# Area of Triangle

The total space occupied by the three sides of a triangle in a 2D plane is called an Area of a Triangle. The 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. Scalene triangle, an isosceles triangle, or an equilateral, this formula applies to all types of Triangles. The height and the base of a triangle are perpendicular to each other.

In this blog, we’ll learn about the area of triangle formulas for different types of triangles, along with some examples.

## What is the Area Of the Triangle?

The region enclosed within the sides of the triangle is called an Area Of The Triangle.  The length of the sides and the internal angles of the area of a triangle varies from one triangle to another. Square units, like m2, cm2, in2, and so on are the units for an area of a triangle.

## The formula used to find The Area Of A Triangle?

Various formulas are used to find an area of a triangle. For instance, when we know the length of all three sides of a triangle then Heron’s Formula is used to find an area of a triangle. To find the area of a triangle when we know two sides and the angle formed between them, various trigonometric functions are used. However, the main formula to find the area of the triangle remains-  Area of triangle = 1/2 × base × height.

The classification of angles of the triangle is based on whether it is an acute, obtuse, or right triangle. When classified based on their sides They can be scalene, isosceles, or an equilateral triangle.

## Let’s understand the Area of the Triangle by using Heron’s Formula:

When we want to find the area of a triangle whose length of the 3 sides of the triangle is known, we use Heron’s formula. If we want to apply this formula, we should 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. We can apply this formula with the help of two steps:

• Step 1: By adding all three sides and dividing them by 2 we can find the semi-perimeter(half-perimeter) of the given triangle.
• Step 2: Applying the value of the semi-perimeter of the triangle in the main formula is called ‘Heron’s Formula’.

Consider the triangle PQR with side lengths p, q, and r. Here we use Heron’s formula:

Area = s(s−p)(s−q)(s−r)

So, (p+q+r) is the perimeter of the triangle. Therefore, ‘s’ is the semi-perimeter which is: (p+q+r)/2.

## How to find an Area of Triangle With 2 Sides and Included Angle (SAS)

When 2 sides and the included angle of a triangle are already mentioned, we use a formula that has 3 variations.

When 2 sides ‘b’ and ‘c’ and included angle A is known, the area of the triangle is:

Area (∆ABC) = 1/2 × bc × sin(A).

When 2 sides ‘a’ and ‘b’ and included angle C is known, the area of the triangle is:

Area (∆ABC) = 1/2 × ab × sin(C).

When 2 sides ‘a’ and ‘c’ and included angle B is known, the area of the triangle is:

Area (∆ABC) = 1/2 × ac × sin(B).

For instance, In ∆PQR, angle P = 30°, side ‘q’ = 4 units, side ‘r’ = 6 units.

Area (∆PQR) = 1/2 × bc × sin P

= 1/2 × 4 × 6 × sin 30º

= 12 × 1/2 (since sin 30º = 1/ 2)

Area = 6 square units.

## Calculating the area of a triangle

Depending upon the type of triangle and the given dimensions, the area of the triangle can be calculated.

## Formulas for Area of Triangle

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

• Right-Angled Triangle: When one angle is equal to 90° and the other two acute angles sum up to 90° then it’s called a right-angled triangle or a right triangle. Hence, the length of the perpendicular side is the height of the triangle.

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

• Equilateral-Angled Triangle: When all the sides of a triangle are equal then it’s called an equilateral triangle. The perpendicular drawn from the base to the vertex of the triangle divides the base into two equal parts.

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

• Isosceles Triangle: If any 2 sides of the triangle are equal and the angles opposite the equal sides are also equal.

Area of an Isosceles Triangle = A = 1/4b 4a²−b².

where ‘b’ is the triangle’s base and ‘ a’ is the measure of 1 of the equal sides.

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