# Area of Triangle

Area of Triangle
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The area of a triangle is defined as the total space occupied by the three sides of a triangle in a 2-dimensional plane. The area of a triangle is equal to half of the base times height, i.e. A = 1/2 x b x h. Hence, to find the area of a triangle, we need to know the base (b) and height (h) of it

This formula is applicable to all types of triangles, whether it is scalene, isosceles or equilateral. Remember the base and height of a triangle are perpendicular to each other. The unit of area is measured in square units, for example m2, cm2, in2, etc.

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 x b x h = 1/2 x 4 cm x 2 cm = 4 cm2

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.

In this lesson, we will learn the area of triangle formulas for different types of triangles, along with some examples.

## What Is the Area of a Triangle?

The area of a triangle is the region enclosed between the sides of the triangle. Every possible triangle will have some area. Depending on the length of the sides and the internal angles, the area of a triangle varies from one triangle to another.

## What Is the Area of a Triangle Formula?

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. Also, trigonometric functions are used to find the area when we know two sides and the angle formed between them in a triangle.

We will calculate the area for all the conditions in this article.

## 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. The perimeter of a triangle is the distance covered around the triangle and is calculated by adding lengths of all the three sides of a triangle.

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

Consider the triangle ABC with side lengths A, B, and C. Heron's formula to find the area of the triangle is:

Area = $$\sqrt {S(S - a)(S - b)(S - c)}$$

Note that (A + B + C) is the perimeter of the triangle.

'S' is the semi-perimeter which is given by (A + B + C)/2

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

There are three variations to the same formula based on which sides and included angle are given.

Consider the triangle,

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

Area (∆ABC) = 1/2 x bc x sin(A)

When sides 'b' and 'a' and included angle B is known, the area of the triangle is:

Area (∆ABC) = 1/2 x ab x sin(C)

When sides 'a' and 'c' and included angle C is known, the area of the triangle is:

Area (∆ABC) = 1/2 x ac x sin(B)

These formulas are very easy to remember and also to calculate.

Example: In ∆ABC,  A = 30° and b = 4, c = 6 in units. Then the area will be;

Area (∆ABC) = 1/2 bc sin A

= 1/2 (4) (6) sin 30º

= 12 x 1/2 (since sin 30º = 1/2)

= 6 sq.unit.

## How To Calculate the Area of a Triangle?

The area of the triangle can be calculated using the formulas as discussed above or using the area of triangle calculator. The area formulas for all the different types of triangles like an area of an 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 at 90° and the other two acute angles sums to 90°. Therefore, the height of the triangle will be the length of the perpendicular side.

Area of a Right Triangle = A = 1/2 x Base x Height (Perpendicular distance)

### 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 have to know the measurement of its sides.

Area of an Equilateral Triangle = A = (√3)/4 x side2

### Area of an Isosceles Triangle:

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

Area of an Isosceles Triangle = A = $$\frac{1}{4}b\sqrt {4{a^2} - {b^2}}$$

where 'b' is the base and 'a' is the measure of equal side.

A table with all the formulas for area of a triangle is given below:

 Area of a Triangle Formula Base and height of a triangle are given A = 1/2 (base x height) Sides of a triangle A, B, and C are given Heron's formula: A = $$\sqrt {S(S - A)(S - B)(S - C)}$$ where a,b, and c are the sides and S is the semi-perimeter; S = (A+B+C)/2 Two sides and the included angle is given A = side1 x side2 x sin(θ) where θ is the angle between the given two sides Area of a right triangle A = 1/2 x Base x Height(Perpendicular distance) Area of an equilateral triangle A = (√3)/4 x side2 Area of an isosceles triangle A = 1/4 x b$$\sqrt {4{a^2} - {b^2}}$$ where 'b' is the base and 'a' is the measure of equal side.

## FAQs on Area Of Triangle

### 1. What is the Area of a Triangle?

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

### 2. How to Find the Area of a Triangle Given three Sides?

Heron's formula is used to find the area of a triangle when the length of the 3 sides of the triangle are known.

Heron's formula:A = $$\sqrt {S(S - A)(S - B)(S - C)}$$ where A,B, and C are the sides and S is the semi-perimeter; S = (A+B+C)/2.

### 3. 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 x h) square unit.

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

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

### 5. 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 x h).

The perimeter of a triangle can be calculated by adding the lengths of three sides of a triangle.

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

Heron's formula can be used to find the area of a triangle when only the length of the 3 sides of the triangle are known without height. Heron's formula:A = $$\sqrt {S(S - A)(S - B)(S - C)}$$ where A,B, and C are the sides and S is the semi-perimeter; S = (A+B+C)/2.

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

The area of a triangle is half the product of the given two sides and sine of the included angle. For a detailed explanation refer to the area of the triangle with 2 Sides and included angle (SAS). In ∆ABC, when sides 'b' and 'c' and included angle A is known, the area of the triangle is:

Area (∆ABC) = 1/2 x bc x sin(A)

### 8. 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: A = $$\sqrt {S(S - A)(S - B)(S - C)}$$ where A, B, and C are the sides and S is the semi-perimeter; S = (A + B + C)/2.

### 9. 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 the Heron's formula: $$\sqrt {S(S - A)(S - B)(S - C)}$$.

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

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

## Solved Examples

Example 1:

Find the area of an acute triangle with a base of 10 inches and a height of 5 inches.

Solution:

A = (1/2) x b x h sq.units

A = (1/2) x (10 in) x (5 in)

A = (1/2) x (50 in2)

A = 25 in2

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) x b x h sq.units

A = (1/2) x (9 cm) x (11 cm)

A = (1/2) x (99 cm2)

A = 49.5 cm2

Example 3:

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

Solution:

A = (1/2) x b x h sq.units

A = (1/2) x (8 cm) x (7 cm)

A = (1/2) x (56 cm2)

A = 28 cm2

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the 'Check Answer' button to see the result.

Mensuration
grade 9 | Questions Set 1
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