Area of Triangle: Definition, Formula, Examples

What is Area of Triangle?

Area of a triangle is the amount of space enclosed within its three sides. For example, If you draw a triangle on a piece of paper and then paint inside it, the part you color is its area.  

The basic formula for calculating the area of a triangle is 

Area = ½ × base × height.

This formula can be used for all types of triangles, including a scalene triangle, an isosceles triangle, and an equilateral triangle.

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Area of Triangle Definition

The area of a triangle is the space inside its three sides. It shows how much surface the triangle covers and helps determine its size.  The area of a triangle is always measured in square units such as square centimeters (cm²), square meters (m²), square millimeters (mm²), or square inches (in²).

Area of a Triangle Formula

The most common area of the triangle formula is:

Area = ½ × base × height 

In this formula:

• Base (b) is the length of one side of the triangle, usually the bottom.

• Height (h) is the perpendicular distance from the base to the top vertex of the triangle.

Read More: Perimeter of a Triangle

Area of Triangle Heron’s Formula

Students can also calculate the area of a triangle using Heron’s Formula when they know the lengths of all three sides. This method is especially useful when the height of the triangle is not given or is difficult to measure. 

To use Heron’s Formula, first calculate the semi-perimeter of the triangle by adding the three sides and dividing the sum by two. Then, apply the values into the formula:

Area =  √s(s−a)(s−b)(s−c)

Here, s is the semi-perimeter, and a, b, and c are the lengths of the triangle’s sides. This method helps to find the area accurately, even for triangles that do not have a right angle or equal sides.

Read More: Median of a Triangle

Area of Triangle Using Two Sides and the Included Angle (SAS Method)

When two sides of a triangle and the included angle between them are known, the area of the triangle is calculated using a trigonometric formula. This method is especially useful when the height of the triangle is not provided.

The formula varies slightly depending on which sides and angle are given:

If sides b and c and the included angle A are known:

Area = ½ × b × c × sin⁡(A)

If sides a and b and the included angle C are known:

Area = ½ × a × b × sin⁡(C)

If sides a and c and the included angle B are known:

Area = ½ ​× a × c × sin(B)

This approach uses the sine of the included angle and works for any triangle, regardless of its type. 

Read More: How to Find the Angle of a Triangle?

Area of Equilateral Triangle

An equilateral triangle is a triangle where all three sides are equal in length, and all three angles are equal, each measuring 60 degrees.

The formula to calculate the area of an equilateral triangle is:

Area = √3/ × side2

Let the side of the triangle be represented by a.

Area of Scalene Triangle

A scalene triangle has three sides of different lengths and three angles of different measures. To calculate its area, you can use the formula:

Area = ½ × base × height, if the height is known.

If only the side lengths are given, use Heron’s Formula, which uses the semi-perimeter and all three sides to find the area without needing the height.

Area of Triangle Examples

Below are some area of triangle examples for different triangle types, which help students practice and understand how to apply the formulas correctly.

Example 1: Find the area of an equilateral triangle having side ‘a’ equal to 7 cm.

Solution:

Given,

Side of the triangle (a) = 7

We use the formula:

Area = √3/4a2

Step 1: Substitute the value of side a:

Area = √3/4 ​​×72 = √3/4​​ × 49

Step 2: Using √3 ≈ 1.732

Area ≈ 1.732 ​× 49 ≈ 0.433 × 49 ≈ 21.217

Example 2: Find the area of a right-angled triangle with base a = 5 and height c = 3

Solution:

Given,

Base (a) = 5 cm

Height (c) = 3 cm

We use the formula:

Area = ½ × base × height

Area = ½ × 5 × 3 = 15/2 = 7.5 cm2

Example 3. Find the area of a scalene triangle with sides a =7 cm, b = 8, and c = 9 cm.

Solution:

Given,

Sides: a = 7 cm, b = 8 cm, c = 9 cm

Step 1: Find the semi-perimeter

S = a + b + c 

2 = 7 + 8 + 9/2  = 24/2 = 12

Step 2: Apply Heron’s Formula

√(12)(12 - 7)(12-8)(12-9)

Area =   (12)(12 - 7) (12 - 8) (12 -9)

 = 12 543

 = 720 

Area ≈ 26.83cm2

4. In triangle ABC, the length of side b is 4 units, the length of side c is 6 units, and the included angle A is 30∘ . Find the area of triangle ABC using the SAS formula.

Solution:

Given:
Side b = 4 units

Side c = 6 units

Angle A = 30∘

We use the SAS formula:

Area = 1/2 × b × c × sin⁡(A)

Step 1: Substitute the known values into the formula

Area = 1/2 × 4 × 6  × sin⁡(30∘)

Step 2: Use the value of sin⁡(30∘)

Area = 1/2 × 4 × 6 × 1/2

Step 3: Calculate

Area = ½ × 24 × 12 = 12 × 12 = 6

The area of triangle ABC is 6 square units.

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