Question

The circumference of the base of a $16m$ high solid cone is $33m$ . Find the surface area of the cone.

Answer 1

Cone is the solid figure with circular base and two sloping heights.
Here we are going to use formula for the surface area of the cone = $\pi r(r + l)$ and circumference of the circle= $2\pi r$

Complete step by step solution:
Given: Circumference of the circle = $33m$
Height, $h = 16m$
Now, Circumference = $33m$
$2\pi r = 33$
Putting value, $\pi = \frac{{22}}{7}$
$\Rightarrow 2 \times \frac{{22}}{7} \times r = 33$
Taking all the values on Right hand Side, making subject,
$\begin{array}{l} r = \frac{{7 \times 33}}{{2 \times 22}}\\\ \Rightarrow r = \frac{{21}}{4}\\\ \Rightarrow r = 5.25m \end{array}$
Now,
Slant height, $l = \sqrt {{h^2} + {r^2}}$
Substituting values of $h\& r$
$\begin{array}{l} l = \sqrt {{{16}^2} + {{(5.25)}^2}} \\\ l = \sqrt {256 + 27.5625} \\\ l = \sqrt {283.5625} \\\ l = 16.839\\\ l = 16.84m \end{array}$
Total Surface area of cone= $\pi r(r + l)$
Substituting all the values of $r = 5.25m$ and $l = 16.84m$ in the above equation
$\begin{array}{l} A = \frac{{22}}{7} \times 5.25(5.25 + 16.84)\\\ A = \frac{{22}}{7} \times 5.25(22.09)\\\ A = 364.485{m^2} \end{array}$

Hence, the required answer is 364.485 meter square.

Additional Information: volume of the cone, $V = \frac{1}{3}\pi {r^2}h$
Where r= radius of the circular base of the cone
h = height of the cone

Note: Always remember the standard basic formula for the solid figures and always double check the units given metre or centimetres and convert it accordingly. Units of all the given parameters should be the same