How to find the area of a triangle with one side given?
A triangle is a closed figure composed of three sides. An equilateral triangle is a geometrical 2D figure which has all three sides equal. All the edges subtend an angle of 60° at the corners. Therefore, the sum of all the angles of the triangle is 180°
Properties of an equilateral triangle
- All three sides of an equilateral triangle are equal.
- All the three angles are equal i.e 60° each.
- A regular polygon has three equal sides.
How to find the area of a triangle with one side given?
Solution:
Let us assume a to be the side of the triangle.
If we divide the bottom side in two parts, we get,
By Pythagoras theorem, the altitude can be obtained by,
Perpendicular2 + Base2 = Hypotenuse2
Perpendicular2 + (a/2)2 = a2
Perpendicular2 = a2 – (a/2)2 Â
Perpendicular2 = 3a2/4
Taking square root,
Perpendicular = √3/2a
Now,
Area of triangle = 1/2 x Perpendicular x Base
= 1/2 x √3/2a x a
On solving, we obtain,
Area = a2(√3/4)
Sample Questions
Question 1. Find the area of the equilateral triangular signboard of side 16 cm and also find the cost of painting the signboard at ₹2 per cm2?(Assume √3 = 1.732)
Solution:
Here we have to find the cost of painting an equilateral triangular signboard.
Given:
Side of equilateral triangular signboard = 16 cm
First finding the area of the equilateral triangular sign board
As we know that
Area of equilateral triangle = √3/4 × a2
Here a = 16 cm
Area of equilateral triangle = √3/4 × 162
Area of equilateral triangle = √3/4 × 16 × 16
Area of equilateral triangle = 64√3
Now,
Finding the cost of painting
Given
Cost of painting = ₹2 per cm2
Cost of painting = Area of equilateral triangle × 2
Cost of painting = 64√3 × 2
Cost of painting = 128√3
Using √3 = 1.732 (Given)
Cost of painting = ₹221.696
Therefore,
Cost of painting the equilateral triangular signboard is ₹221.696.
Question 2. Find the area of an equilateral triangular park of side 32 m?
Solution:
Here we have to find the area of the equilateral triangular park.
Here we are given that,
Side of the triangular park = 32 m
As we know that
Area of equilateral triangle = √3/4 × a2
Area of equilateral triangular park = √3/4 × a2
Here ‘a’ is the side of the equilateral triangular park is 32 m
Area of equilateral triangular park = √3/4 × 322
Area of equilateral triangular park = √3/4 × 32 × 32
Area of equilateral triangular park = √3 × 8 × 32Â
Area of equilateral triangular park = 256√3 m2
Therefore,
The area of the equilateral triangular park is 256√3 m2
Question 3. Find the cost of carpeting equilateral triangular yoga ground with the altitude 10 m and base 20 m at the rate of ₹20 pr m2?
Solution:
Here we have to find the cost of carpeting equilateral triangular yoga ground
Given:
Altitude = 10 m
Base = 20 m
As we know that
Area of equilateral triangle = 1/2 × base × height
Area of equilateral triangle = 1/2 × 20 × 10
Area of equilateral triangle = 100 m2
Now finding the cost of carpeting
Cost of carpeting yoga ground at ₹20 pr m2
Cost of carpeting yoga ground = Area of yoga ground × ₹20
Cost of carpeting yoga ground = 100 × ₹20
Cost of carpeting yoga ground = ₹2000
Therefore,
The cost of carpeting the yoga ground is ₹2000.
Question 4. Find the area of an equilateral triangle having a perimeter of 18 m?
Solution:
Here we have to find the area of the equilateral triangle
Given:
Perimeter of equilateral triangle = 18 m
The Perimeter of equilateral triangle = Side + Side + Side
As all the sides of the equilateral triangle are equal
Perimeter of equilateral triangle = 3 × Side
18 = 3 × Side
Side = 18/3
Side = 6 m
Side of the equilateral triangle is 6 m
Further,
Finding the area of the equilateral triangle
Area of equilateral triangle = √3/4 × a2
a = Â side of the equilateral triangle
Area of equilateral triangle = √3/4 × 62
Area of equilateral triangle = √3/4 × 6 × 6
Area of equilateral triangle = 9√3 m2
Question 5. If the side of an equilateral triangle is 10 m and its area is 75 m2 Â then find the height of the equilateral triangle?
Solution:
Here we have to find the height of the equilateral triangle
Given:
Side of equilateral triangle = Base of equilateral triangle = 10 m
Area of equilateral triangle = 75 m2
As we know that
Area of equilateral triangle = 1/2 × base × height
75 = 1/2 × 10 × height
75 = 5 × height
Height = 75/5
Height = 15 m
Therefore,
The height of equilateral triangle is 15 m.
Last Updated :
03 Jan, 2024
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