What is the Diameter of a Circle?
A diameter is a line segment that passes through the center and its endpoints are on the circumference of a circle. In other words, the longest chord of a circle is called the diameter. The distance from the center to any point on the circle is called the radius of the circle with two radii that form the diameter. This shows us that the diameter of a circle is twice the length of its radius.
Formulas for the Diameter of a Circle
- When the radius is given:
d = 2r
- When the circumference C of the circle is given:
C = \(\pi\)d
d = \(\frac{C}{\pi}\)
- When the area of the circle A is given:
A = \(\pi r^2\)
A = \(\pi(\frac{d}{2})^2\)
A = \(\pi\frac{d^2}{4}\)
d\(^2\) = \(\frac{4A}{\pi}\)
d = 2\(\sqrt{\frac{A}{\pi}}\)
Thus, there are three formulas to find the diameter of a circle:
d = 2r
d = \(\frac{C}{\pi}\)
d = 2\(\sqrt{\frac{A}{\pi}}\)
Solved Examples
Example 1: The radius of a circle is 16 centimeters. Find the area and diameter of the circle.
Solution:
Radius of circle, r = 16 cm
d = 2r [Relation between diameter and circle]
d = 2 x 16 [Substitute 16 for r]
d = 32 cm
So, the diameter of a circle is 32 centimeters.
A = \(\pi r^2\) [Write the formula for the area of a circle]
A = 3.14 x 16\(^2\) [Substitute 3.14 for pi and 16 for r]
A = 3.14 x 256 [Find the square of 16]
A = 803.84 cm\(^2\) [Multiply]
Hence, the area of the circle is 803.84 square centimeters and the diameter of the circle is 32 centimeters.
Example 2: The area of a circle is 12.56 square centimeters. Find the diameter of the circle. (Take the value of \(\pi\) to be 3.14)
Solution:
Use the formula for the area of a circle to find the diameter.
A = \(\frac{\pi d^2}{4}\) [Formula of area]
12.56 = \(\frac{3.14~\times~d^2}{4}\) [Substitute 12.56 for A and 3.14 for \(\pi\)]
50.24 = 3.14 x d\(^2~\) [Multiply each side by 4]
16 = d\(^2\) [Divide each side by 3.14]
\(\sqrt{16}\) = d [Apply square root to each side]
4 = d [Take positive root of 16]
Hence, the diameter of the circle is 4 centimeters.
Example 3: Roland made a circle with the help of a rope. The perimeter of the circle is 314 centimeters. Help him to find the diameter of the circle.
Solution:
Use the formula for the circumference of the circle to find the diameter.
C = πd [Write the formula for circumference]
314 = 3.14 x d [Substitute 314 for C and 3.14 for π]
10 = d [Divide each side by 3.14]
So, the diameter of the circle of rope that Roland created is 10 centimeters.
Yes, the diameter of a circle is twice the radius. Hence, dividing the diameter by two gives us the radius of a circle.
Yes, a chord is a line segment formed by joining any two points on the circle. Additionally, the diameter is also the longest chord of a circle.
Pi is an irrational number. For ease of mathematical calculation, we use the value of pi as 3.14 as a decimal, or as a fraction as 22/7.
The ratio of circumference and diameter of a circle gives us the value of pi.