Chapter 12, Problem 19SGR

To determine

To find the lateral and surface area of the given cone.

Answer to Problem 19SGR

Lateral surface $=376$ square centimeters

Surface area $=455$ square centimeters

Explanation of Solution

Given:

  

A cone

Diameter, $(d)=10$ centimeters

Radius, $(r)=5$ centimeters

Height, ( $h$ ) $=22$ centimeters

Formula used:

Lateral surface area of a cone $=\pi rl$

where $r$ is the radius and $l$ is the slant height of the cylinder.

Total surface area of a cone $=\pi r(l+r)$

where $r$ is the radius and $l$ is the slant height of the cylinder.

Pythagoras theorem:

In $\Delta ABC$ right angled at $B$ , then

  $A{B}^2+B{C}^2=C{A}^2$

Calculation:

  

In $\Delta ABC$,$A{B}^2+B{C}^2=C{A}^2$

Putting the values,

  $\Rightarrow {10}^2+{22}^2=C{A}^2$

  $\Rightarrow 100+484=C{A}^2$

  $\Rightarrow C{A}^2=584$

Taking square root on both sides,

  $\Rightarrow CA=l=24$ , where $l$ is the slant height of the cone.

To find the lateral surface area of the cone, use the following formula.

Lateral surface area $=\pi rl$

Now putting the values,

  $\Rightarrow 3.14\times 5\times 24$

  $\Rightarrow 376$ square centimeters

So, the lateral surface area of the cone is 376 square centimeters.

To find the surface area of the cone, use the following formula.

Lateral surface area $=\pi r(l+r)$

Now putting the values,

  $\Rightarrow 3.14\times 5\times (24+5)$

  $\Rightarrow 3.14\times 5\times 29$

  $\Rightarrow 455$ square centimeters

Conclusion:

Hence, the lateral surface area is 376 square centimeters and surface area is 455 square centimeters.