Scalene Triangle Formula

A scalene triangle is one of the three types of triangles which is classified based on their sides. The other triangles based on their sides are the isosceles triangle and equilateral triangle. A scalene triangle is a triangle that has all its sides of different lengths. It means all the sides of a scalene triangle are unequal and all the three angles are also of different measures. However, the sum of all the interior angles is always equal to 180 degrees. Thus, it meets the angle sum property condition of the triangle. 

Some of the important properties of the scalene triangle are as follows:

• It has no equal sides.
• It has no equal angles.
• It has no line of symmetry.
• It has no point symmetry.
• The angles inside this triangle can be acute, obtuse, or right angle.
• If all the angles of the triangle are less than 90 degrees(acute), then the center of the circumscribing circle will lie inside a triangle.
• In a scalene obtuse triangle, the circumcenter will lie outside the triangle.
• A scalene triangle can be an obtuse-angled, acute-angled, or right-angled triangle.

Let's look into the Scalene Triangle Formula in detail.

What Is Scalene Triangle Formula?

The scalene triangle formulas refer to formulas that help calculate the area and perimeter of the given scalene triangle. We will be learning about the following scalene triangle formulas as listed below:

• Area of a Scalene Triangle
• Perimeter of a Scalene Triangle

Scalene Triangle Formulas

As the triangle has 6 quantities namely 3 sides and 3 angles, the area of a triangle is calculated via various formulas depending upon the known quantities of the triangle.

Formula to Calculate the Area of a Scalene Triangle

The formula of the area of the scalene triangle is used to find the area occupied by the scalene triangle within its boundary. 

• Area of a triangle with base and height

When the base and height of the scalene triangle is known, then the area of a triangle is:

Area of a triangle =  1/2 × (Base(b) × Height(h))

where b and h are the base and height of the triangle respectively.

• Area of Triangle Using Heron's Formula

Heron's formula is applicable when all three sides of the triangle are known to us. Consider a triangle ABC with sides a, b, and c has shown in the image.

Heron’s formula is:

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

where a, b, c are the side length of the triangle and s is the semi-perimeter and equals (a+b+c)/2.

• Area of the scalene triangle with 2 sides and included angle (SAS)

We can find the scalene triangle's area when the length of its two sides and the included angle are given.

1. When two sides b and c and included angle A is known, the area of the triangle is:       

\(\text{Area} = \dfrac{1}{2} \ bc \times \sin A\) 

2. When sides a and c and included angle B is known, the area of the triangle is:

\(\text{Area} = \dfrac{1}{2} \ ac \times \sin B\)

3. When sides a and b and included angle C is known, the area of the triangle is:

\(\text{Area} = \dfrac{1}{2} \ ab \times \sin C\)

Formula to Calculate the Perimeter of a Scalene Triangle

The perimeter of a triangle is equal to the sum of the length of sides of a triangle. Consider the scalene triangle as shown below.

The perimeter is given as:

Perimeter  = a + b + c units

Let us see the applications of the scalene triangle formula in the following solved examples.

Examples Using Scalene Triangle Formulas

Example 1: Robert was given the two sides of the triangle and the angle between them as 14 units, 28 units, and 30 degrees respectively. Find the area of this triangle using the scalene triangle formula.

Solution: 

The first side of the triangle, a = 14

The second side of the triangle, b = 28

The angle between the sides, C = 30 degrees

Using the scalene triangle formula: \(\text{Area (A)} = \dfrac{1}{2} \ ab \times \sin C\)

A = \(\dfrac{1}{2} \times 28 \times 14 \times \sin 30\)

A = \(\dfrac{1}{2} \times 28 \times 14 \times \dfrac 12\)

A = 7 × 14

A = 98

Answer: So, The area of the triangle is 98 squared units.

Example 2: If the sides of a triangle are 10cm, 11cm, and 15cm. Find its perimeter.

Solution:

Given:

Length of sides of the triangle are 10cm, 11cm and 15cm. 

Using Scalene Triangle Formula, Perimeter = 10 + 11 + 15 = 36 cms

Answer: Thus, the perimeter is 36 cm.

Example 3: The length of the sides of a triangle ABC are 7 units, 9 units, and 12 units. Calculate its area.

Solution:

Given:

Sides of the triangle

Let a = 7 units, b = 9 units, c = 12 units .

Using Heron's Formula: Area of triangle = √(s(s-a)(s-b)(s-c) square units.

We will first find s, s = (a+b+c)/2 ⇒ s =(7+9+12)/2 ⇒ s = 14.

Now, put the values.

Thus, A = √(14(14-7)(14-9)(14-12)) 

= √(14(7)(5)(2))

= √(980)

= 31.304 units². 

Answer: The area of the scalene triangle is 31.304 square units"