Area of Triangle: Definitions, Examples & 6+Formulas

Finding the area of a triangle is one of the first steps to mastering geometry. When we know the base and height it is easy. It is simply half of b times h. The universal formula is: Area = 1/2 bh

While this is the most common method, math problems in school often give you different clues, like only the side lengths or angles. In this guide, WuKong Education will show you how to use this basic formula and what to do when the height is missing.

What is the Area of a Triangle?

The area of a triangle is the two-dimensional space encompassed by its three sides. It is usually measured in square units, like square centimeters (cm²), square meters (m²), or square inches. The area of a triangle is proportional to the lengths of its sides and the angles between them, as demonstrated in the different formulas and methods presented in this page.

How to Find the Area of a Triangle?

To calculate the area of a triangle, you must know at least one of the following: the base and height, the lengths of all three sides, or the lengths of two sides and their angle. Let us look at how to use these various ways to determine the area of different sorts of triangles.

Area of Equilateral Triangle

An equilateral triangle is a triangle with all three sides of equal length. To find the area of an equilateral triangle with a side length of ‘s’, you can use the formula:

Area = (√3/4) × s²

Here’s a step-by-step explanation with an example:

Steps to find the area of an equilateral triangle:

1. Identify the side length of the equilateral triangle.
2. Plug the side length into the formula: Area = (√3 / 4) × side length^2.
3. Calculate the result to find the area of the equilateral triangle.

Example: Find the area of an equilateral triangle with a side length of 8 cm.

Step 1: The side length of the equilateral triangle is 8 cm.

Step 2: Plug the side length into the formula:

Area = (√3 / 4) × 8^2 = (√3 / 4) × 64

A = (1.732 / 4) × 64 = 0.433 × 64

= 27.73 square cm

Therefore, the area of the equilateral triangle with a side length of 8 cm is 27.73 square cm.

Explanation:

The formula for the area of an equilateral triangle is derived from the general triangle area formula:

Area = (1/2) × base × height

For an equilateral triangle, the base and height are related by the fact that the height is the perpendicular distance from the base to the opposite vertex. This height can be calculated using trigonometry as:

height = (side length × √3) / 2

Substituting this into the general triangle area formula, we get:

Area = (1/2) × side length × (side length × √3 / 2)

= (√3 / 4) × side length^2

This is the simplified formula used to find the area of an equilateral triangle given the side length.

The key steps are to identify the side length, plug it into the formula, and calculate the final area. This provides a straightforward way to determine the area of any equilateral triangle.

Area of Right Triangle

A right triangle is a triangle with one 90-degree angle. If you know the lengths of the base (b) and height (h) of a right triangle, you can use the formula:

Area = 1/2 × b × h

Steps to find the area of a right triangle:

1. Identify the base and height of the right triangle.
2. Plug the base and height values into the formula: Area = 1/2 × base × height.
3. Calculate the result to find the area of the right triangle.

Example: Find the area of a right triangle with a base of 6 cm and a height of 8 cm.

Step 1: The base of the right triangle is 6 cm, and the height is 8 cm.

Step 2: Plug the base and height values into the formula:

Area = 1/2 × 6 cm × 8 cm

= 1/2 × 48 cm²

= 24 cm²

Therefore, the area of the right triangle with a base of 6 cm and a height of 8 cm is 24 square cm.

Explanation:

The formula for the area of a right triangle is derived from the general triangle area formula:

Area = 1/2 × base × height

For a right triangle, the base and height are perpendicular to each other, forming a right angle. This allows us to use the simpler formula of 1/2 × base × height to calculate the area.

Area of the Isosceles Triangle

An isosceles triangle is a triangle with at least two sides of equal length. If you know the length of the base (b) and the height (h), you can use the formula:

Area = 1/2 × b × h

Steps to find the area of an isosceles triangle:

1. Identify the base and height of the isosceles triangle.
2. Plug the base and height values into the formula: Area = 1/2 × base × height.
3. Calculate the result to find the area of the isosceles triangle.

Example: Find the area of an isosceles triangle with a base of 10 cm and a height of 8 cm.

Step 1: The base of the isosceles triangle is 10 cm, and the height is 8 cm.

Step 2: Plug the base and height values into the formula:

Area = 1/2 × 10 cm × 8 cm

= 1/2 × 80 cm²

= 40 cm²

Therefore, the area of the isosceles triangle with a base of 10 cm and a height of 8 cm is 40 square cm.

Area of Triangle with 3 Sides

If you know the lengths of all three sides of a triangle (a, b, and c), you can use Heron’s formula to find the area:

Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2

Where:

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

Here’s a step-by-step explanation with an example:

Steps to find the area of a triangle with three known side lengths:

1. Identify the lengths of the three sides of the triangle (a, b, and c).
2. Calculate the semi-perimeter (s) using the formula: s = (a + b + c) / 2.
3. Plug the side lengths and the semi-perimeter into Heron’s formula: Area = √(s(s-a)(s-b)(s-c))
4. Calculate the result to find the area of the triangle.

Example: Find the area of a triangle with side lengths of 6 cm, 8 cm, and 10 cm.

Step 1: The lengths of the three sides are: a = 6 cm, b = 8 cm, and c = 10 cm.

Step 2: Calculate the semi-perimeter (s):

s = (a + b + c) / 2

s = (6 cm + 8 cm + 10 cm) / 2

= 24 cm / 2

= 12 cm

Step 3: Plug the side lengths and the semi-perimeter into Heron’s formula:

Area = √(s(s-a)(s-b)(s-c))

A= √(12 cm (12-6)(12-8)(12-10) = √(12 cm × 6 × 4 × 2)

= √(576 cm²) = 24 cm²

Therefore, the area of the triangle with side lengths of 6 cm, 8 cm, and 10 cm is 24 square cm.

Formulas for Triangle Area

Depending on the information given, a triangle’s area can be calculated using one of numerous formulas. The most widely used formulas include:

This table summarizes the several formulas for estimating the area of a triangle:

Five Practical and Solved Examples for Triangles

Example 1 Finding Area with Known Base and Height

The base of a triangle is 10 cm and the height is 6 cm. Find the area of the triangle.

Solution:

The formula for the area of a triangle is: Area = 1/2 × base × height

Substituting the given values:

Area = 1/2 × 10 cm × 6 cm = 30 square cm

Example 2: Calculating the Area of a Right-Angled Triangle

A right-angled triangle has perpendicular sides of 8 cm and 6 cm. Find the area of the triangle.

Solution:

For a right-angled triangle, the area formula is: Area = 1/2 × base × height

Here, the base is 8 cm and the height is 6 cm. Substituting these values:

Area = 1/2 × 8 cm × 6 cm = 24 square cm

Example 3: Using Heron’s Formula for a Triangle with Three Known Sides

The sides of a triangle are 8 cm, 6 cm, and 10 cm. Find the area of the triangle.

Solution:

For a general triangle, we can use Heron’s formula to calculate the area:

s = (a + b + c) / 2 (where a, b, c are the side lengths)

Area = √(s(s-a)(s-b)(s-c))

Substituting the given values:

s = (8 + 6 + 10) / 2 = 12

Area = √(12 × (12-8) × (12-6) × (12-10)) = 24 square cm

Example 4: How to Find the Area of an Equilateral Triangle

An equilateral triangle has a side length of 12 cm. Find the area of the triangle.

Solution:

For an equilateral triangle, the area formula is: Area = √3/4 × side length^2

Substituting the given side length:

Area = √3/4 × 12^2 = 62.43 square cm

Example 5: Calculating Area in a Coordinate Plane

A triangular-shaped farmland has vertices at the coordinates (5,3), (10,7), and (2,10). Find the area of the farmland.

Solution: We can use the coordinate formula to calculate the area of the triangle: Area = 1/2 × |(x1y2 – x2y1) + (x2y3 – x3y2) + (x3y1 – x1y3)|

Substituting the given coordinates:

x1=5, y1=3

x2=10, y2=7

x3=2, y3=10

• Area = 1/2 × |(57 – 10(1010 – 2(23 – 510)| = 1/2 × |35 – 30 + 100 – 14 – 15 – 50| = 1/2 × 26 = 13 square units

In summary, these 5 examples cover various methods for calculating the area of a triangle, including:

1. Known base and height
2. Known perpendicular sides of a right-angled triangle
3. Known side lengths
4. Known side length of an equilateral triangle
5. Known coordinates of the vertices

Master More Shapes: Your Area Calculation Cheat Sheet

Did you know that the triangle is actually the building block of geometry? Every multi-sided shape—from rectangles to complex polygons—can be broken down into a collection of triangles.

Once you master the triangle, you unlock the key to understanding how all area formulas work. Use the table below to see how triangle math connects to other shapes, or jump straight to our other deep-dive guides:

FAQ on Area of a Triangle

Delvair | WuKong Math Teacher

Delvair, a graduate of the Federal University of Maranhão in Brazil, is a dedicated educator with over six years of experience in school-based mathematics instruction. She specializes in advanced math pedagogy, with a particular expertise in Math Kangaroo competition coaching. Driven by the belief that education is the bedrock of a thriving society, Delvair is committed to creating an empowering environment where every child can excel. She holds the firm conviction that with the right guidance, every student possesses the potential to master complex mathematical concepts.