Tents, hollows and cut-outs

A tent

A circus tent has straight walls and a pointed roof. The walls are the curved surface of a cylinder. The roof is the curved surface of a cone. A tent has no floor, and no cloth is needed between the walls and the roof.

If the cylinder part has radius $r$ and height $h$, and the cone has slant height $l$, then

Example 1. A tent has a cylinder part of radius $3.5\ \text{m}$ and height $3\ \text{m}$. The cone on top has height $1.2\ \text{m}$. Find the canvas needed, with $\pi = \frac{22}{7}$. Canvas costs $90$ rupees per square metre. What is the cost?

1. The slant height. $l = \sqrt{3.5^2 + 1.2^2}\ \text{m} = \sqrt{13.69}\ \text{m} = 3.7\ \text{m}$.
2. The cylinder part. $2 \times \dfrac{22}{7} \times 3.5\ \text{m} \times 3\ \text{m} = 66\ \text{m}^2$.
3. The cone. $\dfrac{22}{7} \times 3.5\ \text{m} \times 3.7\ \text{m} = 40.7\ \text{m}^2$.
4. The area. $66\ \text{m}^2 + 40.7\ \text{m}^2 = 106.7\ \text{m}^2$.
5. The cost. $106.7 \times 90 = 9603$ rupees.

Cloth in strips. Canvas is sold in a strip of a fixed width. The area of a strip is length times width. So

If the strip is $2\ \text{m}$ wide, the tent needs $106.7\ \text{m}^2 \div 2\ \text{m} = 53.35\ \text{m}$.

Scooping out a hollow

Take a block of wood and drill a bowl into it. The block now has a hollow. How does its surface area change?

The rule. Remove the face where the hollow starts. Add the surface of the inside of the hollow.

A cube with a hemispherical hollow. A hemispherical bowl of radius $r$ is cut into the top face of a cube of edge $a$. The circle $\pi r^2$ is removed from the top face. The inside of the bowl adds $2\pi r^2$. So

The area goes up by $\pi r^2$.

A cylinder with a hemispherical hollow at each end. Both flat circles are removed, and two bowls are added. The surface area is

The circles that were removed ($2\pi r^2$) were part of the total surface, $2\pi r h + 2\pi r^2$. They are replaced by $4\pi r^2$.

A cylinder with a conical hollow. A trophy is a cylinder with a cone cut into the top. The cone has the same radius as the cylinder, so the whole top face is gone. What is outside? The bottom circle, the curved side and the inside of the cone:

Example 2. A solid cylinder of height $12\ \text{cm}$ and radius $5\ \text{cm}$ has a hemispherical hollow of radius $5\ \text{cm}$ at each end. Find the total surface area in terms of $\pi$.

1. The side. $2\pi \times 5\ \text{cm} \times 12\ \text{cm} = 120\pi\ \text{cm}^2$.
2. The two bowls. Each bowl is $2\pi \times (5\ \text{cm})^2 = 50\pi\ \text{cm}^2$. Two bowls give $100\pi\ \text{cm}^2$.
3. The total. $120\pi + 100\pi = 220\pi\ \text{cm}^2$, about $691.2\ \text{cm}^2$.

There are no flat faces left, since both circles became bowls.

Practise with new numbers

Try it in Python

This program finds the canvas of a tent and the cost. The slant height is worked out from the height of the cone.

Common mistakes

• Adding a floor to a tent. A tent has none, and the canvas is only the walls and the roof.
• Adding the circle where the cone meets the cylinder. It is not a cloth surface. It is inside the tent.
• Forgetting to remove the circle when a hollow is cut into a face.
• Counting a hollow twice. A bowl scooped from the top of a cylinder replaces the top face. It does not sit on top of it.

Remember these

• The canvas of a tent is $2\pi r h + \pi r l$.
• For a hollow, remove the face where it starts and add the inside surface.
• A hemispherical hollow in a flat face changes the area by $+\pi r^2$.
• The length of a strip of cloth is the area divided by the width.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A conical tent has radius $4\ \text{m}$ and slant height $5\ \text{m}$. How much canvas is needed, in terms of $\pi$?

2. A tent needs $120\ \text{m}^2$ of canvas, sold in strips $1.5\ \text{m}$ wide. What length of strip is needed?

3. A hemispherical bowl of radius $3\ \text{cm}$ is scooped from the top of a cube of edge $10\ \text{cm}$. By how much does the surface area change, in terms of $\pi$?

4. Which faces make up the surface of a cylinder with a conical hollow in its top, when the hollow has the same radius as the cylinder?

5. A solid cylinder with a hemispherical hollow at each end has radius $r$ and height $h$. Why is there no flat face in its surface?

References

• NCERT, Mathematics for Class 10, the chapter on Surface Areas and Volumes, for the surface areas of combinations of solids.
• Cambridge IGCSE Mathematics (0580), syllabus topic Mensuration, for the surface areas of cylinders, cones, spheres and hemispheres.