Surface Area of Solids Joined Together: Quick Revision

A brief note on Surface Area of Solids Joined Together. Read it before a test. Try each question at the end before you open its answer.

The basic solids

• Cube: lateral $4a^2$, total $6a^2$. Cuboid: lateral $2(a + b)c$, total $2(ab + bc + ca)$.
• Cylinder: curved $2\pi r h$, total $2\pi r(r + h)$.
• Cone: $l = \sqrt{r^2 + h^2}$, curved $\pi r l$, total $\pi r(r + l)$.
• Sphere: $4\pi r^2$. Hemisphere: curved $2\pi r^2$, total $3\pi r^2$.
• The flat net of a cone is a sector of radius $l$ and angle $\dfrac{r}{l} \times 360^\circ$.

Joined and hollowed solids

• Add only the faces that stay on the outside. Never add two total surface areas.
• Cone on a hemisphere: $\pi r l + 2\pi r^2$.
• Capsule (a cylinder with two hemispheres): $2\pi r h + 4\pi r^2$. Its whole length is $h + 2r$.
• Tent: $2\pi r h + \pi r l$, with no floor.
• A hollow: remove the face where it starts and add the inside surface. A hemispherical hollow in a flat face adds $\pi r^2$.

A worked example

A capsule is $24\ \text{mm}$ long in all, and its radius is $5\ \text{mm}$. Find its surface area in terms of $\pi$.

1. The cylinder part is $24\ \text{mm} - 2 \times 5\ \text{mm} = 14\ \text{mm}$ long.
2. Its curved surface is $2\pi \times 5\ \text{mm} \times 14\ \text{mm} = 140\pi\ \text{mm}^2$.
3. The two ends make a sphere: $4\pi \times (5\ \text{mm})^2 = 100\pi\ \text{mm}^2$.
4. The total is $240\pi\ \text{mm}^2$.

Practice with new numbers

A program to try

The program finds the surface area of a toy in three ways. Only the first and the last agree.

Mistakes to avoid

• Adding the total surface areas of the parts. The joining circle is hidden.
• Using the height in place of the slant height in $\pi r l$.
• Using the whole length of a capsule as the height of its cylinder part.
• Adding a floor to a tent.

Check yourself

1. A cone has radius $9\ \text{cm}$ and height $12\ \text{cm}$. What is its curved surface area?

2. A hemisphere has radius $10\ \text{cm}$. What is its total surface area?

3. A tent has a cylinder part with $2\pi r h = 50\pi\ \text{m}^2$ and a cone part with $\pi r l = 30\pi\ \text{m}^2$. How much canvas is needed?

4. A hemispherical bowl of radius $2\ \text{cm}$ is cut into the top face of a cube. By how much does the surface area change?