Surface Areas and Volumes | Exercise 12.1

Question 3

A toy is in the form of a cone of radius $3.5\text{ cm}$ mounted on a hemisphere of same radius. The total height of the toy is $15.5\text{ cm}$ . Find the total surface area of the toy.

Solution

Understand the Question

The toy is formed by joining a cone and a hemisphere with a common radius $r = 3.5\text{ cm}$ .
The height of the hemisphere equals its radius ( $3.5\text{ cm}$ ). Thus, the vertical height of the cone is $h = 15.5 - 3.5 = 12\text{ cm}$ .
Since the circular base is inside where the two solids join, the total surface area of the toy is the sum of the curved surface areas of the cone and the hemisphere: $\text{Total Surface Area} = \pi r l + 2\pi r^2$ where $l = \sqrt{h^2 + r^2}$ is the slant height of the cone.

Step 1 · Find the Height of the Cone

Given:

Radius of cone and hemisphere, $r = 3.5\text{ cm}$
Total height of the toy $= 15.5\text{ cm}$
Height of hemisphere $= r = 3.5\text{ cm}$

Height of the cone, $h$ :

\begin{aligned} h &= \text{Total height} - \text{Height of hemisphere} \\[0.6em] &= 15.5 - 3.5 \\[0.6em] &= 12\text{ cm} \end{aligned}

Step 2 · Find the Slant Height of the Cone

By Pythagoras theorem:

\begin{aligned} l &= \sqrt{h^2 + r^2} \\[0.6em] &= \sqrt{(12)^2 + (3.5)^2} \\[0.6em] &= \sqrt{144 + 12.25} \\[0.6em] &= \sqrt{156.25} \\[0.6em] &= 12.5\text{ cm} \end{aligned}

Step 3 · Calculate the Curved Surface Areas

Using $\pi = \dfrac{22}{7}$ :

CSA of cone:

\begin{aligned} \text{CSA of cone} &= \pi r l \\[0.6em] &= \dfrac{22}{7} \times 3.5 \times 12.5 \\[0.6em] &= 22 \times 0.5 \times 12.5 \\[0.6em] &= 137.5\text{ cm}^2 \end{aligned}

CSA of hemisphere:

\begin{aligned} \text{CSA of hemisphere} &= 2\pi r^2 \\[0.6em] &= 2 \times \dfrac{22}{7} \times (3.5)^2 \\[0.6em] &= 2 \times \dfrac{22}{7} \times 12.25 \\[0.6em] &= 2 \times 22 \times 1.75 \\[0.6em] &= 77\text{ cm}^2 \end{aligned}

Step 4 · Find the Total Surface Area of the Toy

\begin{aligned} \text{Total Surface Area} &= \text{CSA of cone} + \text{CSA of hemisphere} \\[0.6em] &= 137.5 + 77 \\[0.6em] &= 214.5\text{ cm}^2 \end{aligned}

Answer

$214.5\text{ cm}^2$

Common Mistakes

Adding Flat Base Areas: Confusing total surface area of individual solids with the combined solid. The circular base is internal, so use only the curved surface areas ( $\text{CSA} = \pi r l + 2\pi r^2$ ), not $\text{TSA}$ .
Confusing Height and Slant Height: Using vertical height $h = 12\text{ cm}$ directly in $\pi r h$ instead of calculating the slant height $l = \sqrt{h^2 + r^2} = 12.5\text{ cm}$ for $\pi r l$ .

Question 3

More questions in Exercise 12.1