## Introduction

- A cone is a
**three-dimensional****(solid)**figure having a circular base and a pointed edge at the top called the vertex. - A right circular cone is similar to a regular pyramid except that its base is a circle.

- The altitude of a right circular cone $\left(h\right)$ is the measure of the perpendicular line segment joining the vertex and the center of the base.
- The slant height $\left(l\right)$ of a right circular cone is the measure of the line segment joining the vertex and any point on the circumference of the base.
- The mathematical relationship between the altitude, the radius of the base, and slant height is given by: $l=\sqrt{{r}^{2}+{h}^{2}}$.
- The area of the base is $B={\mathrm{\pi r}}^{2}$.

## Volume of a Cone

- The volume (capacity) of a cone is the amount of space occupied by it in a three-dimensional plane.
- The volume of a cone is measured in
**cubic units.** - If we know the radius of the base and altitude of a right circular cone, its volume can be calculated using the formula mentioned below.

$Volum{e}_{cone}=\frac{1}{3}{\mathrm{\pi r}}^{2}\mathrm{h}$ where $\mathrm{\pi}=\frac{22}{7}\approx 3.14$, $r=$the radius of the base, and $h=$altitude

## Surface Area of a Cone

- There are two types of surface areas associated with a right circular cone.

1. **Lateral Area:** It is the region occupied by the curved surface of the right circular cone.

$L{A}_{cone}=\mathrm{\pi rl}$, where $r=$the radius of the base, and $l=$slant height

2. **Total Area:** It is the total area occupied by a right circular cone, including the base area.

$T{A}_{cone}=L{A}_{cone}+B$

$=\mathrm{\pi rl}+{\mathrm{\pi r}}^{2}$

$=\mathrm{\pi r}\left(\mathrm{l}+\mathrm{r}\right)$, where $B=$the area of the base

## Solved Examples

Question 1: Find the volume of a right circular cone if the radius of its base is 4 cm and its altitude is 15 cm.

Solution: $Volum{e}_{cone}=\frac{1}{3}{\mathrm{\pi r}}^{2}\mathrm{h}$

$=\frac{1}{3}\mathrm{\pi}\times {\left(4\right)}^{2}\times 15$

$=80\mathrm{\pi}{\mathrm{cm}}^{3}$

Question 2: Find the lateral area of a right circular cone in terms of $\mathrm{\pi}$ if the radius of its base is 3 ft and the altitude is 4 ft.

Solution: First we need to find the measure of slant height.

$l=\sqrt{{r}^{2}+{h}^{2}}$$=\sqrt{{\left(3\right)}^{2}+{\left(4\right)}^{2}}$$=\sqrt{25}$$=5ft$

$L{A}_{cone}=\mathrm{\pi rl}$

$=\pi \times 3\times 5$

$=15\mathrm{\pi}{\mathrm{ft}}^{2}$

Question 3: Find the total surface area of a right circular cone if radius is 3 m and the slant height is15 m.

Solution: $T{A}_{cone}=\mathrm{\pi r}\left(\mathrm{l}+\mathrm{r}\right)$

$=\pi \times 3\left(15+3\right)$

$=\pi \times 3\times 18$

$=54\pi {m}^{2}$

## Cheat Sheet

- Area of the base, $B={\mathrm{\pi r}}^{2}$
- Relation between the radius of the base, height, and slant height: $l=\sqrt{{r}^{2}+{h}^{2}}$
- $Volum{e}_{cone}=\frac{1}{3}{\mathrm{\pi r}}^{2}h$
- Lateral Surface Area, $L{A}_{cone}=\mathrm{\pi rl}$
- Total Surface Area, $T{A}_{cone}=L{A}_{cone}+B$$=\mathrm{\pi r}\left(\mathrm{l}+\mathrm{r}\right)$

## Blunder Areas

- The volume of the cone = $\frac{1}{3}$volume of the cylinder.
- All the formulas mentioned in the above sections are applicable to right circular cones only.

- Fiona Wong
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