: 14pt;\"><span style=\"font-weight: 400;\">The density of water at 4\u00b0C = 1000 kg\/m<\/span><sup><span style=\"font-weight: 400;\">3<\/span><\/sup><\/span><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The formula in use is<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>p = \u03c1gh<\/b><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Thus,\u00a0<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">p = (1000 kg\/m<sup>3<\/sup>)(9.81 m.s<sup>-2<\/sup>)(1 m) = 9810 Pa<\/span><\/strong><\/span><\/p>\n<ul>\n<li><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Consider the automobile hydraulic system shown in the figure below, where a force of 100 N is applied to the brake pedal, which acts on the pedal cylinder through a lever.<br \/>\nA force of 500 N is exerted on the pedal cylinder. Pressure created in the pedal cylinder is transmitted to four wheel cylinders. The pedal cylinder has a diameter of 0.500 cm, and each wheel cylinder has a diameter of 2.50 cm.<br \/>\nCalculate the force F2 created at each of the wheel cylinders.<\/span><\/li>\n<\/ul>\n<figure id=\"attachment_5022\" aria-describedby=\"caption-attachment-5022\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5022 size-medium\" src=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-8-300x240.webp\" alt=\"Pascal's Law and Pascal Brakes\" width=\"300\" height=\"240\" srcset=\"https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-8-300x240.webp 300w, https:\/\/praxilabs.com\/en\/blog\/wp-content\/uploads\/2025\/08\/Pascals-Law-8.webp 768w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-5022\" class=\"wp-caption-text\"><\/span> <span style=\"font-family: tahoma, arial, helvetica, sans-serif;\">How does Pascal&#8217;s law affect the Pascal brakes? (Source: <a href=\"https:\/\/openstax.org\/books\/college-physics\/pages\/11-5-pascals-principle\" target=\"_blank\" rel=\"noopener\">OpenStax<\/a>)<\/span><\/figcaption><\/figure>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b><br \/>\nSolution Strategy<\/b><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">The cross-sectional areas A1 and A2 can be calculated from their given diameters. Since the force F1 that is applied to the pedal cylinder is given, the law F1*A1 = F2*A2 can be used to find the force F2. By manipulating the equations algebraically, we can deduce the value of F2 through substitution with the values.<\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Solution<\/b><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Pascal\u2019s principle applied to hydraulic systems is given by<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>F1 * A1 = F2 * A2<\/b><\/span><\/p>\n<p><span style=\"font-weight: 400; font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">Thus,\u00a0<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif;\"><strong><span style=\"font-size: 14pt;\">F2 = (A2\/A1)*F1 = ((\u03c0 (r2)<sup>2<\/sup>) \/ (\u03c0 (r1)<sup>2<\/sup>)) * F1<br \/>\nF2 = ((1.25 cm)<sup>2<\/sup> \/ (0.250 cm)<sup>2<\/sup>) \u00d7 500 N<br \/>\nF2 = 1.25 \u00d7 10<sup>4<\/sup> N<\/span><\/strong><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Discussion<\/b><\/span><\/p>\n<p><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">This value (1.25 \u00d7 10<sup>4<\/sup> N) is the amount of force exerted by each of the four wheel cylinders. Note that if we added more wheels and each has a 2.50 cm diameter, each will exert the same amount of force.<\/span><\/p>\n<h2 style=\"text-align: left;\"><span class=\"ez-toc-section\" id=\"Applications_of_Pascals_Law\"><\/span><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\"><b>Applications of Pascal\u2019s Law<\/b><\/span><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><iframe loading=\"lazy\" title=\"Principle of hydraulic lift\" width=\"618\" height=\"464\" src=\"https:\/\/www.youtube.com\/embed\/A3ormYVZMXE?start=1&#038;feature=oembed&#038;enablejsapi=1&#038;origin=https:\/\/praxilabs.com\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<p style=\"text-align: left;\"><span style=\"font-size: 14pt; font-family: tahoma, arial, helvetica, sans-serif;\">A <a href=\"https:\/\/praxilabs.com\/en\/blog\/2018\/02\/08\/virtual-labs-features-benefits\/\">virtual lab<\/a> can be of a great help to you! Where the finely-created experiments and 3D simulations will guide you through the topics of your interest, only with a click. <strong>PraxiLabs<\/strong> offer you many experiments to apply Pascal&#8217;s law by yourself, and tackle many of the pressure&#8217;s problems.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: tahoma, arial, helvetica, sans-serif; font-size: 14pt;\"><a href=\"https:\/\/praxilabs.com\/en\/sign-up\"><strong>Join PraxiLabs for FREE and Set the Pressure Yourself!<\/strong><\/a><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Pressure is an intrinsic property of matter. You can exert pressure on a solid, a liquid, or on a gas, and so can solids, liquids, and gasses according to the surrounding conditions. A famous example in this context is the cautionary signs on your perfume bottle, saying \u201cthe bottle should not be left in the &hellip;<\/p>\n","protected":false},"author":11,"featured_media":4518,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[4],"tags":[],"class_list":["post-1863","post","type-post","status-publish","format-standard","has-post-thumbnail","","category-physics"],"modified_by":"Muhamed Elmesery","_links":{"self":[{"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/posts\/1863","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/comments?post=1863"}],"version-history":[{"count":16,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/posts\/1863\/revisions"}],"predecessor-version":[{"id":5436,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/posts\/1863\/revisions\/5436"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/media\/4518"}],"wp:attachment":[{"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/media?parent=1863"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/categories?post=1863"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/praxilabs.com\/en\/blog\/wp-json\/wp\/v2\/tags?post=1863"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}