The Pythagorean Theorem or Pythagoras’ theorem, was given by Pythagoras of Samos, who was an ancient Ionian Greek Philosopher. Pythagoras’ Theorem is a Mathematical relation between the sides of a right-angled Triangle.
Right Angled Triangle: A right angle Triangle or right-angled triangle is a triangle that has one angle is a right angle or a 90° angle. The side opposite to the angle is called the hypotenuse, and the sides adjacent to the right angle are called the base and height.
This is a right-angled triangle, and from the figure, we can see that the side opposite to the right angle is the hypotenuse.
A right-angled triangle is the basis of all trigonometric operations.
Formula:
(Hypotenuse)2 =(Base) 2+ (Height)2
For Example, a right-angle triangle with a height of 4cm and a base of 3 cm, find the hypotenuse of the triangle.
So we a triangle with base = 3 cm and height=4cm,
Therefore, using Pythagoras’ theorem
(Hypotenuse)2 =(Base) 2+ (Height)2
(Hypotenuse)2 =(3) 2+ (4)2
(Hypotenuse)2 = 9 + 16
(Hypotenuse)2 =25
Hypotenuse = 5
Examples
🔹 Example 1: Find the Hypotenuse
Given:
- a=3
- b=4
c2=(3)2+(4)2
c2 =9+16
c2=25
c=5
Answer: c=5
🔹 Example 2: Find a Missing Leg
Given:
- c=13
- b=5
a2=c2−b2
a2=(13)2−(5)2
a2=169−25
a2=144
a=12
Answer: a=12
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Real-Life Applications
1. Construction & Architecture
Builders use the Pythagorean theorem to ensure walls are square and rooms are correctly proportioned. For instance, measuring 3 feet along one wall and 4 feet along another, the diagonal between them should be 5 feet for a perfect right angle.
2. Navigation & Mapping
To calculate the shortest distance between two points (e.g., a direct line across a field or body of water), the theorem helps find the straight-line distance (hypotenuse) from two perpendicular paths.
3. Aviation
Pilots use it for calculating direct flight paths, determining how far they’ve flown north and east to calculate their actual straight-line distance from the starting point.
4. Computer Graphics
Used in rendering 3D objects and calculating distances between pixels on the screen.
5. Surveying
Surveyors apply it to determine distances and property boundaries.
FREQUENTLY ASKED QUESTIONS
1. What is Pythagoras’ Theorem?
Pythagoras’ Theorem is a relationship between the sides of a right-angle triangle. The formula for the Pythagorean theorem
(Hypotenuse)2 =(Base) 2+ (Height)2
2. How can we find the base of a right triangle if we have the hypotenuse and height of the Triangle?
To find the base of a triangle whose hypotenuse and base are known ,we can use the formula for the Pythagorean theorem in such a way.
For eg. we have a triangle with hypotenuse 7 and height 5, then the base of the triangle is given by:
(Base)2 =(Hypotenuse) 2– (Height)2
(Base)2 =(7) 2– (5)2
(Base)2 = 49 – 25
(Base)2 =24
Base =√24
Base =4.8
3. What is the formula to determine the area of a right-angle triangle?
The Area of a right-angle triangle is given by:
Area=(Base * Height)/2
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September 12, 2025 
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