Triangle Formulas

Triangle formulas are the basic formulas that are used to find the unknown dimensions and parameters of a triangle. The main formulas of a triangle are related to its area and perimeter. Let us learn more about all the triangle formulas in detail.

What are Triangle Formulas?

The two important triangle formulas are the formulas for the area of a triangle and the perimeter of a triangle. There are different triangle formulas that are applicable to different types of triangles. Let us learn about all the basic formulas related to a triangle.

All Triangle Formulas

All triangle formulas mainly include the formulas related to the area and perimeter of a triangle. Let us learn about them in the following sections.

Area of Triangle Formulas

• The area of a triangle is equal to half the product of the base and height of the triangle and it is expressed as, Area of triangle = ½ × base × height
• While the above formula is used for any triangle, sometimes, in the case of a scalene triangle, we use the Heron's formula to find its area.
• The area of a triangle using Heron's Formula is given as, Area of scalene triangle = \(\sqrt{s(s-a)(s-b)(s-c)}\), where 's' is the semi-perimeter of the triangle. So, semi-perimeter = Perimeter/2 = (a + b + c)/2
• In case of an equilateral triangle, the area can be calculated using the formula, Area of equilateral triangle = (√3/4)a², where 'a' is the side of the triangle.
• In the case of an isosceles triangle, the isosceles triangle formula for area is, Area of isosceles triangle = 1/2 × Base × Height, 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.)
• The Pythagoras theorem can be used to find the unknown side of a right-angled triangle if the other two sides are known. This theorem 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. Once all the sides are known, the area of the triangle can be calculated using the formulas given above.

Perimeter of Triangle Formulas

• The perimeter of a triangle is equal to the sum of all the sides of the triangle, and the formula is expressed as, Perimeter of a triangle formula, P = (a + b + c), where 'a', 'b', and 'c' are the three sides of the triangle.
• The equilateral triangle formula for perimeter is, Perimeter of equilateral triangle = (a +a + a) = 3a. Here 'a' is a side of an equilateral triangle. (Note: In an equilateral triangle, all three sides are equal)
• The isosceles triangle formula for perimeter is Perimeter of isosceles triangle = (s + s + b) = (2s + b). Here 's' is the length of the two equal sides, and 'b' is the base of an isosceles triangle.

Observe the figure given below which shows the basic triangle formulas.

Examples on Triangle Formulas

Example 1: Find the area of a triangle whose base is 40 units and whose 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

= ½ × 40 × 25

= 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)

= 5 + 10 + 6

= 21 units

Answer: The perimeter of the given 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

Let us use the triangle formula (Heron's Formula)
Area of triangle = √(s(s-a)(s-b)(s-c)), where 's' is the semi-perimeter. First, let us find the semi-perimeter of the triangle.

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.