How to find the perimeter of an equilateral triangle?
Last Updated : 27 Aug, 2024
The perimeter of an equilateral triangle is equal to 3 x a, where a is the length of any side
An Equilateral triangle is a triangle in which all three sides are equal and the angles are also equal. The value of each angle of an equilateral triangle is 60 degrees therefore, it is also known as an equiangular triangle. The equilateral triangle is considered a regular polygon or a regular triangle as the angles are equal and sides are also equal.
For instance, in the triangle, ABC are equal i.e.
AB = BC = CA = a unit.
Also, ∠A, ∠B, and, ∠C = 60°
Properties of Equilateral Triangle
- All three sides are equal.
- All three angles are equal to 60°
Perimeter of Equilateral Triangle
Semi Perimeter of an Equilateral Triangle
Semi Perimeter of an Equilateral Triangle = \frac{Perimeter\ of\ an\ Equilateral\ Triangle}{2}
Let us assume a to be the side of an equilateral triangle.
In other words, we have,
Semi Perimeter of an Equilateral Triangle = \frac{3a}{2}
Perimeter of an Equilateral Triangle when the area is given
Let us assume a to be the side of an equilateral triangle.
Perimeter of an equilateral triangle can be computed using its area, which is given by,
Area=\frac{\sqrt3}{4}a^2
Now,
We know,
Perimeter of an equilateral triangle = Side + Side + Side
Perimeter of an equilateral triangle, P is given by = 3 × a
Therefore, the values a can be replaced by P/3.
Area=\frac{\sqrt3}{4}(\frac{P}{3})^2
Perimeter of an Equilateral Triangle when Altitude is given
The perimeter of an equilateral can be calculated when the altitude (height) of the triangle is given.
We have,
Height of an Equilateral Triangle = \frac{\sqrt3}{2}a
Upon substituting the values of the perimeter of the equilateral triangle, we have,
Perimeter of an equilateral triangle = Side + Side + Side
Perimeter of an equilateral triangle = 3 × a
Therefore,
Height (or Altitude ) = \frac{\sqrt3}{2}(\frac{P}{3})
Sample Questions
Question 1. Calculate the perimeter of an equilateral triangle if the side of the triangle is 30√3 cm.
Solution:
Here we have to find the perimeter of an equilateral triangle
We are given that the side of the equilateral triangle is 30√3 cm
As we know that
Formula for perimeter of an equilateral triangle
Perimeter of an equilateral triangle = Side + Side + Side
Perimeter of an equilateral triangle = 3 × a
where a is side of an equilateral triangle
Perimeter of an equilateral triangle = 3 × 30√3
Perimeter of an equilateral triangle = 90√3 cm
Therefore,
Perimeter of an equilateral triangle is 90√3 cm.
Question 2. If the side of an equilateral triangle is 90 m, then find the perimeter and semi-perimeter of the triangle?
Solution:
Here we have to find the perimeter and semi-perimeter of an equilateral triangle,
First finding the perimeter of an equilateral triangle
We are given that the side of the equilateral triangle is 90 m
Formula for the perimeter of an equilateral triangle
Perimeter of an equilateral triangle = Side + Side + Side
The perimeter of an equilateral triangle = 3 × a
Substituting the value of a in the formula
Perimeter of an equilateral triangle = 3 × 90
Perimeter of an equilateral triangle = 270 m
Further finding the semi-perimeter
Formula for semi-perimeter of an equilateral triangle = \frac{3a}{2}
Where a is the side of an equilateral triangle
Substituting value of a in the formula
Semi-perimeter of an equilateral triangle = \frac{3\times90}{2}
Semi-perimeter of an equilateral triangle = 135 m
Therefore,
Perimeter of an equilateral triangle is 270 m and semi-perimeter of an equilateral triangle is 135 m.
Question 3. Consider that the area of an equilateral triangle is 100√3 cm2, Then calculate its perimeter?
Solution:
Here we have to find the perimeter of equilateral triangle using its area
Formula for equilateral triangle area = \frac{\sqrt3}{4}a^2
Area of equilateral triangle = 100√3
100√3= \frac{\sqrt3}{4}a^2
a2 = 100\sqrt3\times\frac{4}{\sqrt3}
a = √400
a = 20
Therefore,
Side of the equilateral triangle is 20 cm
Now further finding perimeter of equilateral triangle
Perimeter of equilateral triangle = side + side + side = 3a
Perimeter of equilateral triangle = 3 × 20
Perimeter of equilateral triangle = 60 cm
Question 4. Find the perimeter of an Equilateral triangle if the height of the triangle is 35√3 m.
Solution:
Here we have to find the Perimeter of the equilateral triangle with the height 35√3 m
Formula for calculating perimeter using height is given below
Height = \frac{\sqrt3}{2}a
Here a is the side of the equilateral triangle
35√3 = \frac{\sqrt3}{2}a
a = 35\sqrt3\times\frac{2}{\sqrt3}
a = 70 m
Now further finding perimeter of the equilateral triangle
Perimeter of equilateral triangle = side + side + side = 3a
Perimeter of equilateral triangle = 3 × 70
Perimeter of equilateral triangle = 210 m
Question 5. If the side of an equilateral triangle is 23 cm, then find the perimeter and height of the equilateral triangle?
Solution:
Here we have to find the perimeter of an equilateral triangle
We are given that the side of the equilateral triangle is 23 cm
As we know that
Formula for the perimeter of an equilateral triangle
Perimeter of an equilateral triangle = Side + Side + Side
Perimeter of an equilateral triangle = 3 × a
where a is side of an equilateral triangle
Perimeter of an equilateral triangle = 3 × 23
Perimeter of an equilateral triangle = 69 cm
Further finding the height of the triangle
Height = \frac{\sqrt3}{2}a
Here a is the side of the equilateral triangle
Substituting the value of a in the formula
Height = \frac{\sqrt3}{2}\times23
Height = 11.5√3
Therefore,
Perimeter of an equilateral triangle is 69 cm and the height of an equilateral triangle is 11.5√3 cm.
Question 5. If the side of an equilateral triangle is 3 cm, then find the semi perimeter of the equilateral triangle?
Solution:
Side of equilateral traingle is= 3cm.
Perimeter of Traingle= 3 x Side of Equilateral Triangle
= 3 x 3cm
= 9cm
Semi Perimeter of Equilateral Triangle= Perimeter/2
= 9cm/2
= 4.5cm
Question 6. If the side of an equilateral triangle is 4 cm, then find the height of the equilateral triangle?
Solution:
Height of Equilateral triangle is = √3 x side/2
= √3 x 4/2
= 2√3
Question 7. If the side of an equilateral triangle is 8 cm, then find the area of the equilateral triangle?
Solution:
Area of Equilateral triangle is = √3 x side2/4
= √3 x 64/4
= 16√3
Question 8. If the perimeter of an equilateral triangle is 18 cm, then find the area of the equilateral triangle?
Solution:
Side of Equilateral triangle is = perimeter/3
= 18cm/3
= 6cm
Area of Equilateral triangle is = √3 x side2/4
= √3 x 36/4
= 9√3
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