Area of Triangle
The area of a triangle is defined as the total space occupied by the three sides of a triangle in a 2dimensional plane. The basic formula for the area of a triangle is equal to half the product of its base and height, i.e., A = 1/2 × b × h. This formula is applicable to all types of triangles, whether it is a scalene triangle, an isosceles triangle or an equilateral triangle. It should be remembered that the base and the height of a triangle are perpendicular to each other.
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. 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.
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. 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 semiperimeter 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:
Area = \(\sqrt {s(s  a)(s  b)(s  c)}\)
Note that (a + b + c) is the perimeter of the triangle. Therefore, 's' is the semiperimeter which is: (a + b + c)/2
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, rightangled triangle, and isosceles triangle are given below.
Area of a RightAngled Triangle:
A rightangled 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.
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 one of the equal sides.
Observe the table given below which shows all the formulas for the area of a triangle.
Given Dimensions  Area of a Triangle Formula 
When the base and height of a triangle are given.  A = 1/2 (base × height) 
When the sides of a triangle are given as a, b, and c. 
(Heron's formula) Area of a scalene triangle = \(\sqrt {s(s  a)(s  b)(s  c)}\) 
When two sides and the included angle is given.  A = 1/2 × side 1 × side 2 × sin(θ) where θ is the angle between the given two sides 
When base and height is given.  Area of a right triangle = 1/2 × Base × Height 
When one side is given.  Area of an equilateral triangle = (√3)/4 × side^{2} 
When an equal side and the base is given.  Area of an isosceles triangle = 1/4 × b\(\sqrt {4{a^2}  {b^2}}\) where 'b' is the base and 'a' is the measure of equal side. 
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 rightangled 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 cm^{2}

Example 3: Find the area of an obtuseangled 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 cm^{2}
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: A = \(\sqrt {s(s  b)(s  b)(s  c)}\) where a, b, and c are the sides and 's' is the semiperimeter; s = (a + b + c)/2.
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: A = \(\sqrt {s(s  a)(s  b)(s  c)}\) where a, b, and c are the sides and 's' is the semiperimeter; 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: \(\sqrt {s(s  a)(s  b)(s  c)}\) where a, b, and c are the sides and 's' is the semiperimeter; s = (a + b + c)/2.