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Area of a Triangle

In this section, we will learn triangle definition, types of triangles, area of triangle formula, derivation, and how to find the area of a trainable along with examples in detail.

Triangle

A triangle is a polygon having three vertices and three edges. The sum of three interior angles is 180 degrees. The following diagram represents a triangle.

Area of a Triangle

Types of triangle

There are three types of triangles:

  • Equilateral Triangle
  • Isosceles Triangle
  • Scalene Triangle

Equilateral Triangle

An equilateral triangle has three sides of equal length and three equal angles. Each angle is of 60 degrees. It has three lines of symmetry.

Area of a Triangle

Isosceles Triangle

An isosceles triangle has two sides of equal length and two equal angles. It has one line of symmetry.

Area of a Triangle

Scalene Triangle

A scalene triangle has no sides of equal length and no equal angles. It has no line of symmetry.

Area of a Triangle

Area of a Triangle

The area of a triangle is the region covered by a triangle. The area of a triangle is equal to half of the base multiplied by the perpendicular height of the triangle.

If the given triangle is a right triangle, multiply the two sides together, which are adjacent to the right angle.

Area of a Triangle

If the given triangle is not a right triangle, first find the perpendicular height of the triangle by dropping a vertical line down from the highest point on the triangle to the base, as we have shown in the following figure.

Area of a Triangle

Area of Triangle Formula

Two formulas are used to find the area of a triangle.

It is the half of the base multiplied by the height. The formula given below works for all triangles.

Area of Triangle (A) = Area of a Triangle bh

Where b denoted the base and h denote the height of the given triangle.

Note: We can choose any side of the triangle as a base but make sure that height is measured at right angles to the base.

Heron's Formula

Heron's formula applied when all three sides of a triangle are given, or the three sides are unequal.

Area of a Triangle

It includes two important steps:

Step 1: First, calculate the semi-perimeter (s) by adding the length of three sides of the triangle and divide the sum by 2.

semi-perimeter (s) = (a + b + c) / 2

Where a, b, c is the length of the sides.

Step 2: Apply the semi-perimeter and three sides a, b, c of the triangle in the formula to find the area of a triangle.

Area of a Triangle (A) = √(s (s-a)(s-b)(s-c) )

We can also find the area of a triangle when two sides and an angle is known. But the Heron's formula does not work for the same. The formula depends on which sides and angles are given. Let's see the formulas for different sides and angles.

  • When the sides a, b and the angle C is given then the formula for area of a triangle will be:
Area of a Triangle

Area of a Triangle (A) = ½ ab sin C

  • When the sides b, c, and the angle A is given then the formula for the area of a triangle will be:
Area of a Triangle

Area of a Triangle (A) = ½ bc sin A

  • Similarly, when the sides c, a, and the angle B is given, then the formula for the area of a triangle will be:
Area of a Triangle

Area of a Triangle (A) = ½ ca sin B

Area of an Equilateral Triangle

An equilateral triangle has three sides of equal length.

Area of a Triangle

Area of an equilateral triangle (A) = Area of a Triangle a2

Where a is the length of a triangle.

Area of an Isosceles Triangle

Area of a Triangle
Area of an Isosceles triangle (A) = ½ (base × height)

Derivation

Let's see why the area of a triangle is half of b*h.

  • Draw a triangle whose base is b and height is h, as shown below.
Area of a Triangle
  • Double the triangle.
Area of a Triangle
  • In the newly created triangle, draw a perpendicular.
Area of a Triangle
  • Cut a triangle (yellow part) and move it to the right, as we have in the following image.
Area of a Triangle

We see that the triangle is now converted into a rectangle. We know the formula of area of a rectangle:

Area of rectangle (A) = length * width

i.e. is (b*h).

The base multiplied by height is twice the area of the triangle (represented in blue color). But we have to find the area of a triangle i.e.

A = Area of a Triangle bh

Therefore, the area of a triangle is half of the base*height.

Therefore, the area of a triangle is half of the base*height.

How to Find Area of a triangle

When base and height is given

Example 1: Find the area of a triangle whose height is 20 cm and the base is 15 cm.

Solution:

Given, base (b) = 15 cm, height (h) = 20 cm
Area of a Triangle

We know that,

Area of Triangle (A) = ½ bh

Putting the values, we get:

A = ½ 15 * 20
A = 150 sq. cm.

The area of the triangle is 150 sq. cm.

Example 2: Find the area of the triangle given below.

Area of a Triangle

Solution:

Given, base (b) = 9 m, height (h) = 13 m

We know that,

Area of Triangle (A) = ½ bh

Putting the values, we get:

A = ½ 9 * 13
A = 58.5 m2

The area of triangle is 58.5 m2.

Example 3: The base of an obtuse triangle is 6.6 inches, and the height is 14.7 inches. Find the area of the triangle.

Solution:

Given, base (b) = 6.6 inches, height (h) = 14.7 inches

Area of a Triangle

We know that,

Area of Triangle (A) = ½ bh

Putting the values, we get:

A = ½ 6.6 * 14.7
A = 48.51 inches2

The area of the triangle is 48.51 inches2.

Example 4: Find the area of the triangle given below.

Area of a Triangle

Solution:

From the above figure, base (b) = 177 cm.

Note: We will consider the height that makes the right angle on the base that is 100 cm. We will not consider 130 as height because it doesn't make the right angle.

We know that,

Area of Triangle (A) = ½ bh

Putting the values, we get:

A = ½ 177 * 100
A = 8850 sq. cm

The area of the triangle is 8850 sq. cm.

When the length of three sides is given

Example 5: Find the area of a triangle whose all sides are 7 cm long.

Solution:

Area of a Triangle

Given, a = 7 cm, b = 9 cm, c = 11 cm

In this question, the length of all three sides is given. So, we will apply Heron's formula.

Area of a Triangle (A) = √(s (s-a)(s-b)(s-c) )

First, we will calculate semi-perimeter (s).

s = (7 + 9 + 11) / 2

s = 13.5 cm

Putting the values in the above formula, we get:

A = √(13.5 (13.5-7)(13.5-9)(13.5-11) )

A = √(13.5 (6.5)(4.5)(2.5) )

A = √987.1875

A = 31.41 sq. cm.

The area of the triangle is 31.41 sq. cm.

When SAS (side-angle-side) is given

Example 6: Find the area of a triangular land whose side AB is 125 m and side BC is 220 m. The angle between AB and BC is .

Solution:

First, draw a triangle according to the given information:

Area of a Triangle

Given, BC = a= 220 m, AB = c = 125 m, and angle B = 123°

We know that,

Area of triangle (A) = ½ ac sin B

A = ½ (220 * 125) * (sin )

A = ½ (27500) * (0.83867)

A = ½ (23063.4407)

A = 11,532 m2

The area of triangle is 11,532 m2.

Example 7: Find the area of the triangle given below.

Area of a Triangle

Solution:

Given, CB = a = 9 cm, AC = b = 14 cm, and angle C is 25°

We know that,

Area of triangle (A) = ½ ab sin C

A = ½ (9 * 14) * (sin 25°)

A = ½ (126) * (0.42261)

A = ½ (53.24886)

A = 27 cm2

The area of triangle is 27 cm2.

Area of an equilateral triangle

Example 8: In an equilateral triangle, a side is ten cm long, find the area of the triangle.

Solution:

Given, a = 10 cm

Area of a Triangle

We know the formula of area of an equilateral triangle:

Area of a Triangle




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