{"id":825714,"date":"2024-05-08T18:29:12","date_gmt":"2024-05-08T12:59:12","guid":{"rendered":"https:\/\/leverageedu.com\/discover\/?p=825714"},"modified":"2024-05-08T18:29:12","modified_gmt":"2024-05-08T12:59:12","slug":"basic-concept-pythagoras-theorem","status":"publish","type":"post","link":"https:\/\/leverageedu.com\/discover\/school-education\/basic-concept-pythagoras-theorem\/","title":{"rendered":"Pythagoras Theorem: Proof, Formula, Derivation, Examples \u00a0"},"content":{"rendered":"\n<p>The Pythagoras theorem is also known as the Pythagorean theorem. It is a basic concept in mathematics that describes the relationship between the lengths of the sides of a right-angled triangle. Furthermore, this theorem is named after the ancient Greek philosopher and mathematician Pythagoras, who is credited with its discovery as well as proof. In addition, the Theorem declares that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Read on to learn more about the father of the Pythagoras Theorem, the Rule for the theorem, the Formula and the Examples along with their solutions.&nbsp;<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>GRE Geometry: How Important is it?<\/strong><\/p>\n\n\n\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-who-is-the-father-of-the-pythagoras-theorem\">Who is the Father of the Pythagoras Theorem?<\/h2>\n\n\n\n<p>Pythagoras from c. 570 to 495 BCE was a Greek philosopher and mathematician who founded the Pythagorean school in Croton, Italy.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>He is remembered as one of the most significant mathematicians of all time.&nbsp;<\/li>\n\n\n\n<li>Moreover, his contributions to mathematics include the development of the Pythagorean Theorem.&nbsp;<\/li>\n\n\n\n<li>Furthermore, Pythagoras was a polymath who made noteworthy contributions to different fields, such as mathematics, music, and philosophy.&nbsp;<\/li>\n\n\n\n<li>His teachings emphasised the importance of harmony and balance, which are reflected in the mathematical principles he developed, including the Pythagorean theorem.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Conic Sections<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-what-is-the-rule-for-the-pythagorean-theorem\">What is the Rule for the Pythagorean Theorem?<\/h2>\n\n\n\n<p>Additionally, the Pythagorean theorem is a rule that applies specifically to right-angled triangles. Let us see what exactly is the Pythagoras Theorem statement.&nbsp;&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.&nbsp;<\/li>\n\n\n\n<li>This relationship is a fundamental property of right triangles and is used extensively in diverse mathematical and real-world applications.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><strong>Branches of Mathematics: Arithmetic, Algebra, Geometry, Calculus, Trigonometry, Topology, Probability and Statistics<\/strong>&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-pythagoras-theorem-formula-nbsp\">Pythagoras Theorem Formula&nbsp;<\/h2>\n\n\n\n<p>Therefore, the Formula for the Pythagoras Theorem is:<\/p>\n\n\n\n<p><strong>c\u00b2 = a\u00b2 + b\u00b2<\/strong><br><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img  loading=\"lazy\"  decoding=\"async\"  width=\"1024\"  height=\"768\"  src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABAQMAAAAl21bKAAAAA1BMVEUAAP+KeNJXAAAAAXRSTlMAQObYZgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAApJREFUCNdjYAAAAAIAAeIhvDMAAAAASUVORK5CYII=\"  alt=\"\"  class=\"wp-image-825753 pk-lazyload\"  style=\"width:534px;height:auto\"  data-pk-sizes=\"auto\"  data-ls-sizes=\"auto, (max-width: 1024px) 100vw, 1024px\"  data-pk-src=\"https:\/\/cdnblogdiscover.leverageedu.com\/discover\/wp-content\/uploads\/2024\/05\/Pythagoras-Theorem-1.png\"  data-pk-srcset=\"https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110127\/Pythagoras-Theorem-1.png 1024w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110127\/Pythagoras-Theorem-1-300x225.png 300w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110127\/Pythagoras-Theorem-1-768x576.png 768w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110127\/Pythagoras-Theorem-1-150x113.png 150w\" ><\/figure>\n\n\n\n<p>where c is the length of the hypotenuse and a and b are the lengths of the other two sides.&nbsp;<\/p>\n\n\n\n<p>Additionally, this Formula can be used to find the length of the hypotenuse if the lengths of the other two sides are known, or to find the lengths of the other two sides if the length of the hypotenuse is known.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><a href=\"https:\/\/leverageedu.com\/blog\/wp-content\/uploads\/2021\/10\/Pythagoras-Theorem-Practice-Worksheet-1.pdf\"><strong>Pythagoras Theorem Practice Sheet PDF<\/strong><\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-proof-of-pythagoras-theorem\">Proof of Pythagoras Theorem<\/h2>\n\n\n\n<p>Read to find out how you can prove the Pythagoras theorem using simple steps.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Let us take a right triangle having points &#8216;ABC&#8217; with a right angle at point C.<\/li>\n\n\n\n<li>a, b, and c are the lengths of sides that are opposite, adjacent, and hypotenuse to the right angle, respectively.<\/li>\n\n\n\n<li>Now let us draw squares on each side of the triangle, where side a has points DEFG, side b has points KLMN, and side c has points WXYZ.<\/li>\n\n\n\n<li>After that, we will dissect them into smaller squares based on side length.<\/li>\n\n\n\n<li>We can rearrange the smaller squares from DEFG and KLMN and find that they can perfectly fill the square WXYZ.<\/li>\n\n\n\n<li>Therefore, the area of squares DEFG (a\u00b2) and KLMN (b\u00b2) combined equals the area of square WXYZ (c\u00b2), proving a\u00b2 + b\u00b2 = c\u00b2.<\/li>\n<\/ol>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-lateral-area-of-cylinder\/\"><strong>Lateral Area of Cylinder: Formula, Example and More!&nbsp;<\/strong><\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-pythagoras-theorem-examples\">Pythagoras Theorem Examples<\/h2>\n\n\n\n<p>Moreover, here are some examples of the Pythagoras Theorem:<\/p>\n\n\n\n<p><strong>Example 1:<\/strong> Given a triangle with sides 5, 12, and 13 units, applying the Pythagorean Theorem confirms it has a right angle.<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img  loading=\"lazy\"  decoding=\"async\"  width=\"1024\"  height=\"768\"  src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABAQMAAAAl21bKAAAAA1BMVEUAAP+KeNJXAAAAAXRSTlMAQObYZgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAApJREFUCNdjYAAAAAIAAeIhvDMAAAAASUVORK5CYII=\"  alt=\"\"  class=\"wp-image-825754 pk-lazyload\"  style=\"width:660px;height:auto\"  data-pk-sizes=\"auto\"  data-ls-sizes=\"auto, (max-width: 1024px) 100vw, 1024px\"  data-pk-src=\"https:\/\/cdnblogdiscover.leverageedu.com\/discover\/wp-content\/uploads\/2024\/05\/Pythagoras-Theorem-1-1.png\"  data-pk-srcset=\"https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110128\/Pythagoras-Theorem-1-1.png 1024w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110128\/Pythagoras-Theorem-1-1-300x225.png 300w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110128\/Pythagoras-Theorem-1-1-768x576.png 768w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110128\/Pythagoras-Theorem-1-1-150x113.png 150w\" ><\/figure>\n\n\n\n<p><strong>Example 2:<\/strong> In a right triangle with sides 7 and 24 units, find the length of the hypotenuse (AC) using the theorem.<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img  loading=\"lazy\"  decoding=\"async\"  width=\"720\"  height=\"540\"  src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABAQMAAAAl21bKAAAAA1BMVEUAAP+KeNJXAAAAAXRSTlMAQObYZgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAApJREFUCNdjYAAAAAIAAeIhvDMAAAAASUVORK5CYII=\"  alt=\"\"  class=\"wp-image-825756 pk-lazyload\"  style=\"width:597px;height:auto\"  data-pk-sizes=\"auto\"  data-ls-sizes=\"auto, (max-width: 720px) 100vw, 720px\"  data-pk-src=\"https:\/\/cdnblogdiscover.leverageedu.com\/discover\/wp-content\/uploads\/2024\/05\/Pythagoras-Theorem-2-1.png\"  data-pk-srcset=\"https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110131\/Pythagoras-Theorem-2-1.png 720w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110131\/Pythagoras-Theorem-2-1-300x225.png 300w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110131\/Pythagoras-Theorem-2-1-150x113.png 150w\" ><\/figure>\n\n\n\n<p><strong>Solution 2:<\/strong> Given that the sides of the right triangle are 7 units and 24 units, we can substitute these values into the formula:<\/p>\n\n\n\n<p>c\u00b2 = 7\u00b2 + 24\u00b2<\/p>\n\n\n\n<p>c\u00b2 = 49 + 576<\/p>\n\n\n\n<p>c\u00b2 = 625<\/p>\n\n\n\n<p>Taking the square root of both sides to solve for c (the length of the hypotenuse):<\/p>\n\n\n\n<p>c = \u221a625<\/p>\n\n\n\n<p>c = 25 units<\/p>\n\n\n\n<p>Therefore, the length of the hypotenuse (AC) in the right triangle with sides 7 and 24 units is 25 units.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-integration-formulas\/\"><strong>Integration Formulas: Examples and Solutions<\/strong><\/a><\/p>\n\n\n\n<p><strong>Example 3:<\/strong> Calculating the length of the diagonal of a square with a side length of 4 cm using the Pythagorean Theorem.<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img  loading=\"lazy\"  decoding=\"async\"  width=\"720\"  height=\"540\"  src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABAQMAAAAl21bKAAAAA1BMVEUAAP+KeNJXAAAAAXRSTlMAQObYZgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAApJREFUCNdjYAAAAAIAAeIhvDMAAAAASUVORK5CYII=\"  alt=\"\"  class=\"wp-image-825757 pk-lazyload\"  style=\"width:539px;height:auto\"  data-pk-sizes=\"auto\"  data-ls-sizes=\"auto, (max-width: 720px) 100vw, 720px\"  data-pk-src=\"https:\/\/cdnblogdiscover.leverageedu.com\/discover\/wp-content\/uploads\/2024\/05\/Pythagoras-Theorem-3.png\"  data-pk-srcset=\"https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110133\/Pythagoras-Theorem-3.png 720w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110133\/Pythagoras-Theorem-3-300x225.png 300w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110133\/Pythagoras-Theorem-3-150x113.png 150w\" ><\/figure>\n\n\n\n<p><strong>Solution 3: <\/strong>To calculate the length of the diagonal of a square with a side length of 4 cm using the Pythagorean Theorem, we can apply the formula c = \u221a(a\u00b2 + b\u00b2), where c represents the length of the diagonal, and a and b are the lengths of the sides of the square.<\/p>\n\n\n\n<p>Given that the side length of the square is 4 cm, we can substitute this value into the formula:<\/p>\n\n\n\n<p>c = \u221a(4\u00b2 + 4\u00b2)<\/p>\n\n\n\n<p>c = \u221a(16 + 16)<\/p>\n\n\n\n<p>c = \u221a32<\/p>\n\n\n\n<p>c \u2248 5.65 cm<\/p>\n\n\n\n<p>Therefore, the length of the diagonal of a square with a side length of 4 cm, calculated using the Pythagorean Theorem, is approximately 5.65 cm.<\/p>\n\n\n\n<p class=\"has-very-light-gray-to-cyan-bluish-gray-gradient-background has-background\"><strong>Also Read: <\/strong><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concept-types-of-angles\/\"><strong>Different Types of Angles with Formulas and Examples&nbsp;<\/strong><\/a><\/p>\n\n\n\n<p><strong>Example 4:<\/strong> Verifying if a triangle with sides 11 cm, 60 cm, and 61 cm forms a right-angled triangle using the theorem.<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img  loading=\"lazy\"  decoding=\"async\"  width=\"1024\"  height=\"768\"  src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABAQMAAAAl21bKAAAAA1BMVEUAAP+KeNJXAAAAAXRSTlMAQObYZgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAApJREFUCNdjYAAAAAIAAeIhvDMAAAAASUVORK5CYII=\"  alt=\"\"  class=\"wp-image-825758 pk-lazyload\"  style=\"width:664px;height:auto\"  data-pk-sizes=\"auto\"  data-ls-sizes=\"auto, (max-width: 1024px) 100vw, 1024px\"  data-pk-src=\"https:\/\/cdnblogdiscover.leverageedu.com\/discover\/wp-content\/uploads\/2024\/05\/Pythagoras-Theorem-4.png\"  data-pk-srcset=\"https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110134\/Pythagoras-Theorem-4.png 1024w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110134\/Pythagoras-Theorem-4-300x225.png 300w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110134\/Pythagoras-Theorem-4-768x576.png 768w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/05\/14110134\/Pythagoras-Theorem-4-150x113.png 150w\" ><\/figure>\n\n\n\n<p><strong>Solution 4: <\/strong>According to the theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.<\/p>\n\n\n\n<p>Given sides:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Side a = 11 cm<\/li>\n\n\n\n<li>Side b = 60 cm<\/li>\n\n\n\n<li>Side c = 61 cm<\/li>\n<\/ul>\n\n\n\n<p>To verify if this triangle is a right-angled triangle, we can apply the Pythagorean Theorem:<\/p>\n\n\n\n<p>c\u00b2 = a\u00b2 + b\u00b2<\/p>\n\n\n\n<p>Substitute the given values:<\/p>\n\n\n\n<p>61\u00b2 = 11\u00b2 + 60\u00b2<\/p>\n\n\n\n<p>3721 = 121 + 3600<\/p>\n\n\n\n<p>3721 = 3721<\/p>\n\n\n\n<p>Since the equation holds true, with 3721 on both sides, the sides 11 cm, 60 cm, and 61 cm form a right-angled triangle. This confirms that the triangle is a right triangle, satisfying the Pythagorean Theorem.<\/p>\n\n\n\n<p class=\"has-text-align-center has-vivid-red-color has-text-color has-link-color has-medium-font-size wp-elements-9408409346b43890cd1d5a3a77f43026\"><strong>Related Blogs&nbsp;<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-types-of-fractions\/\"><strong>7 Types of Fractions with Examples<\/strong><\/a><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-what-are-co-prime-numbers\/\"><strong>What are Co Prime Numbers?<\/strong><\/a><\/td><\/tr><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-how-to-find-percentage-of-marks\/\"><strong>How to Find Percentage of Marks?<\/strong><\/a><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-tables-1-to-20\/\"><strong>Multiplication Tables of 1 to 20<\/strong><\/a><\/td><\/tr><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-ordinal-numbers\/\"><strong>Ordinal Numbers from 1 to 100!<\/strong><\/a><strong>&nbsp;<\/strong><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/civics-and-polity-table-of-17\/\"><strong>Table of 17: Multiples up to 20 &amp; a Trick!<\/strong><\/a>&nbsp;&nbsp;<\/td><\/tr><tr><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concepts-table-of-12\/\"><strong>Table of 12: Multiples up to 20!<\/strong><\/a><\/td><td><a href=\"https:\/\/leverageedu.com\/discover\/school-education\/basic-concept-hcf-of-two-consecutive-odd-numbers\/\"><strong>What is the HCF of Two Consecutive Odd Numbers?<\/strong><\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>I hope this helps! Did you like learning about the Pythagoras Theorem? Keep reading our blogs to learn more about the <a href=\"https:\/\/leverageedu.com\/discover\/category\/school-education\/basic-concepts\/\"><strong>Basic Concepts of Maths<\/strong><\/a>!<\/p>\n","protected":false},"excerpt":{"rendered":"The Pythagoras theorem is also known as the Pythagorean theorem. It is a basic concept in mathematics that&hellip;\n","protected":false},"author":106,"featured_media":825717,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"editor_notices":[],"footnotes":""},"categories":[423,476,389],"tags":[],"class_list":{"0":"post-825714","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-basic-concepts","8":"category-maths","9":"category-school-education"},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.3 (Yoast SEO v27.3) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Pythagoras Theorem: Proof, Formula, Derivation, Examples \u00a0 I Leverage Edu Discover<\/title>\n<meta name=\"description\" content=\"Read this blog to know some of the most important information about the Pythagoras Theorem along with formulas and examples!\" \/>\n<meta name=\"robots\" content=\"index, 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