Formula of frustum of cone

we will check out Formula of frustum of cone in this post. we will also derive the formula so that it is easy to understand.

What is Frustum of cone

A frustum of cone is obtained by removing the upper part of the cone

Lets take a look at the Frustum below

H is the height of the Frustum
R is the radius of the Base
r is the radius of the Top
l is the lateral height or slant height of the frustum

Formula for Volume of the Frustrum

$V = \frac {1}{3} \pi H(r^2 + R^2 + rR)$

Derivation

Lets take the full cone from where frustum is taken

Volume of Frustum

$= \frac {1}{3} \pi (h+H)R^2 – \frac {1}{3} \pi h r^2$

Now from similar triangle theorem for right triangle for small clone and big close
$\frac {h}{h+H} = {r}{R}$
or $h = H \frac { r}{R-r}$

Substituting this in Volume

$= \frac {1}{3} \pi (H \frac { r}{R-r}+H)R^2 – \frac {1}{3} \pi H \frac { r}{R-r} r^2$
$=\frac {1}{3} \pi H ( \frac {R^3}{R-r} – \frac {r^3}{R-r})$
$= \frac {1}{3} \pi H \frac { R^3-r^3}{R-r}$
Now $a^3 -b^3 = (a-b) (a^2 + b^2 +a b)$

Therefore
$= \frac {1}{3} \pi H (r^2 + R^2 + rR)$

Formula for Curved Surface Area of the Frustrum

$S = \pi l ( r + R)$

where l is the slant height and it is given by

$l= \sqrt { H^2 + (R-r)^2}$

Derivation

Lets take the full cone from where frustum is taken

Curved surface area of frustum
$S= \pi R (s + l) – \pi r s$

Now from similar triangle theorem for right triangle for small clone and big close
$\frac {s}{s+l} = {r}{R}$
or $s = l \frac { r}{R-r}$

Therefore surface area is

$= \pi R (l \frac { r}{R-r} + l) – \pi r l \frac { r}{R-r}$
$= \pi l \frac {R^2}{R-r} – \pi l \frac {r^2}{R-r}$
$=\pi l \frac { R^2 -r^2}{R-r} = \pi l (R+r )$

Formula for Total Surface Area of the Frustrum

Formula for Total surface = Curved Surface Area + Area of Top + Area of base
$= \pi l (R+r ) + \pi (r^2 + R^2)$

Solved Examples

Question 1
The radii of the top and bottom of a bucket of slant height 45 cm are 28 cm and 7 cm, respectively. Find the curved surface area of the bucket
Solution

Here l=45 cm, r=7 cm and R=28 cm
Curved Surface area= $\pi l(R+r) = \pi \times 45 (28 + 7) =\frac {22}{7} \times 45 \times 35 = 4950 cm^2$

I hope this article on Formula of frustum of cone helps you.

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