A wire is bend to form an equilateral triangle . if the triangle is 4 root 3 cm², what is the area of a circle formed ( in cm²)by the same wire?
Answers
Given,
A wire is bent to form an equilateral triangle.
The area of the equilateral triangle formed = 4√3 cm^2
To find,
The area of a circle formed (in cm²) by the same wire.
Solution,
We can simply solve this mathematical problem using the following process:
Let us assume that the length of each equal side of the equilateral triangle is x cm and the radius of the circle formed is y cm, respectively.
As per mensuration;
Area of an equilateral triangle= √3/4 (each equal side)^2
The perimeter of an equilateral triangle= 3×(each equal side)
Area of a circle = π(radius)^2
The perimeter of a circle = 2π(radius){Statement-1}
Now, according to the question;
The same wire is used to make an equilateral triangle and also a circle
=> the perimeter of the equilateral triangle = the perimeter of the circle
{Equation-1}
Now, according to the statement-1;
Area of the equilateral triangle = 4√3 cm^2
=> √3/4 (each equal side)^2 = 4√3 cm^2
=> (each equal side)^2 = 4cm^2 × 4
=> each side of the equilateral triangle = 4 cm
Now, according to statement-1;
The perimeter of the equilateral triangle formed
= 3×(each equal side)
= 3×(4 cm)
= 12 cm
=> the perimeter of the circle formed by the same wire = 12 cm
{according to equation-1}
=> 2π(radius) = 12 cm
=> radius of the circle formed = (6/π) cm
Now, the area of the circle formed
= π(radius)^2
{according to statement-1}
= π(6/π)^2 cm^2
= 36/π cm^2
= (36×7)/22 cm^2
= (252/22) cm^2
= 11.45 cm^2
Hence, the area of the circle formed is equal to 11.45 cm^2.