Right Angled Triangle | Properties and Formula

Right Angled Triangle: A triangle is a polygon with three sides, three vertices, and three angles thus, the name Triangle. A right-angled triangle is a triangle with one right angle (90°). Right Angle Triangle plays a very important role in trigonometry.

In this article, we will learn about the right-angled triangle, including its definition, perimeter, area, right-angled triangle formula, and properties in detail.

Right Angled Triangle

What is a Right Angled Triangle?

A right-angled triangle is a type of triangle that has one of its angles measuring exactly 90 degrees. This 90-degree angle, also known as a right angle, gives the right-angled triangle its name and distinct properties.

Right Angled Triangle Definition

A triangle with any interior angle equal to 90° is called a Right Triangle.

The sum of all the interior angles of the triangle is 180° which is called the Angle Sum Property of a Triangle. So if any one triangle is 90° the sum of the other two angles is also, 90°.

Properties of Right Angled Triangle

A Right Angled Triangle has the following key properties :

• One of the angles in a right-angled triangle is exactly 90 degrees.
• The side opposite the right angle is the longest side of the triangle and is called the hypotenuse.
• For triangles with the same angles, the sides are in a consistent ratio. For example, in a 45-45-90 right triangle, the sides are in the ratio 1:1:√2​, and in a 30-60-90 triangle, the sides are in the ratio 1:√3​/2.
• The altitude drawn to the hypotenuse of a right triangle creates two smaller right-angled triangles, each of which is similar to the original right-angled triangle.
• Every right-angled triangle has a circumcircle (circle passing through all three vertices) with the hypotenuse as its diameter. It also has an incircle (circle tangent to all three sides), with the center at the intersection of the angle bisectors.

Right Triangle Formula

The formula for the right-angled triangle is given by the Pythagoras Theorem. According to the pythagoras theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

(Hypotenuse)² = (Perpendicular)² + (Base)²

Perimeter of Right Angled Triangle

The perimeter of the right triangle shown above is equal to the sum of the sides,

BC + AC + AB = (a + b + c) units.

The perimeter is a linear value with a unit of length. Therefore,

Right Angled Triangle Perimeter Formula

Perimeter of Triangle = (a + b + c) units

Right Angled Triangle Area Formula

The area of a right triangle is the space occupied by the boundaries of the triangle.

The area of a right triangle is given below,

Area of a right triangle = (1/2 × base × height) square units.

Derivation of Right Angled Triangle Area Formula

For any right triangle, PQR right angled at Q with hypotenuse as, PR

Now if we flip the triangle over its hypotenuse a rectangle is formed which is named PQRS. The image given below shows the rectangle form by flipping the right triangle.

As we know, the area of a rectangle is given as the product of its length and width, i.e. Area = length × breadth

Thus, the area of Rectangle PORS = b x h

Now, the area of the right triangle is twice the area of the rectangle then,

Thus,

Area of ∆PQR = 1/2 × Area of Rectangle PQRS

A = 1/2 × b × h

Hypotenuse of Right Angled Triangle

For a right triangle, the hypotenuse is calculated using the Pythagoras Theorem,

H = √(P² + B²)

where,
H is the Hypotenuse of the Right Triangle,
P is the Perpendicular of the Right Triangle,

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Solved Examples on Right Angled Triangle

Let’s solve some example problems on right angled triangles.

Example 1: Find the area of a triangle if the height and hypotenuse of a right-angled triangle are 10 cm and 11 cm, respectively. 

Solution: 

Given: 

Height = 10 cm
Hypotenuse = 11 cm

Using Pythagoras’ theorem,

(Hypotenuse)² = (Base)² + (Perpendicular)²

(11)² = (Base)² + (10)²

(Base)² = (11)² – (10)²
            = 121 – 100 

Base = √21 
        = 4.8 cm

Area of the Triangle = (1/2) × b × h

Area = (1/2) × 4.8 × 10

Area = 24 cm²

Example 2: Find out the area of a right-angled triangle whose perimeter is 30 units, height is 8 units, and hypotenuse is 12 units.

Solution: 

Perimeter = 30 units
Hypotenuse = 12 units
Height = 8 units

Perimeter = base + hypotenuse + height

30 units = 12 + 8 + base

Base = 30 – 20
        = 10 units

Area of Triangle = 1/2×b×h 
                          = 1/2 ×10 × 8
                          = 40 sq units

Example 3: If two sides of a triangle are given find out the third side i.e. if Base = 3 cm and Perpendicular = 4 cm find out the hypotenuse.

Solution:

Given: 
Base (b) = 3 cm 
Perpendicular (p) = 4 cm
Hypotenuse (h) = ?

Using Pythagoras theorem,

(Hypotenuse)² = (Perpendicular)² + (Base)²
= 4² + 3²
= 16 + 9
                       = 25 cm²

Hypotenuse = √(25)

Hypotenuse = 5 cm

FAQs on Right Angled Triangle

What are Right Triangle formulas in Geometry?

Right triangle formulas are used to calculate the perimeter, area, height, etc. of the right triangle. The formulas of right triangles are,

• Pythagoras Theorem (Formula):  (Hypotenuse)² =  (Perpendicular)² + (Base)²
• Area of Right Triangle Formula: Area =  1/2 × Base × Height
• Perimeter of Right Triangle Formula: Sum of lengths of 3 sides

What is the formula of Right Triangle Area?

The formula for the Area of a Right Triangle is,

Area = 1/2 × Base × Height

What are the Different Types of Right Triangles?

The different types of Right Triangles are :

1. Scalene Right Triangle:
   • Different lengths for all three sides.
   • Example: Sides of 3 cm, 4 cm, and 5 cm.
2. Isosceles Right Triangle:
   • Two equal sides (legs) and one right angle.
   • Example: Legs of 1 cm each, hypotenuse 22​ cm.
3. Special Right Triangles:
   • 45-45-90 Triangle: Equal legs, each acute angle is 45 degrees.
   • 30-60-90 Triangle: Angles of 30, 60, and 90 degrees; side lengths in a specific ratio.

What are the Applications of Right Triangle Formula?

Right Triangle Formula is widely used in mathematics, some of the applications of the right triangle formula are,

• Right Triangle Formula is used in studying Triangles and their properties.
• Right Triangle Formula is used in the study of Trigonometry, etc

How to find the Height of Right Triangle?

The height of the Right Triangle is calculated using, the Pythagoras theorem, i.e., (H)² = (P)² + (B)² where P is the perpendicular of the triangle and which is also called the height of a Right Triangle.