Question

The volume and the total surface area of a cone whose height and slant height are $12cm$ and $13cm$ respectively are ________.

• A. 80π $c{m}^3$ and 90π $c{m}^2$
• B. 100π $c{m}^3$ and 90π $c{m}^2$
• C. 100π $c{m}^3$ and 80π $c{m}^2$
• D. 85π $c{m}^3$ and 95π $c{m}^2$

Answer 1

The correct option is B 100π $c{m}^3$ and 90π $c{m}^2$

Slant height of the cone is given by

$l^2=(h^2+r^2)$
$\Rightarrow r=\sqrt{l^2-h^2}$
$\Rightarrow r=\sqrt{{13}^2-{12}^2}$
$\Rightarrow r=5cm$

So, volume of cone
$=\frac{1}{3}\pi r^{2}h$
$=\frac{1}{3}\times \pi \times 5^2\times 12$
$=\frac{1}{3}\times \pi \times 25\times 12$
$=100\pi c{m}^3$

Now, total surface area
$=(\pi \times r\times l)+(\pi \times r^2)$
$=\pi \times 5\times 13+\pi \times 25$
$=90\pi c{m}^2$