Chapter 12.3, Problem 5ST1

To determine

To find: the lateral surface area, total surface area and volume of a right circular cone.

Answer to Problem 5ST1

The lateral surface area, total surface area and cone are ${\text{135π unit}}^{\text{2}}{\text{,216π unit}}^{\text{2}}\text{ and 324}\pi {\text{ unit}}^{\text{3}\text{.}}$ .

Explanation of Solution

Given information:

Radius of right-circular cone, $r=9$

Height of right-circular cone, $h=12$

Calculation:

Let’s calculate the latent height of cone, $l$ :

  $\begin{array}{l}l=\sqrt{h^2+r^2}\\ =\sqrt{{12}^2+9^2}\\ =\sqrt{144+81}\\ =\sqrt{225}\end{array}$

  $\text{ =15}$

Now

Calculate lateral surface area of cone.

  $\begin{array}{l}\text{Lateral Surface Area}=\pi rl\\ =\pi \times 9\times 15\\ \text{ =135}\pi {\text{ unit}}^2\end{array}$

And

Calculate total surface area of cone.

  $\begin{array}{l}\text{Total Surface Area}=LA+\pi r^2\\ =135\pi +\pi \left(9^2\right)\\ =135\pi +\pi \left(81\right)\\ =216\pi {\text{ unit}}^2\end{array}$

And

Calculate volume of cone.

  $\begin{array}{l}Volume=\frac{1}{3}\pi r^{2}h\\ =\frac{1}{3}\pi \left(9^2\right)\times 12\\ =324\pi {\text{ unit}}^3\end{array}$

Therefore, the lateral surface area, total surface area and cone are ${\text{135π unit}}^{\text{2}}{\text{,216π unit}}^{\text{2}}\text{ and 324}\pi {\text{ unit}}^{\text{3}\text{.}}$ .