Area of a Triangle: Geometry

February 3, 2025

The area of a triangle is the amount of space enclosed within its three sides. It is measured in square units (e.g., cm², m², in²). The most common formula for finding the area of a triangle is:

Area = (1/2) × Base × Height

Where:

Base (b) is the length of the triangle’s bottom side.
Height (h) is the perpendicular distance from the base to the opposite vertex.

1. Basic Formula: Using Base and Height

Example 1:

Find the area of a triangle with a base of 10 cm and a height of 5 cm.

Area = (1/2) × 10 × 5
= 50 / 2 = 25 cm²

✅ Final Answer: 25 cm²

2. Area of a Triangle Using Heron’s Formula

When all three sides are known but the height is unknown, use Heron’s Formula:

Area = √[s(s-a)(s-b)(s-c)]

Where:

a, b, c are the three sides of the triangle.
s is the semi-perimeter, calculated as:
s = (a + b + c) / 2

Example 2:

Find the area of a triangle with sides a = 7 cm, b = 8 cm, c = 9 cm.

Find the semi-perimeter:
s = (7 + 8 + 9) / 2 = 12
Apply Heron’s Formula:
Area = √[12(12-7)(12-8)(12-9)]
= √[12 × 5 × 4 × 3]
= √720 ≈ 26.83 cm²

✅ Final Answer: 26.83 cm²

3. Area of a Right-Angled Triangle

For a right-angled triangle, the two legs serve as the base and height.

Area = (1/2) × Leg₁ × Leg₂

Example 3:

A right-angled triangle has legs of 6 cm and 8 cm. Find its area.

Area = (1/2) × 6 × 8
= 48 / 2 = 24 cm²

✅ Final Answer: 24 cm²

4. Area of an Equilateral Triangle

For an equilateral triangle (all sides equal), use the formula:

Area = (√3 / 4) × s²

Where s is the side length.

Example 4:

Find the area of an equilateral triangle with side length 6 cm.

Area = (√3 / 4) × (6²)
= (√3 / 4) × 36
≈ 62.35 / 4 = 15.59 cm²

✅ Final Answer: 15.59 cm²

5. Real-Life Applications of Triangle Area

Construction & Architecture – Used in designing roofs, buildings, and plots of land.
Engineering – Helps in determining the load-bearing capacity of structures.
Art & Design – Used in patterns, origami, and graphic design.
Geography & Maps – Used to calculate land areas in different shapes.

Practice Problems

Find the area of a triangle with base = 12 cm and height = 7 cm.
Calculate the area of an equilateral triangle with side length 10 cm.
A triangle has sides 5 cm, 6 cm, and 7 cm. Find its area using Heron’s Formula.
A right-angled triangle has legs of 9 cm and 12 cm. What is its area?
A triangular garden has a base of 15 m and a height of 9 m. Find its area.