The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs of the right triangle. This same relationship is often used in the construction industry and is referred to as the 3-4-5 Rule.
The right triangle below has one leg with a length of three, another leg with a length of four and a hypotenuse with a length of five.
Given the lengths of any two sides of a right triangle, the length of the third side can be calculated using the Pythagorean theorem. In the example above, there are three possible unknowns. Each case is outlined below.
There are many ways to prove the Pythagorean theorem. One such proof is given here.
in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides. prove it
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A = area
What is the Pythagorean Theorem?
The Pythagorean Theorem states that the sum of the squared sides of a right triangle equals the length of the hypotenuse squared.
You might recognize this theorem in the form of the Pythagorean equation:
\[ a^{2} + b^{2} = c^{2} \]If you know the length of any 2 sides of a right triangle you can use the Pythagorean equation formula to find the length of the third side.
Calculator Use
This calculator solves the Pythagorean Theorem equation for sides a or b, or the hypotenuse c. The hypotenuse is the side of the triangle opposite the right angle.
For right triangles only, enter any two values to find the third. See the solution with steps using the Pythagorean Theorem formula.
This calculator also finds the area A of the right triangle with sides a and b. The formula for area of a right triangle is:
Pythagorean Theorem Formula
Using the Pythagorean Theorem formula for right triangles you can find the length of the third side if you know the length of any two other sides. Read below to see solution formulas derived from the Pythagorean Theorem formula:
\[ a^{2} + b^{2} = c^{2} \]Solve for the Length of the Hypotenuse c
The length of the hypotenuse is the square root of the sum of the sides squared.
\[ c = \sqrt{a^{2} + b^{2}} \]Solve for Length of Side a
The length of side a is the square root of the squared hypotenuse minus the square of side b.
\[ a = \sqrt{c^{2} - b^{2}} \]Solve for the Length of Side b
The length of side b is the square root of the squared hypotenuse minus the square of side a.
\[ b = \sqrt{c^{2} - a^{2}} \]Solve for Area A of the Right Triangle
The area of a right triangle is side a multiplied by side b divided by 2.
\[ A = \dfrac {ab}{2} \]What are Pythagorean Triples?
A Pythagorean triple is a set of 3 positive integers for sides a and b and hypotenuse c that satisfy the Pythagorean Theorem formula a2 + b2 = c2
The smallest known Pythagorean triple is 3, 4, and 5. Showing the work:
\[ a^{2} + b^{2} = c^{2} \] \[ 3^{2} + 4^{2} = 5^{2} \] \[ 9 + 16 = 25 \] \[ 25 = 25 \]References:
Weisstein, Eric W. "Pythagorean Theorem" From MathWorld--A Wolfram Web Resource. Pythagorean Theorem.
Wikipedia "Pythagorean Theorem" at //en.wikipedia.org/wiki/Pythagorean_theorem last accessed May 4, 2020.