Cone: Surface Area and Volume

Overview

**Surface Area and Volume of cone**:
Total surface area of a cone

$=$ area of the base-surface$+$ sum of the area of the side-faces

$=2\pi r(r+l)$
Volume of a cone

$=\frac{1}{3}\times \phantom{\rule{1ex}{0ex}}\text{area of the base-face}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{height}$

$=\frac{1}{3}\pi {r}^{2}h$

cone

The solid shape shown in the figure is "a cone". Cone is a 3D solid shape with a circular-base at the bottom with a curved surface converging to a single point on the top.

We consider only right-circular-cones with its axis at right angle to the base-surface.
Other cones are

(1) elliptical-cones with an ellipse at the base-surface

(2) oblique-cones with the angle, between its axis and the base, not a right-angle.

A right-circular-cone is shown in orange. An oblique-cone is shown in blue.

surface area

The surface area of the cone of radius $r$ and height $h$ is

$\phantom{\rule{1ex}{0ex}}\text{area of base}\phantom{\rule{1ex}{0ex}}$ $+\phantom{\rule{1ex}{0ex}}\text{area of sector}$

$\phantom{\rule{1ex}{0ex}}\text{area of base}\phantom{\rule{1ex}{0ex}}$ $+\pi \text{radius}\times \left(slantheight\right)$

$\pi r(r+s)$

Note: A cone consists of

circular base

a sector from a circle with radius of slant-height $s$ and arc length $2\pi r$.

$s$ is the slant height computed as $\sqrt{{r}^{2}/4+{h}^{2}}$.

A cone of height $h$ is shown in orange. The curved lateral surface is unraveled to a sector as shown in blue. The radius of the sector is the slant-height of the cone $s$. And the arc-length of the sector is the perimeter of the base circle of the cone.

Total surface area of the cone

$=$ area of the base-surface $+$ the area of the curved surface

$=\pi {r}^{2}+areaofarc$

$=\pi {r}^{2}+\frac{2\pi r}{2\pi s}\times \pi {s}^{2}$

$=\pi {r}^{2}+\pi rs$

`= pi r (r + s)

volume

The volume of the cone of radius $r$ and height $h$ is

volume of the modified-square-pyramid

$\frac{1}{3}\times \phantom{\rule{1ex}{0ex}}\text{area of base-surface}\phantom{\rule{1ex}{0ex}}$ $\times \phantom{\rule{1ex}{0ex}}\text{height}$

$\frac{1}{3}\pi {r}^{2}h$

The cone is shown in orange. As per the Cavalieri's principle in 3D, the cone is equivalently represented by the oblique-pyramid shown in blue.

The modified-pyramid has

• the area at the base surface equal to the area of base-surface of the cone

• the height equal to the height of the cone.

• the cross-sectional area along the vertical axis equal to that of the cone

Volume of the cone

$=$ volume of the pyramid with identical cross-sectional area along the vertical axis

$=\frac{1}{3}\times \phantom{\rule{1ex}{0ex}}\text{area of the base}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{height}$

$=\frac{1}{3}\times \pi {r}^{2}\times h$

$=\frac{1}{3}\pi {r}^{2}h$

What is the volume of a cone of height $7$ cm and radius $3$ cm?

The answer is "$66c{m}^{3}$"

summary

**Surface Area and Volume of cone**:
Total surface area of a cone

$=$ area of the base-surface$+$ sum of the area of the side-faces

$=2\pi r(r+l)$
Volume of a cone

$=\frac{1}{3}\times \phantom{\rule{1ex}{0ex}}\text{area of the base-face}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{height}$

$=\frac{1}{3}\pi {r}^{2}h$

Outline

The outline of material to learn *Mensuration : Length, Area, and Volume* is as follows.

Note 1: * click here for the detailed overview of Mensuration High *

Note 2: * click here for basics of mensuration, which is essential to understand this. *

• ** Basics of measurement**

→ __Summary of Measurement Basics__

→ __Measurement by superimposition__

→ __Measurement by calculation__

→ __Measurement by equivalence__

→ __Measurement by infinitesimal pieces__

→ __Cavalieri's Principle (2D)__

→ __Cavalieri's Principle (3D)__

• **Perimeter & Area of 2D shapes**

→ __Circumference of Circles__

→ __Area of Circles__

• **Surface area & Volume of 3D shapes**

→ __Prisms : Surface Area & Volume__

→ __Pyramids : Surface Area & Volume__

→ __Cone : Surface Area & Volume__

→ __Sphere : Surface Area & Volume__

• **Part Shapes **

→ __Understanding part Shapes__

→ __Circle : Sector and Segment__

→ __Frustum of a Cone__