Table of Contents

Last modified on March 28th, 2023

A cone is a unique three-dimensional shape with a flat circular face at one end and a pointed tip at another end. The word ‘cone’ is derived from the Greek word ‘konos’, meaning a peak or a wedge.

A traffic signal cone, an ice-cream cone, or a birthday hat are some common examples of a cone.

- Its circular face is the
**base**. - Above the circular base is the curved surface that narrows to a pointed tip called the
**vertex**(or apex). - Has no edge.
- Since it has a curved surface, it is not a polyhedron.
- The shape is achieved by rotating a triangle. So it can be said to be a rotated triangle.

**Radius (r) –**It is the distance between the center of its circular base to any point on the circumference of the base.**Height (h) –**It is the perpendicular distance between its vertex to the base center. We consider the height of a right circular cone as its axis.**Slant Height (s) –**It is the distance from its vertex to the point on the outer edge of its circular base.

Based on the position of the vertex with respect to its base, a cone is of two types, as shown in the figure.

It has its vertex aligned right above the center of the base. The axis coincides with the height and makes a right angle at the base center.

It does not have its vertex aligned perpendicular to its base. It looks tilted.

Since, in practical life, a cone means a right circular cone, here, we will learn the formulas related to it.

The formula of the surface area (or total surface area) of a right circular cone is:

Surface Area (SA) = πr^{2} + πrs, here r = radius, s = slant height, π = 3.141, πr^{2} = base area, πrs = lateral (curved) surface area of a cone (LSA)

So, we can rewrite the formula as SA = πr^{2} + LSA

**Find the surface area of a right circular cone with a slant height of 6 cm and radius of 4 cm.**

Solution:

As we know,**Surface Area (SA) = πr ^{2} + πrs**, here r = 4 cm, s = 6 cm, π = 3.141

∴ SA = 3.141 × 4

= 125.7 cm

**Calculate the lateral surface area of a cone with a radius of 3 in and slant height of 8 in.**

Solution:

As we know,**Lateral Surface Area (LSA) = πrs, **here r = 3 in, s = 8 in, π = 3.141

∴ LSA = 3.141 × 3 × 8

= 75.3 cm^{2}

**Find the base area of a cone with a radius of 7.5 cm.**

Solution:

**As we know,**

Base area = πr^{2}, here π = 3.141, r = 7.5 cm

= 3.141 × (7.5)^{2}

= 176.7 cm^{2}

The formula is:

**Volume** ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = radius, h = height, π = 3.141

[Volume (V) = 1/3 πr^{2}h]

We can relate the formula of the volume of a cylinder which is πr^{2}h

Therefore, the volume of a cone is precisely one-third of that of a cylinder.

**Calculate the volume of a cone with a radius of 7 mm and height of 12 mm.**

Solution:

As we know,**Volume** ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = 7 mm, h = 12mm, π = 3.141

∴ ${V=\dfrac{1}{3}\times \pi \times 7^{2}\times 12}$

= 615.7 mm^{3}

The formula is:

Slant Height ${\left( s\right) =\sqrt{r^{2}+h^{2}}}$, here r = radius, h = height

**Find the slant height of a cone with a radius of 6 mm and height of 11 mm.**

Solution:

As we know,

Slant Height ${\left( s\right) =\sqrt{r^{2}+h^{2}}}$, here r = 6 mm, h = 11 mm

${\therefore s=\sqrt{6^{2}+11^{2}}}$

= 12.53 mm

Finding the **HEIGHT **of a cone when the **VOLUME and RADIUS **are known

**Find the height of a cone with a volume of 65.9 mm ^{3} and a radius of 3 mm..**

Solution:

**∵** **Volume** ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = radius, h = height, π = 3.141

So, by rearranging the equation, we get, **Height **${h=\dfrac{3V}{\pi r^{2}}}$, here V = 65.9 mm^{3}, r = 3 mm

${= \dfrac{3\times 65\cdot 9}{3.141\times 3^{2}}}$

= 6.99 mm

Finding the **RADIUS** when the **VOLUME** **and HEIGHT** are known

**Find the radius of a cone with a volume of 165.4** **cm ^{2} and a height of 7.8 mm.**

Solution:

**∵** **Volume** ${\left( V\right) =\dfrac{1}{3}\pi r^{2}h}$, here r = radius, h = height, π = 3.141

So, by rearranging the equation, we get, **Radius **${\left( r\right) =\sqrt{\dfrac{3V}{\pi h}}}$, here V = 165.4 cm^{2}, h = 7.8 cm

${= \sqrt{\dfrac{3\times 165\cdot 4}{3\cdot 141\times 7\cdot 8}}}$

= 4.5 cm

Last modified on March 28th, 2023