Triangle Formulas

A triangle formula is used to calculate the perimeter and area of a three-sided polygon which is also known as trigon. Let us learn the triangle formulas in detail.

What Are Triangle Formulas?

The two important triangle formulas are the area of a triangle formula and the perimeter of a triangle formula. Further, the triangle formulas are applicable to different types of triangles. 

Area of Triangle Formula

The area of the triangle is equal to half of the product of the base and height of the triangle, it is given as,

Area of triangle, A = [(½) base × height] square units.

There are two important triangle formulas related to the area of a triangle, i.e., the Herons formula and the Pythagoras theorem.

The area of a triangle using Heron's Formula is given as,

Area of triangle ABC = \(\sqrt{s(s-a)(s-b)(s-c)}\),

where s is the semi-perimeter = Perimeter/2 = (a + b + c)/2

Pythagoras theorem is used to find the side of the right-angled triangle which is mathematically expressed as, h² = p² + b². Here, 'h' is the hypotenuse (longest side of a right triangle), p is the perpendicular side, and b is the Base.

In the case of an equilateral triangle, the equilateral triangle formula for area is, A = (√3/4)a² square units, where a is the side of the triangle.

In the case of an isosceles triangle, the isosceles triangle formula for area is, A = 1/2 × Base × Height square units, where, height = \(\sqrt{\text{a}^2 - \dfrac{b^2}{4}}\). (Here 'a' is the equal side, and 'b' is the base of the isosceles triangle.)

Perimeter of Triangle Formula

The perimeter of the triangle is equal to the sum of all the sides of the triangle, it is given as

General perimeter of a triangle formula, P = (a + b + c) units.

The equilateral triangle formula for perimeter is (a +a + a) = 3 a units. Here 'a' is a side of an equilateral triangle. (Note: in equilateral triangle all three sides are equal)

The isosceles triangle formula for perimeter is (s + s + b) = (2s + b) units, here s is a measurement of two equal sides, and b is the base of an isosceles triangle.

Examples on Triangle Formulas

Example 1: Find the area of a triangle whose base is 40 units and its height is 25 units.

Solution:

To find: The area of a triangle

The base of a triangle = 40 units (given)

Height of triangle = 25 units (given)

Using triangle formulas,

Area of triangle, A = [(½) base × height] square units

= [(½) 40 × 25] square units                                

= 500 square units

Answer:  The area of the triangle is 500 square units.

Example 2: A triangle has sides a = 5 units, b = 10 units, and c = 6 units. What is the perimeter of this triangle?

Solution:

To find: The perimeter of a triangle                 

Three sides of a triangle  = 5, 10, 6. (Given)

Using triangle formulas,

Perimeter of a triangle, p = (a + b + c) units.

= ( 5+10+6) units.

=  21 units

Answer: The perimeter of a triangle is 21 units.

Example 3: If the lengths of the sides of a triangle are 4 in, 7 in, and 9 in, calculate its area using Heron's formula.

Solution:

To find: Area of the triangle 
Given that, Side a = 4 in, Side b = 7 in, Side c = 9 in

Using  triangle formula (Heron's Formula),
A = √(s(s-a)(s-b)(s-c))

As, s = (a+b+c)/2
s = (4+7+9)/2
s = 20/2 = 10 inches

Substituting the values in the Heron's formula,
A = √(10(10-4)(10-7)(10-9))
⇒ A =√(10(6)(3)(1))
⇒ A = √(180) = 13.416 in²

∴ The area of the triangle is 13.416 in².