# Triangles: Area

A triangle is a polygon with three sides.

We know that the area of a rectangle is b×h , where bb is the base and h is the height of the rectangle. Using the formula for the area of a rectangle, we can find the area of a triangle.

A diagonal of a rectangle separates the rectangle into two congruent triangles. The area of each triangle is one-half the area of the rectangle.

So, the area A of a triangle is given by the formula A=^{1}/_{2}bh where bb is the base and h is the height of the triangle.

**Example:**

Find the area of the triangle.

## What Is the Area of a Triangle?

The area of a triangle is the region enclosed between the sides of the triangle. Depending on the length of the sides and the internal angles, the area of a triangle varies from one triangle to another. The unit of area is measured in square units, for example, m^{2}, cm^{2}, in^{2}, etc.

There are many ways to find the area of a triangle. Apart from the above formula, Heron’s formula is used to calculate the triangle’s area, when we know the length of all three sides. Trigonometric functions are also used to find the area of a triangle when we know two sides and the angle formed between them.

**Example: **What is the area of a triangle with base ‘b’ = 2 cm and height ‘h’ = 4 cm?

Using the formula: Area of a Triangle, A = 1/2 × b × h = 1/2 × 4 × 2 = 4 cm^{2}

Triangles can be classified based on their angles as acute, obtuse, or right triangles. They can be scalene, isosceles, or equilateral when classified based on their sides.

## Area of Triangle Using Heron’s Formula

Heron’s formula is used to find the area of a triangle when the length of the 3 sides of the triangle is known. To use this formula, we need to know the perimeter of the triangle which is the distance covered around the triangle and is calculated by adding the length of all three sides. Heron’s formula has two important steps.

- The first step is to find the semi perimeter (half perimeter) of the given triangle by adding all three sides and dividing it by 2.
- The next step is to apply the value of the semi-perimeter of the triangle in the main formula called “Heron’s Formula”.

Consider the triangle ABC with side lengths a, b, and c. To find the area of the triangle we use Heron’s formula:

Note that (a + b + c) is the perimeter of the triangle. Therefore, ‘s’ is the semi-perimeter which is: (a + b + c)/2

Where, s is half the perimeter,

We can also determine the area of a triangle by the following methods:

- In this method two Sides, one included Angle is given

Where a, b, c are the lengths of the sides of a triangle

In this method we find area of an Equilateral Triangle

In this method we find area of a triangle on a coordinate plane by Matrices

Where, *(x*_{1}*, y*_{1}*), (x _{2}, y_{2}), (x_{3}, y_{3})* are the coordinates of the three vertices

- In this method, we find area of a triangle in which two vectors from one vertex is there.

## Area of Triangle With 2 Sides and Included Angle (SAS)

When two sides and the included angle of a triangle are given, we use a formula that has three variations according to the given dimensions. For example, consider the triangle given below.

When sides ‘b’ and ‘c’ and included angle A is known, the area of the triangle is:

Area (∆ABC) = 1/2 × bc × sin(A)

When sides ‘a’ and ‘b’ and included angle C is known, the area of the triangle is:

Area (∆ABC) = 1/2 × ab × sin(C)

When sides ‘a’ and ‘c’ and included angle B is known, the area of the triangle is:

Area (∆ABC) = 1/2 × ac × sin(B)

**Example:** In ∆ABC, angle A = 30°, side ‘b’ = 4 units, side ‘c’ = 6 units.

Area (∆ABC) = 1/2 × bc × sin A

= 1/2 × 4 × 6 × sin 30º

= 12 × 1/2 (since sin 30º = 1/2)

Area = 6 square units.

## How to Calculate the Area of a Triangle?

The area formulas for all the different types of triangles like the equilateral triangle, right-angled triangle, and isosceles triangle are given below.

### Area of a Right-Angled Triangle:

A right-angled triangle, also called a right triangle, has one angle equal to 90° and the other two acute angles sum up to 90°. Therefore, the height of the triangle is the length of the perpendicular side.

Area of a Right Triangle = A = 1/2 × Base × Height

### Area of an Equilateral Triangle:

An equilateral triangle is a triangle where all the sides are equal. The perpendicular drawn from the vertex of the triangle to the base divides the base into two equal parts. To calculate the area of the equilateral triangle, we need to know the measurement of its sides.

Area of an Equilateral Triangle = A = (√3)/4 × side^{2}

### Area of an Isosceles Triangle:

An isosceles triangle has two of its sides equal and the angles opposite the equal sides are also equal.

where ‘b’ is the base and ‘a’ is the measure of one of the equal sides.

Observe the table given below which shows all the formulas for the area of a triangle.

**Examples on Area of Triangle**

Example 1: Find the area of a triangle with a base of 10 inches and a height of 5 inches.

Solution:

A = (1/2) × b × h

A = 1/2 × 10 × 5

A = 1/2 × 50

A = 25 in^{2}

Example 2: Find the area of a right-angled triangle with a base of 9 cm and a height of 11 cm.

Solution:

A = (1/2) × b × h

A = 1/2 × 9 × 11

A = 1/2 × 99

A = 49.5 cm2

Example 3: Find the area of an obtuse-angled triangle with a base of 8 cm and a height of 7 cm.

Solution:

A = (1/2) × b × h

A = 1/2 × 8 × 7

A = 1/2 × 56

A = 28 cm2

## Area of a Triangle Solved Examples

**Example 1:**

Find the area of an acute triangle with a base of 13 inches and a height of 5 inches.

**Solution:**

A = (½)× b × h sq.units

⇒ A = (½) × (13 in) × (5 in)

⇒ A = (½) × (65 in^{2})

⇒ A = 32.5 in^{2}

**Example 2:**

Find the area of a right-angled triangle with a base of 7 cm and a height of 8 cm.

**Solution:**

A = (½) × b × h sq.units

⇒ A = (½) × (7 cm) × (8 cm)

⇒ A = (½) × (56 cm^{2})

⇒ A = 28 cm^{2}

**Example 3:**

Find the area of an obtuse-angled triangle with a base of 4 cm and a height 7 cm.

**Solution:**

A = (½) × b × h sq.units

⇒ A = (½) × (4 cm) × (7 cm)

⇒ A = (½) × (28 cm^{2})

⇒ A = 14 cm^{2}

**Solved Examples**

**Q.1: **The sides of a right triangle ABC are 5 cm, 12 cm, and 13 cm.

**Solution: **In △ABC in which base= 12 cm and height= 5 cm

Area of △ABC=^{1}/_{2}×B×H

A = ^{1}/_{2}×12×5

A = 30 cm^{2}

**Q.2:** Find the area of a triangle, which has two sides 12 cm and 11 cm and the perimeter is 36 cm.

** Solution: **Here we have perimeter of the triangle = 36 cm, a = 12 cm and b = 11 cm.

Third side c = 36 cm – (12 + 11) cm = 13 cm

So, 2s = 36, i.e., s = 18 cm,

s – a = (18 – 12) cm = 6 cm,

s – b = (18 – 11) cm = 7 cm,

and, s – c = (18 – 13) cm = 5 cm.

**FAQs on Area Of Triangle**

### What is the Area of a Triangle?

The area of a triangle is the region enclosed by its perimeter or the three sides of the triangle.

### How to Calculate the Area of a Triangle?

For any given triangle, where the base of the triangle is ‘b’ and height is ‘h’, the area of the triangle can be calculated by the formula, A = 1/2 (b × h).

### How to Find the Base and Height of a Triangle?

The area of the triangle is calculated with the formula: A = 1/2 (b × h). Using the same formula, the height and base can be calculated when the other dimensions are known.

### How to Find the Area and Perimeter of a Triangle?

The area of a triangle can be calculated with the help of the formula: A = 1/2 (b × h). The perimeter of a triangle can be calculated by adding the lengths of all the three sides of the triangle.

### How to Find the Area of a Triangle Without Height?

When only the length of the 3 sides of the triangle are known and the height is not given, the Heron’s formula can be used to find the area of the triangle. Heron’s formula:

### How to Find the Area of a Triangle Given Two Sides and an Included Angle?

In a triangle, when two sides and the included angle is given, then the area of the triangle is half the product of the two sides and sine of the included angle. For example, In ∆ABC, when sides ‘b’ and ‘c’ and included angle A is known, the area of the triangle is calculated with the help of the formula: 1/2 × b × c × sin(A). For a detailed explanation refer to the area of the triangle with 2 Sides and included angle (SAS).

### How to Find the Area of a Triangle with 3 Sides?

The area of a triangle with 3 sides can be calculated using Heron’s formula. Heron’s formula:

where a, b, and c are the sides and ‘s’ is the semi-perimeter; s = (a + b + c)/2.

### How to Calculate the Area of an Obtuse Triangle?

The area of an obtuse triangle can be calculated using the formula: 1/2 × Base × Height.

### How to Find the Area of an Irregular Triangle/Scalene Triangle?

The area of an irregular triangle (sometimes referred to as a scalene triangle) can be calculated using Heron’s formula:

where a, b, and c are the sides and ‘s’ is the semi-perimeter; s = (a + b + c)/2.

### What is the area when two sides of a triangle and included angle are given?

The area will be equal to half times of the product of two given sides and sine of the included angle.

### How to find the area of a triangle given three sides?

When the values of the three sides of the triangle are given, then we can find the area of that triangle by using Heron’s Formula. Refer to the section ‘**Area of a triangle by Heron’s formula**‘ mentioned in this article to get a complete idea.

### How to find the area of a triangle using vectors?

Suppose vectors **u** and **v** are forming a triangle in space. Then, the area of this triangle is equal to half of the magnitude of the product of these two vectors, such that,

**A = ½ |u × v|**

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