Area of a Scalene Triangle

A scalene triangle is an interesting shape in geometry. With an equilateral triangle, you get three equal sides. With an isosceles triangle, you get two equal sides. But in a scalene triangle, none of the sides are equal. It is because of this unique property that you need to take a different approach to calculate the area of the scalene triangle. From trigonometric formula to Heron’s formula, there are several approaches you can take, depending on the information you have.

In this article, we will be taking a look at the scalene triangle properties, scalene triangle examples, and solved problems for a better understanding.

1.0Scalene Triangle Properties

A scalene triangle is defined as a triangle in which no two sides or angles are equal. Because of its irregular nature, it displays certain properties that set it apart from other triangles.

2.0Key Scalene Triangle Properties

These properties make the scalene triangle versatile in geometry, architecture, and even real-world applications like design and construction.

3.0Formula for Scalene Triangle Area

Since no sides are equal, a scalene triangle does not have a fixed shortcut for areas like equilateral or right triangles. Instead, different formulas for the scalene triangle area can be applied depending on the given data.

Using Heron’s Formula

The most common formula to calculate the area of a scalene triangle is Heron’s formula.

$Area=\sqrt{s(s-a)(s-b)(s-c)}$

where:

• a, b, c lengths of the three sides
• $s=\frac{a+b+c}{2}$

Using Base and Height

If the base and the corresponding height are known:

Area = ½ x base x height

Using Trigonometry

If two sides and the included angle are given:

Area = ½ x a x b x sin C

These three methods ensure that the area of the scalene triangle can always be determined with the available data.

4.0Difference Between Scalene, Isosceles, and Equilateral Triangles

While the scalene triangle has no equal sides, isosceles triangle and equilateral triangles have different symmetries. Understanding this distinction helps in solving problems correctly.

Feature	Scalene Triangle	Isosceles Triangle	Equilateral Triangle
Sides	All unequal	Two equal sides	All sides equal
Angles	All unequal	Two equal angles	All angles 60°
Symmetry	No line of symmetry	One line of symmetry	Three lines of symmetry
Example use	Random plots of land	Roof trusses	Geometric tiling
Diagram

This table highlights why the formula for the scalene triangle area needs special consideration, unlike the equilateral formula, which is fixed.

5.0Solved Problems on a Scalene Triangle

To strengthen understanding, let’s look at some scalene triangle examples and apply the formulas.

Example 1: Find the area of a scalene triangle with sides a = 7 cm, b = 9 cm, and c = 10 cm.

Solution:

s = (a + b + c)/2 = (7 + 9 + 10)/2 = 13

$Area=\sqrt{s(s-a)(s-b)(s-c)}$

$=\sqrt{13(13-7)(13-9)(13-10)}$

$=\sqrt{13\times 6\times 4\times 3}$

= 936

Area = 30.6 cm²

So, the area of the scalene triangle = 30.6 cm².

Example 2: The base of a scalene triangle is 12 cm, and the height is 8 cm. Find the area.

Solution:

Area = ½ × 12 × 8 = 48 cm²

Example 3: In a scalene triangle, two sides are 15 cm and 20 cm with an included angle of 60°. Find the area.

Solution:

Area = ½ × 15 × 20 × sin⁡ 60∘

Area = 150 × 32

Area ≈ 129.9cm²

Example 4:  In PQR, ∠P = 30°, ∠Q = 60°, find the value of ∠R. Also, which type of a triangle is it called?

Solution: In PQR, by angle sum property of a triangle,

∠P + ∠Q + ∠R = 180°

30° + 60° + ∠R = 180°

∠R = 180° − 30° − 60° = 90°

It is a right angled scalene triangle.