# Diameter of a Cone

The diameter of a cone is the diameter of its circular base. It is the distance between any 2 points on the circumference of the base passing through the center. It is denoted by d.

Like radius, it is expressed in mm, cm, m, in, ft, and yd.

## Formula

The formula to calculate the diameter of a cone can be obtained from the formula used to calculate the volume of a cone.

As we know,

Volume ${\left( V\right) =\dfrac{1}{12}\pi d^{2}h}$, here d = diameter, h = height, π = 3.141

Therefore, by rearranging the equation, we can write the formula of diameter as:

${d=2\sqrt{\dfrac{3V}{\pi h}}}$

## Solved Examples

Find the diameter of a cone with a volume of 667 cm3 and a height of 13 cm.

Solution:

As we know,
${d=2\sqrt{\dfrac{3V}{\pi h}}}$, here V = 667 cm3, h = 13 cm, π = 3.141
${\therefore d=2\times \sqrt{\dfrac{3\times 667}{3\cdot 141\times 13}}}$
= 14 cm

Finding the DIAMETER of a cone with SLANT HEIGHT and HEIGHT

Calculate the diameter of a right circular cone with a slant height of 16 mm and a height of 12 mm.

Solution:

Here we will apply the Pythagorean Theorem considering the slant height as hypotenuse and height as the perpendicular leg and find the diameter as the base of a right triangle.
${d=2\sqrt{s^{2}-h^{2}}}$, here s = 16 mm, h = 12 mm, π = 3.141
${\therefore d=2\sqrt{16^{2}-12^{2}}}$
= 21.16 mm

Finding the DIAMETER of a cone with SLANT HEIGHT and LATERAL SURFACE AREA

Calculate the diameter of a cone with a slant height of 6.5 cm and a lateral surface area of 225 cm2.

Solution:

As we know,
r = d/2, we will replace this in the equation of Lateral Surface Area (LSA):
${\because LSA=\pi \dfrac{d}{2}s}$
Rearranging the equation to find d:
${d=2\dfrac{LSA}{\pi s}}$, here LSA = 225cm2, s = 6.5cm, π = 3.141
${\therefore d=\dfrac{2\times 225}{3.141\times 6\cdot 5}}$
= 22.04 cm