Triangles are a fundamental component of geometry and are often examined in US competitive tests such as the SAT and ACT. Triangles are categorized into three types based on their sides: 1) Equilateral triangle – all sides are equal in length, 2) Isosceles triangle – two sides are equal in length, and 3) Scalene triangle – all sides are of different lengths. It is important to understand these categorizations to efficiently solve geometry problems. In this article, we will thoroughly study Scalene Triangles, including their features, types, and comparison with isosceles and equilateral triangles, along with a couple of solved examples to make your problem-solving better. Triangular properties are easy to master, and doing so can make you stand out in standardized tests!

Scalene Triangle

A scalene triangle is a triangle where the length of each side and the three angles has a different measure. An example of a scalene triangle is shown above. We have used three different symbols to mark the three sides as they all are of different lengths.

Scalene Triangle Formula

In this article we will discuss two formulas for a scalene triangle : area and perimeter of a scalene triangle.

Area of a Scalene Triangle

The area of a scalene triangle is given by the formula

Area =\(\frac{1}{2}\) x base x height (square units).

If the base and height of a triangle are not given and only the measurements of the sides are given, then we can calculate the area of a scalene triangle by applying the Heron’s Formula

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

Where s is the semi perimeter of the triangle, which is given by the formula,

\(s=\frac{a+b+c}{2}\)

Where a, b and c are the sides of the triangle.

Perimeter of a Scalene Triangle

The perimeter of a scalene triangle is equal to the sum of all the sides of the triangle, given by the formula,

Perimeter (P) = a+b+c (units)

Where a, b and c are the sides of the triangle.

Types of Scalene Triangles

Scalene Triangles can be divided into three types

1. Acute angled scalene triangle.
2. Obtuse angled scalene triangle.
3. Right angled scalene triangle.

Acute Angled Scalene Triangle

In an acute angled scalene triangle all the interior angles of that triangle are acute, with each side of different measurements.

Despite having different angle measurements, the sum of the interior angles is always equal to 180 degrees.

Referring to the diagram above, triangle ABC with angle measurements as : 80°, 65° and 35°

Sum of all the angles = 80°+65°+35° = 180°.

Obtuse Angled Scalene Triangle

In an obtuse scalene triangle, one of the angles of the triangle is greater than 90°, with varying lengths of all the three sides.

It generally has two acute angles and one obtuse angle as interior angles and the sum of the interior angles will always be equal to 180°.

Right Angled Scalene Triangle

A right angled scalene triangle is the one in which one of the angles measures 90° and the other two angles are acute angles.

They follow the relationship : hypotenuse side² = perpendicular² + base² (pythagoras theorem) and the sum of the interior angles is equal to 180°.

Difference between Scalene, Isosceles and Equilateral Triangle

Characteristics of Scalene Triangle

The following are the characteristics of a Scalene Triangle

1. A scalene triangle has no equal sides.
2. The measurement of all the angles in a scalene triangle also differ.
3. It has no line or point of symmetry.
4. Despite all angles being different the sum of the interior angles is always equal to 180°.

Scalene Triangle Solved Examples

Example 1. Consider a Scalene triangle, where the length of the sides are given by, 5cms, 7cms and 12 cms. Find the area of the triangle.

Solution 1.

Given data,

Side 1 (a) = 5cm.

Side 2 (b) = 7cm.

Side 3 (c) = 6cm.

The area of a scalene triangle is given by,

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

Where s is the semi perimeter of the triangle, which is given by the formula,

\(s=\frac{a+b+c}{2}\)

Where a, b and c are the sides of the triangle.

Semi perimeter of the triangle is given by (s) = \(\frac{5+7+6}{2}\)

Semi perimeter (S) = 9cm

The area of a scalene triangle is given by,

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

\(A=\sqrt{9(9-5)(9-7)(9-6)}\)

\(A=\sqrt{9(4)(2)(3)}\)

\(A=\sqrt{9\times 24}\)

A = 14.69 cm².

Area of the given triangle is 14.69 cm².

Example 2. In a scalene triangle its semi perimeter is 24 cm, the length of the two sides are 8 cm and 6 cm, what will be the length of the third side?

Solution 2.

Given data,

In a scalene triangle the formula for semi perimeter is given by,

\(s=\frac{a+b+c}{2}\)

Where a, b and c are the three sides of a triangle.

Side 1 (a) = 8cm,

Side 2 (b) = 6 cm,

Let the third side be “c”

\(24=\frac{8+6+c}{2}\)

\(24=\frac{14+c}{2}\)

48 = 14 + c

c = 34 cm.

The length of the third side is 34 cm.

Example 3. Find the base of a scalene triangle whose height is 6 cm and area is 12 cm².

Solution 3.

Given data,

Area of a scalene triangle : 12 cm².

Height : 6 cm

Area of a triangle : \(\frac{1}{2}\times base\times height\)

\(12=\frac{1}{2}\times base\times 6\)

\(\frac{12\times 2}{6}\)

Base length = 4 cm.

The length of the base for the given triangle is 4cm.

We hope this article gave you useful insights and improved your knowledge of the subject, preparing you well for your exams. Remain connected with the Testbook app to learn more about mathematics and other necessary subjects. The app provides a range of practice tests, live classes, and expert-recommended study materials to prepare you well and increase your confidence. Don't forget to check out the test series available to gauge your knowledge and monitor your progress so that you are adequately prepared for any future exams.