# Area of Scalene Triangle

A scalene triangle is a triangle whose all the three sides are of unequal length and all the three angles are of different measures. However, the sum of all the three interior angles is always equal to 180° degrees.

In this article, you will learn about various methods to find the area of a scalene triangle.

The area of a scalene triangle is the amount of space that it occupies in a two-dimensional surface. So, the area of a scalene triangle can be calculated if the length of its base and corresponding altitude (height) is known or the length of its three sides is known or length of two sides and angle between them is given.

Method 1:

To find area of a scalene triangle if the length of its base and corresponding altitude (height) is given (image will be updated soon)

The area of a scalene triangle = $\frac{1}{2}$ × (base) × (height) sq. units

= $\frac{1}{2}$ × (b) × (h) sq. units

Method 2:

To find area of a scalene triangle if the the length of its three sides is given (image will be updated soon)

The area of scalene triangle using Heron’s formula = $\sqrt{s(s-a)(s-b)(s-c)}$ sq. units

Where, ‘a’, ‘b’ and ‘c’ are the length of sides of the scalene triangle

And, s = semi-perimeter of triangle = $\frac{a+b+c}{2}$

Method 3:

To find the area of a scalene triangle if the length of two sides and angle between them is given. (image will be updated soon)

The area of the scalene triangle if the length of its two sides and angle between them is given.

Area of scalene triangle = $\frac{1}{2}$ × a × b × sinC sq. units

where ‘a’ and ‘b’ are the length of  two sides and C is the angle between them.

### Solved Examples:

Q.1. Find the height of the scalene triangle whose area is 12 sq. cms and one of its sides length is 6cm.

Solution: let the base of the scalene triangle be 6cm and corresponding height be ‘h’ cm.

Given, area of scalene triangle = 12 sq. cms (image will be updated soon)

⇒ $\frac{1}{2}$ × (base) × (height) =  12 sq. cms

⇒ $\frac{1}{2}$ × 6 × h  =  12 sq. cms

⇒ h = $\frac{12(2)}{6}$ = 4 cm

Q.2. Find the area of a triangular plot whose sides are in the ratio of 3:5:7 and have perimeter of 300m.

Solution:

Given, ratio of sides of triangular plot is 3:5:7

Let the sides of triangular plot be a = 3x, b = 5x and c = 7x

It is given that its perimeter = 300m

⇒ a + b + c = 300

⇒ 3x + 5x + 7x = 300

⇒ 15x = 300

⇒ x = 20

So, sides of rectangular plot are: (image will be updated soon)

a = 3x = 3 × 20 = 60m

b = 5x = 5 × 20 = 100m

c = 7x = 7 × 20 = 140m

And, semi perimeter = s = $\frac{parameter}{2}$ = $\frac{300}{2}$ = 150m

Now, the area of scalene triangle using Heron’s formula = $\sqrt{s(s-a)(s-b)(s-c)}$ sq. units

Putting the respective values in the above formula,

Area of scalene triangle = $\sqrt{150(150-60)(150-100)(150-140)}$sq. mts

=  $\sqrt{150(90)(50)(10)}$ sq. mts

=  1500$\sqrt 3$ sq. mts

Therefore, the required area of triangular plot = 1500$\sqrt 3$ sq. mts

Q.3. Find the area of a scalene triangle whose two adjacent sides are 8cm and 10cm and the angle between the sides is 30o .

Solution: Let the two adjacent sides of the scalene triangle be a = 8cm and b = 10cm, the angle included between these two sides,  ∠C =30o . (image will be updated soon)

So, the area of scalene triangle = $\frac{1}{2}$ × a × b × sinC sq. units

= $\frac{1}{2}$ × 8 × 10 × sin30o sq. cms

= $\frac{1}{2}$× 8 × 10 × $\frac{1}{2}$ sq. cms                       (∵ sin30o = $\frac{1}{2}$)

= 40 sq. cms