This Pythagorean theorem calculator will calculate the length of any of the missing sides of a right triangle, provided you know the lengths of its other two sides. This includes calculating the hypotenuse. The hypotenuse of the right triangle is the side opposite the right angle, and is the longest side. This side can be found using the hypotenuse formula, another term for the Pythagorean theorem when it's solving for the hypotenuse. Recall that a right triangle is a triangle with an angle measuring 90 degrees. The other two angles must also total 90 degrees, as the sum of the measures of the angles of any triangle is 180. Read on to answer "what is the Pythagorean theorem and how is it used?"
What is the Pythagorean theorem?
The Pythagorean theorem describes how the three sides of a right triangle are related in Euclidean geometry. It states that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. You can also think of this theorem as the hypotenuse formula. If the sides of a right triangle are
b and the hypotenuse is
c, the formula is
a² + b² = c²
The theorem was credited to the ancient Greek philosopher and mathematician Pythagoras, who lived in the sixth century BC. Although it was previously used by the Indians and Babylonians, Pythagoras (or his students) were credited to be the first to prove the theorem. It should be noted that there is no concrete evidence that Pythagoras himself worked on or proved this theorem.
How to use the Pythagorean theorem
- Input the two lengths that you have into the formula. For example, suppose you know
a = 4,
b = 8and we want to find the length of the hypotenuse
- After the values are put into the formula we have
4²+ 8² = c²
- Square each term to get
16 + 64 = c²
- Combine like terms to get
80 = c²
- Take the square root of both sides of the equation to get
c = 8.94. Go ahead and check it with our Pythagorean theorem calculator!
Note that if you are solving for a or b, rearrange the equation to isolate the desired variable before combining like terms and taking the square root
The Pythagorean theorem calculator will solve for the sides in the same manner that we displayed above. We have included the method to show you how you can solve your problem if you prefer to do it by hand.
What is the hypotenuse formula?
The hypotenuse formula is simply taking the Pythagorean theorem and solving for the hypotenuse,
c. Solving for the hypotenuse, we simply take the square root of both sides of the equation
a² + b² = c²and solve for
c. When doing so, we get
c = √(a² + b²). This is just an extension of the Pythagorean theorem and often is not associated with the name hypotenuse formula.
Other considerations when dealing with triangles
Notice the sides of a triangle have a certain degree of gradient or slope. We can use a slope calculator to determine the slope of each side. In a right triangle the sides that form the right angle will have slopes whose product is -1. The formula for slope if you wish to calculate by hand is
(y₂ - y₁)/(x₂ - x₁)
So if the coordinates are
(3,6) and (7,10), the slope of the segment is
(10-6)/(7-3) = 1. If the slope of the other segment forming the angle is
-1 then the lines would be perpendicular since
1 * -1 = -1. Therefore, the triangle is a right triangle.
You can also figure out the missing side lengths and angles of a right triangle using the right triangle calculator. If the angles given in the problem are in degrees and you want to convert to radians or radians to degrees, check out our angle converter. There is an easy way to convert degrees to radians and radians to degrees.
If angle is in radians
- Multiply by
If angle is in degrees
- Multiply by
Sometimes you may encounter a problem where two or all three side lengths missing. In such cases, the Pythagorean theorem calculator won't help - you will use trigonometric functions to solve for these missing pieces. This can be accomplished by hand or by using our triangle calculator.