Surface areas of the basic solids

What a surface area measures

Think of painting a tin of beans, wrapping a gift or covering a football. In each job you ask how much paper or paint the outside needs. That amount is the surface area of the solid. It is measured in square units, such as $\text{cm}^2$ and $\text{m}^2$.

There are three words to keep apart.

• The curved surface area (CSA) is the area of the curved part only. Cylinders, cones, spheres and hemispheres have one. In the United States the name is lateral area for a cylinder or a cone.
• The lateral surface area of a cube or a cuboid is the area of the four side faces. The top and the bottom are left out.
• The total surface area (TSA) is everything on the outside, flat faces included.

Which one to use depends on what the question covers. A tin with a lid has a total surface. A tent has no floor, so it has only the curved part.

The six solids

Let $r$ be the radius and $h$ the height. For a cone, $l$ is the slant height: the length along the side from the rim to the tip.

• Cube of edge $a$: lateral surface area $4a^2$, total surface area $6a^2$.
• Cuboid with edges $a$, $b$ and $c$, where $c$ is the height: lateral surface area $2(a + b)c$, total surface area $2(ab + bc + ca)$.
• Cylinder: curved surface area $2\pi r h$, total surface area $2\pi r(r + h)$.
• Cone: $l = \sqrt{r^2 + h^2}$, curved surface area $\pi r l$, total surface area $\pi r(r + l)$.
• Sphere: surface area $4\pi r^2$.
• Hemisphere: curved surface area $2\pi r^2$, total surface area $3\pi r^2$.

Why do these formulas work? Cut the solid open and lay it flat. The flat shape is called the net.

The cylinder. Peel the label off a tin. It is a rectangle. One side is the height $h$. The other side is the distance round the tin, $2\pi r$. The area is $2\pi r \times h$. Add two circles of area $\pi r^2$ each, and the total is $2\pi r h + 2\pi r^2 = 2\pi r(r + h)$.

The cone. Cut a paper cone along its slant side and flatten it. You get a sector of a circle. The radius of the sector is the slant height $l$. The curved edge of the sector is the rim of the cone, with length $2\pi r$.

A full circle of radius $l$ has circumference $2\pi l$. So the sector is the fraction $\dfrac{2\pi r}{2\pi l} = \dfrac{r}{l}$ of that circle. Its area is $\dfrac{r}{l} \times \pi l^2 = \pi r l$. Its angle is $\dfrac{r}{l} \times 360^\circ$.

The sphere. A sphere cannot be flattened without stretching, so we do not get a net. Archimedes found that a sphere has exactly the area of the curved side of the cylinder around it. That cylinder has radius $r$ and height $2r$, so its curved area is $2\pi r \times 2r = 4\pi r^2$. We accept this result here.

Pi in the answer

A question may say "take $\pi = \frac{22}{7}$". Then use that value. This is quick when the radius is a multiple of $7$. If the question says nothing, leave $\pi$ in the answer, such as $175\pi\ \text{cm}^2$, and also give a decimal from your calculator, $549.8\ \text{cm}^2$. In IGCSE and GCSE papers the calculator value of $\pi$ is the usual choice, and the answer is given to three significant figures.

Worked examples

Example 1. Painting a room. A room is $8\ \text{m}$ long, $5\ \text{m}$ wide and $3\ \text{m}$ high. Find the area of the four walls.

1. The walls. They are the lateral surface of a cuboid, so the area is $2(a + b)c$.
2. Put in the numbers. $2 \times (8\ \text{m} + 5\ \text{m}) \times 3\ \text{m} = 78\ \text{m}^2$.

If the ceiling is painted as well, add $8\ \text{m} \times 5\ \text{m} = 40\ \text{m}^2$. The total is $118\ \text{m}^2$.

Example 2. A closed tin. A cylindrical tin has radius $7\ \text{cm}$ and height $20\ \text{cm}$. Find its total surface area, with $\pi = \frac{22}{7}$.

1. The formula. $2\pi r(r + h)$.
2. Put in the numbers. $2 \times \dfrac{22}{7} \times 7\ \text{cm} \times (7\ \text{cm} + 20\ \text{cm}) = 44\ \text{cm} \times 27\ \text{cm} = 1188\ \text{cm}^2$.

Example 3. A cone. A cone has radius $6\ \text{cm}$ and height $8\ \text{cm}$. Find its curved surface area and total surface area, in terms of $\pi$.

1. The slant height. $l = \sqrt{6^2 + 8^2}\ \text{cm} = 10\ \text{cm}$.
2. The curved surface. $\pi r l = \pi \times 6\ \text{cm} \times 10\ \text{cm} = 60\pi\ \text{cm}^2$.
3. The base. $\pi r^2 = 36\pi\ \text{cm}^2$.
4. The total. $60\pi + 36\pi = 96\pi\ \text{cm}^2$, which is about $301.6\ \text{cm}^2$.

Practise with new numbers

Try it in Python

This program finds all the surface areas of the solids of this lesson for one radius. It uses pi from the math module and gets the slant height with sqrt.

Notice what the program prints for the sphere and the hemisphere. The total surface of the hemisphere is $3\pi r^2$. Half of the sphere gives $2\pi r^2$, and the flat face adds $\pi r^2$.

Common mistakes

• Using the height in place of the slant height. The curved surface of a cone needs $l$, not $h$.
• Giving the curved area when the question asks for the total, or the other way round. Read what is covered: a lid, a base, a floor.
• Mixing units. A radius in $\text{cm}$ and a height in $\text{m}$ give nonsense. Change both to the same unit first.
• Leaving out the flat circle of a hemisphere. The total surface is $3\pi r^2$, not $2\pi r^2$.

Remember these

• The curved surface of a cylinder is $2\pi r h$, and its total surface is $2\pi r(r + h)$.
• The curved surface of a cone is $\pi r l$ with $l = \sqrt{r^2 + h^2}$, and its total surface is $\pi r(r + l)$.
• A sphere has area $4\pi r^2$. A hemisphere has $2\pi r^2$ curved and $3\pi r^2$ in total.
• The lateral surface of a cuboid is $2(a + b)c$, and its total surface is $2(ab + bc + ca)$.

Check yourself

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

1. A cube has edge $5\ \text{cm}$. Find its lateral surface area and its total surface area.

2. A cylinder has radius $3\ \text{cm}$ and height $10\ \text{cm}$. Find its curved surface area in terms of $\pi$.

3. A cone has radius $5\ \text{cm}$ and height $12\ \text{cm}$. Find the slant height and the curved surface area in terms of $\pi$.

4. A ball has radius $10\ \text{cm}$. Find its surface area in terms of $\pi$.

5. A solid hemisphere has radius $4\ \text{cm}$. Find its total surface area in terms of $\pi$, and say which part of it is the flat face.