Pythagorean Theorem

Last updated on 09 February 2025

Table of contents

What is Pythagoras Theorem?
Applications of Pythagoras Theorem
Solved Examples on Pythagorean Theorem
FAQs on Pythagoras Theorem

The Pythagorean Theorem which is also referred to as the Pythagoras Theorem is one of the fundamental theorems in geometry, credited to the ancient Greek mathematician Pythagoras. It relates to the three sides of a right triangle, which is a triangle containing a 90° angle (a right angle). From architectural designs to astronomical calculations, the Pythagorean Theorem serves as a cornerstone for understanding the intrinsic nature of right triangles.

Have you ever wondered how to figure out the length of a missing side in a right triangle? Enter the Pythagorean Theorem, a nifty trick that helps us solve this puzzle. Let's delve into this magical formula that makes understanding right triangles a piece of cake!

What is Pythagoras Theorem?

The Pythagoras Theorem states that in any right triangle, 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.
In other words, if we square the lengths of the two shorter sides of the right trinagle and then add those together, we get the square of the longest side of the right trinagle, also known as the hypotenuse.

Observe the above triangle △PQR, in which QR is the base, PQ is the altitude (height), and PR is the hypotenuse. The Pythagoras theorem equation is expressed as:

c^{2} = a^{2} + b^{2}

where

c

is the length of the hypotenuse, and

a

and

b

are the lengths of the other two sides.

Applications of Pythagoras Theorem

The applications of the Pythagoras theorem can be seen in our day-to-day life. The versatility of the Pythagorean Theorem manifests in its wide-ranging applications. In the field of architecture, it serves as the backbone for determining diagonal measurements of structures, ensuring stability and proportion. Engineers employ it to calculate distances, ensuring accuracy in the design of bridges, roads, and buildings. In the realm of physics, the theorem finds its application in vector analysis, offering a tool for calculating resultant forces and understanding the interplay between different physical quantities.

Solved Examples on Pythagorean Theorem

Example 1: The hypotenuse of a right-angled triangle is

5 cm

and the base is

4 cm

. Find the measure of the height using the Pythagoras theorem formula.

Given:

hypotenuse (c) = 5 cm

base (b) = 4 cm

\begin{aligned} ∵ & c^{2} & = a^{2} + b^{2} \\ ∴ & 5^{2} & = a^{2} + 4^{2} \\ ∴ & 25 & = a^{2} + 16 \\ ∴ & a^{2} & = 25 - 16 \\ ∴ & a^{2} & = 9 \\ ∴ & a^{2} & = 3^{2} \\ ∴ & a & = 3 cm \end{aligned}

Hence, the length of the height of the given right triangle is

3 cm

Example 2: Using the Pythagoras theorem, find the hypotenuse of the right triangle in which the sides are

5 cm

and

12 cm

respectively.

Given:

a = 5 cm

b = 12 cm

\begin{aligned} ∵ & c^{2} & = a^{2} + b^{2} \\ ∴ & c^{2} & = 5^{2} + 12^{2} \\ ∴ & c^{2} & = 25 + 144 \\ ∴ & c^{2} & = 169 \\ ∴ & c^{2} & = 13^{2} \\ ∴ & c & = 13 cm \end{aligned}

Therefore, the length of the hypotenuse of the given right triangle is

13 cm

FAQs on Pythagoras Theorem

What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right triangle, 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. The Pythagoras theorem equation is expressed as:
$c^{2} = a^{2} + b^{2}$
where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the other two sides.
What are the practical uses of the Pythagorean Theorem?
The theorem is widely used in architecture, navigation, construction, engineering, computer graphics, and physics to calculate distances, determine right angles, and solve problems involving right triangles.
Does the Pythagorean Theorem apply to all triangles?
No, the Pythagorean Theorem applies only to right triangles (triangles with one 90° angle).
What is the converse of the Pythagorean Theorem?
The converse of the Pythagorean Theorem states that if the square of the longest side of a triangle equals to the sum of the squares of the other two sides, then the triangle is a right angled triangle.
How to find the length of the hypotenuse of the right angled triangle using Pythagorean Theorem?
To find the hypotenuse using Pythagorean Theorem, just rearrange the formula as:
$c = \sqrt{a^{2} + b^{2}}$
where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the other two sides.
How to find the length of the missing leg of the right angled triangle using Pythagorean Theorem?
To find the length of the missing leg using Pythagorean Theorem, just rearrange the formula as:
$a = \sqrt{c^{2} - b^{2}}$ or $b = \sqrt{c^{2} - a^{2}}$
where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the other two sides.
Can the Pythagorean Theorem have decimal or irrational solutions?
Yes, the side lengths or hypotenuse can result in decimal or irrational numbers, depending on the input values.
What are Pythagorean Triples?
A Pythagorean Triple is a set of three positive integers $(a, b, c)$ that satisfy the Pythagorean Theorem. Examples include (3, 4, 5) and (5, 12, 13).
How do architects use the Pythagorean Theorem?
Architects use it to ensure structures are built with precise right angles and to calculate dimensions for triangular elements.
Can the Pythagorean Theorem be used for circles or other shapes?
Indirectly, it can be used in problems involving circles, ellipses, or polygons, especially when a right triangle is inscribed or drawn within these shapes.
Why does the Pythagorean Theorem work?
It works because the relationship between the areas of squares on a right triangle’s sides is mathematically consistent across all right triangles.
What are the limitations of the Pythagorean Theorem?
It only applies to right triangles. For other triangles, trigonometric rules such as the Law of Cosines are used.
How is the Pythagorean Theorem different from the Law of Cosines?
The Pythagorean Theorem applies only to right triangles, while the Law of Cosines works for any triangle and reduces to the Pythagorean Theorem when the angle is 90°.
Can the Pythagorean Theorem be used to determine if a triangle is right-angled?
Yes, if the side lengths satisfy $c^{2} = a^{2} + b^{2}$ , the triangle is a right triangle.
What are some real-world examples of the Pythagorean Theorem?
Measuring TV or computer screen sizes (diagonal length), ladder placement for stability, designing ramps, and constructing bridges.
What happens if one side of the triangle is zero?
If one side is zero, the triangle collapses into a line, and the Pythagorean Theorem is no longer applicable.

Pythagorean Theorem

What is Pythagoras Theorem?

Applications of Pythagoras Theorem

Solved Examples on Pythagorean Theorem

FAQs on Pythagoras Theorem

What is the Pythagorean Theorem?

What are the practical uses of the Pythagorean Theorem?

Does the Pythagorean Theorem apply to all triangles?

What is the converse of the Pythagorean Theorem?

How to find the length of the hypotenuse of the right angled triangle using Pythagorean Theorem?

How to find the length of the missing leg of the right angled triangle using Pythagorean Theorem?

Can the Pythagorean Theorem have decimal or irrational solutions?

What are Pythagorean Triples?

How do architects use the Pythagorean Theorem?

Can the Pythagorean Theorem be used for circles or other shapes?

Why does the Pythagorean Theorem work?

What are the limitations of the Pythagorean Theorem?

How is the Pythagorean Theorem different from the Law of Cosines?

Can the Pythagorean Theorem be used to determine if a triangle is right-angled?

What are some real-world examples of the Pythagorean Theorem?

What happens if one side of the triangle is zero?

Perimeter of a Triangle

45-45-90 Triangle / Isosceles Right Triangle