Area Of Right Angled Triangle - Formula, Examples, Definition

Area of Right Triangle

The area of a right triangle is the region that is covered inside the boundary of the right triangle. A right-angled triangle is a triangle in which one of the angles is a right angle (90 degrees). Let us learn more about the area of a right-angled triangle, how to find the area of a right-angled triangle with hypotenuse, along with some examples.

1.	What is Area of a Right Triangle?
2.	Area of Right Triangle Formula
3.	Derivation of Area of Right Angled Triangle Formula
4.	Area of Right Triangle with Hypotenuse
5.	FAQs on Area of a Right Triangle

What is Area of a Right Triangle?

The area of a right-angled triangle is the space that is inside it. Before learning about the area of a right angled triangle, let us see a right-angled triangle.

In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called legs. The two legs can be interchangeably called base and height. Observe the following figure which shows a right angled triangle along with its formula.

Area of Right Triangle Formula

The formula for the area of a right-angled triangle is expressed as:

Area of a right triangle = 1/2 × base × height

So, let us use this formula in the following examples.

Example: Find the area of a right triangle with base 4 cm and height 6 cm.

Solution: It is given that base = 4 cm, height = 6 cm

So, let us find the area of the triangle using the formula,

Area of right angled triangle = 1/2 × base × height

After substituting the values in the formula, we get,

Area of right angled triangle = 1/2 × 4 × 6 = 12 square cm

Therefore, the area of a right triangle with base 4 cm and height 6 cm is 12 cm2

Example: Find the area of a right triangle with a base of 3 cm and a height of 1.42 cm.

Solution: It is given that base = 3 cm, height = 1.42 cm

So, let us find the area of the triangle using the formula,

Area of right angled triangle = 1/2 × base × height

After substituting the values in the formula, we get,

Area of right angled triangle = 1/2 × 3 × 1.42 = 2.13 square cm.

Therefore, the area of a right triangle with a base of 3 cm and a height of 1.42 cm is 2.13 cm2

Derivation of Area of Right Angled Triangle Formula

The formula for the area of a right angled triangle can be derived easily using the following steps.

• Step 1: Consider a rectangle of length l and width w and draw a diagonal as shown above. We can see that the diagonal divides the rectangle into two right triangles.
• Step 2: It is to be noted that these two right triangles are congruent. We also know that the formula for the area of a rectangle = length × width.
• Step 3: So, the area of one right triangle will be half the area of a rectangle. This can be expressed as, Area of one right triangle = 1/2 × length × width. Since we usually represent the legs of the right-angled triangle as base and height, the formula for the area of a right triangle can be expressed as,Area of a right triangle = 1/2 × base × height.

Area of Right Triangle With Hypotenuse

The area of a right triangle can be calculated using the hypotenuse as well. Let us recollect the Pythagoras theorem which states that in a right-angled triangle, the square of the hypotenuse is the sum of the squares of the other two sides, i.e., (Hypotenuse)2 = (Base)2 + (Height)2

We can find the area of a right angled triangle if we know the length of one of the legs along with the hypotenuse. Let us understand this with the help of an example.

Example: Find the area of a right triangle with base 6 cm and hypotenuse 10 cm.

Solution: It is given that base = 6 cm, hypotenuse = 10 cm. So, we can find the value of the missing leg (height) using the Pythagoras theorem.

Let us substitute the given values in the Pythagoras theorem,

(Hypotenuse)2 = (Base)2 + (Height)2

102 = 62 + (Height)2

100 = 36 + (Height)2

(Height)2 = 64

Height = √(64) = 8 cm

Now, we can find the area of the given right triangle in which base = 6 cm, height = 8 cm

After substituting the values in the formula, we get,

Area of right angled triangle = 1/2 × base × height = 1/2 × 6 × 8 = 24 cm2.