Area of a Triangle for Class 7: Formula, Steps and Solved Examples

The area of a triangle is a fundamental concept in Class 7 Mathematics that helps measure the space enclosed within a triangle. The basic formula for the area of a triangle is equal to half the product of its base and height. Area of triangles area widely used in geometry and real-life applications such as construction, design, and land measurement. In this guide, you will learn the formula for finding the area of triangles and how to apply it in different types of problems with clear explanations and solved examples.

Table of Contents

• What is the Area of a Triangle?
• Area of a Triangle Formula
• Area of a Right-Angled Triangle
• Area of an Equilateral Triangle
• Area of an Isosceles Triangle
• Solved Examples on Area of a Triangle
• Practice Problems on Area of a Triangle

What is the Area of a Triangle?

The area of a triangle is the total region enclosed within its three sides. It is the amount of flat space that the triangle covers. Area is always measured in square units such as cm², m², mm², km², and so on.
The area of a triangle can vary depending on the lengths of its sides and the angles between them. Different shapes and sizes of triangles will have different areas based on these measurements.

Area of a Triangle Formula

The area of a triangle is the amount of space enclosed within its three sides. It is give by the formula:

Area of a Triangle =  12×Base×Height

A =  12×b×h

Where:
b = length of the base of the triangle

h = height (also called altitude ),the perpendicular distance from that base to the opposite vertex.

A = area of the triangle (in square units)

Why is area of a triangle 12×b×h?

If you fold a square or a rectangle diagonally, you make two triangles.

The area of the triangle must be half of the area of the square or rectangle.

Two identical triangles make a parallelogram with the same base and the same height as the triangle.

area of parallelogram = base × height

area of parallelogram = 2 × area of triangle

area of triangle = area of parallelogram /2 = (base × height)/2 =   12×b×h

Area of a Right-Angled Triangle

A right-angled triangle has one angle equal to exactly 90°. The two sides that form the right angle are called the legs (or the perpendicular and the base), and the longest side (opposite the right angle) is called the hypotenuse.

Area of a Right-Angled Triangle =  12×Base×Height (where base and height are the two legs)

Example: A right triangle has legs of length 8 cm and 5 cm. Find its area.

Given, l = 8 cm and b = 5 cm.

A=12×b×h=12×5×8=20cm2

Therefore, area of the triangle is 20 cm^{2}.

 

Area of an Equilateral Triangle

An equilateral triangle has all three sides equal and all three angles equal (each = 60°). The formula  A=12×b×his used to find the area of triangles when the base and height are known. However, due to its perfect symmetry, there's another formula to find the area of equilateral triangle using just the side length.

A=34×a2

where a is the length of each side of the equilateral triangle.

Example:Each side of an equilateral triangle is 10 cm. Find its area. (Use √3 = 1.73)

Given, a = 10 cm.

A=34×a2=34×(10)2=(1.73/4)×100=43.25cm2.

 

Area of an Isosceles Triangle

An isosceles triangle has two sides equal in length (called the equal sides or legs) and the third side is the base. The formula  A=12×b×hworks for an isosceles triangle as well. However, since two sides are equal, the height can be expressed in terms of the equal sides and the base . Hence area of a isosceles triangle is

A=b4×4a2−b2

where a is the length of the equal sides and b is the base of the isosceles triangle.

Example: An isosceles triangle has equal sides of 13 cm and a base of 10 cm. Find its area.

Given a = 13 cm and b = 10cm.

A=b4×4a2−b2=104×4(13)2−(10)2=104×576=60cm2.

Solved Examples on Area of a Triangle

Example 1: A triangular garden has a base of 20 m and a height of 15 m. What is its area?

Solution: Given, b = 20 m and h = 15 m

Area = 12×b×h=12×20×15=(300/2)=150m2.

Triangular garden with a base of 20 m and a height of 15 m has an area of 150m2.

Example 2: Find the area of an equilateral triangle with side 8 cm. (Use √3 = 1.732)

Solution: Given, a = 8 cm.

A=34×a2=34×(8)2=(1.73/4)×64=27.71cm2.

Example 3: The area of a triangle is 48  cm2and its base is 12 cm. Find the height.

Solution: Give, A = 48 cm2 and b =  12 cm

Area=12×b×h

48 =  12×12×h

h = (48 × 2)/12 = 96/12 = 8 cm.

The height of the triangle is 8 cm.

Example 4:  A triangle has an area of 45 m² and a height of 9 m. Find the base.

Solution: Given, A = 45  cm2 and h =  9 cm

Area=12×b×h

 45=12×b×9 

b = (45 × 2)/9 = 90/9 = 10 cm.

The base of the triangle is 10 cm.

Example 5: Two triangles have the same height of 8 cm. Their bases are 5 cm and 10 cm. Find the ratio of their areas.

Solution: Given: h = 8 cm, b1= 5 cm, b2 = 10 cm.

A1=12×b1×h=12×5×8=20cm2

A2=12×b2×h=12×10×8=40cm2

A1:A2=20:40=1:2

Practice Problems on Area of a Triangle

1. The area of a right-angled triangle is 184  cm2 and one of its legs is 16 cm long. Find the length of the other leg.

2. A right triangle has one leg = 9 cm and area = 27 cm². Find the other leg.

3. Find the area of a triangle with base 18 cm and height 11 cm. 

4. Find the area of a triangle with base = 14 cm and height = 10 cm.

5. Find the area of an equilateral triangle with side 12 cm. (Use √3 = 1.732)

6. An isosceles triangle has equal sides of 17 cm and a base of 16 cm. Find its area.

7. The base of a triangle is 3 times its height. If the area is 54 cm², find the base and height.

8. The area of a triangle is 63 m² and its base is 14 m. Find its height.

9. A triangle has a base of 15 cm and height of 10 cm. Calculate its area.

10. The sides of an equilateral triangle are 6 cm each. Find its area.