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.

Properties of a Scalene Triangle

• Has three unequal sides

• Has no equal angles

• Does not have a point in symmetry

• Does not have a side of the symmetry

• Angles inside it can be acute, obtuse, or right-angled.

• Whenever the angles lying inside are less than 90 degrees, that is, an acute angle.

In this case, the center of the circumscribing will tend to lie inside the triangle.

Types of Scalene Triangle

The scalene triangle also has types, those are given below:

• Acute-angled scalene triangle: when the circumcenter lies inside the triangle.

• Obtuse-angled scalene triangle: when the circumcenter lies outside the triangle.

• Right-angled scalene triangle: when the circumcenter is at the midpoint of the hypotenuse.

In this article, we will get to know about different types of ways through which we can measure the area of the scalene triangle. The area is the total amount of space it occupies. The area can be calculated by the base and altitude or by knowing the length of the three sides or by the length of any two sides and the angle between them.

First Method:-

The first method by which an area can be calculated is if we know its base and altitude.

The area of a scalene triangle is given as =1/2 × base × height (altitude) sq. units

=1/2 × b × h sq. units

Second Method:-

The second method by which area can be calculated is if the length of all three sides is given.

The area is calculated through Heron’s Formula i.e.= √s(s−a)(s−b)(s−c) sq.units.

Here, a, b and c; are the length of the sides of the given triangle and s is the semi-perimeter of the triangle i.e. (a+b+c)/2

Third Method:-

This method is used if we know the length of any two sides of a triangle and the angle between them.

Area of triangle= 1/2 × a × b × sinC sq units. 

Here, a and b are the length of the two sides and c is the given angle between them. These methods are very important in the study of triangles and mostly all the questions are based on this research.

Solved Examples:

1. Find the height of the scalene triangle whose area is 12 sq. cm 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 

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

⇒ \[\frac{1}{2}\] × 6 × h = 12 sq.cms

⇒ h = \[\frac{12\left ( 2 \right )}{6}\] = 4 cm

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

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

It is given that its perimeter = 300m

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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 = \[\sqrt{s\left ( s-a \right )\left ( s-b \right )\left ( s-c \right )}\]

Putting the respective values in the above formula,

Area of scalene triangle = \[\sqrt{150\left ( 150-60 \right )\left ( 150-100 \right )\left ( 150-140 \right )}\] sq. mts

      = \[\sqrt{150\left ( 90 \right )\left ( 50 \right )\left ( 10 \right )}\] sq. mts

    = 1500\[\sqrt{3}\] sq. mts

Therefore, the required area of triangular plot = 1500 \[\sqrt{3}\] sq. mts

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

Solution: Let the two adjacent sides of the scalene triangle be a = 8cm and b = 10cm, the angle included between these two sides,  ∠C =30°. 

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

= \[\frac{1}{2}\] × 8 × 10 × sin30° sq. cms

= \[\frac{1}{2}\] × 8 × 10 × \[\frac{1}{2}\] sq. cms

= 40 sq. cms