 # Area of Triangle Formula

We all know that a triangle is a polygon, which has three sides. The area of a triangle is a measurement of the area covered by the triangle. We can express the area of a triangle in the square units. The area of a triangle is determined by two formulas i.e. the base multiplies by the height of a triangle divided by 2 and second is Heron’s formula. Let us discuss the Area of a Triangle formula in detail. ## Area of Triangle a Formula

### What is an Area of a triangle?

The area of a polygon is the number of square units covered by the polygon. The area of a triangle is determined by multiplying the base of the triangle and the height of the triangle and then divides it by 2. The division by 2 is done because the triangle is a part of a parallelogram that can be divided into 2 triangles.

Area of a parallelogram = B × H

Where,

 B the base of the parallelogram H the height of the parallelogram

As triangle is the one-half of the parallelogram, so the area of a triangle is:

A= $$\frac{1}{2} \times b \times h$$

Where,

 B the base of the triangle H the height of the triangle

### Heron’s Formula for Area of a Triangle

Herons formula is a method for calculating the area of a triangle when the lengths of all three sides of the triangle are given.

Let a, b, c are the lengths of the sides of a triangle.

The area of the triangle is:

Area=$$\sqrt{s(s−a)(s−b)(s−c)}$$

Where, s is half the perimeter,

s= $$\frac{a+ b+ c}{2}$$

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

1. In this method two Sides, one included Angle is given

Area= $$\frac{1}{2} \times a \times b \times \sin c$$

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

1. In this method we find area of an Equilateral Triangle

Area= $$\frac{\sqrt{3} \times a^{2}}{4}$$

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

$$\frac{1}{2} \times \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix}$$

Where, (x1, y1), (x2, y2), (x3, y3) are the coordinates of the three vertices

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

Area of triangle = $$\frac{1}{2} \left( \overrightarrow{u}\times \overrightarrow{v}\right)$$

## Solved Examples

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

Solution: In $$\bigtriangleup$$ABC in which base= 12 cm and height= 5 cm

Area of $$\bigtriangleup ABC = \frac{1}{2} \times B \times H$$

A = $$\frac{1}{2} \times 12 \times 5$$

A = 30 cm2

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.

Area of the triangle = $$\sqrt{s(s−a)(s−b)(s−c)}$$

A= $$\sqrt{18\times 6\times 7\times 5}$$

A= $$6\sqrt{105}$$ cm2

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##### Maths Formulas 4 Followers

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KUCKOO B

I get a different answer for first example.
I got Q1 as 20.5
median 23 and
Q3 26 Guest
Yashitha

Hi
Same Guest
virat

yes

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