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Pythagorean Theorem Calculator

Enter the lengths of any two sides of a right triangle to find the unknown side using Pythagorean theorem equation i.e. \(a^2 + b^2=c^2\).

a = c² - b²

b = c² - a²

c = a² + b²

A = 1 2 ab

Pythagorean Theorem

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“Pythagorean Theorem Calculator”

This calculator uses Pythagorean theorem to determine the unknown side of a right triangle. It shows a step-by-step process for solving a missing side and related values such as Area, Perimeter, Angles, and Height. All the calculations by this tool can be performed using:

Pythagorean Theorem Formula:

\(a^2 + b^2=c^2\)

Where;

  • c = Hypotenuse
  • a = Side of Right Triangle
  • b = Side of Right Triangle

To Find Side ‘c’:

\(c = \sqrt{a^{2} + b^{2}}\)

To Find Side ‘a’:

\(a = \sqrt{c^{2} - b^{2}}\)

To Find Side ‘b’:

\(b = \sqrt{c^{2} - a^{2}}\)

Area:

\(A=\dfrac{a*b}{2}\)

Perimeter:

\(P=a+b+c)\)

∠α:

\(∠α=arcsin\left(\dfrac{a}{c}\right)\)

∠β:

\(∠β=arcsin\left(\dfrac{b}{c}\right)\)

Height:

\(h=\dfrac{a*b}{c}\)

What Is Pythagorean Theorem?

In Euclidean Geometry, the Pythagorean theorem defines a basic relationship among three sides of a right triangle. It states that:

“The square of the hypotenuse (the longest side) is equal to the sum of the square of the other two sides”

The theorem was discovered and popularized by a famous Greek Mathematician ‘Pythagoras’ in the 6th century BC.

Pythagorean Theorem (Examples & Calculations)

Example # 01:

How to find the hypotenuse of a right triangle with the following known sides:

  • a = 4
  • b =16

Calculations:

\(c = \sqrt{a^{2} + b^{2}}\)

\(c = \sqrt{4^{2} + 16^{2}}\)

\(c = \sqrt{16+256}\) \(c = \sqrt{272}\)

\(c = \sqrt{16*17}\) \(c = 4\sqrt{17}\)

If you want to calculate the hypotenuse of a triangle with different measurements for sides and angles, you can use our other hypotenuse calculator that provides you with a complete solution to find it.

Example # 02:

What would be the value of the missing side ‘b’ if a=9 and c=25?

Calculations:

\(b = \sqrt{c^{2} - a^{2}}\)

\(b = \sqrt{25^{2} - 9^{2}}\)

\(b = \sqrt{625 - 81}\)

\(b = \sqrt{544}\)

\(b = 23.32\)

You can solve the same examples or another one by using this phthagoream theorm calculator.

Faqs:

What are the Pythagorean Triples?

It is the set of three positive integers that satisfies the equation ‘a2 + b2 = c2’. The smallest triples are (3, 4, 5) while there is no limit for the largest one.

Where is the Pythagorean Theorem used in real life?

Pythagorean theorem can be used in various real-life scenarios, such as:

  • To find displacement between points in 2D navigation
  • To determine the slope of hills or mountains
  • Helps to calculate the original height of the tree that broke due to heavy rain, etc.
  • Can be used to calculate the length of the longest item in your house

Is there any limitation while using the Pythagorean Theorem calculator?

  • The tool does not show results for non-right triangles.
  • It typically works with real numbers representing side lengths & wouldn't handle complex number inputs for sides.

Can law of cosines be reduced to Pythagorean Theorem?

Yes, Pythagorean theorem is the most special case that can be determined using the law of cosines. The only condition is that the angle between common sides must be right (\(90^\text{o}\)).

Reference:

From the source of Wikipedia: Forms of the theorem, Euclid's proof, Dissection and Rearrangement.

From the source of Math Planet: The Pythagorean Theorem, Dissection without Rearrangement, Consequences and uses of the theorem.

From the source of Cut the Knot: Pythagorean Theorem, Pythagorean proposition, Dissection using inscribed circle.

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