Surface Area of Prisms and Cylinders

A cube has 6 faces, and every face is a square of the same size.

A cuboid has three pairs of matching faces. Work out the area of one face from each pair.

The prism has two triangular faces and three rectangular faces. Each rectangle is 10 cm long.

Unrolled, the curved surface is a rectangle. Its width is the circumference of the circle and its height is the height of the cylinder.

Add the curved surface area to the areas of the two circular ends.

The net is made of three pairs of matching rectangles. Find the area of one rectangle from each pair.

1. The cuboid is 7 cm by 3 cm by 2 cm.
2. Two rectangles 7\times3=21\text{ cm}^2, two rectangles 7\times2=14\text{ cm}^2 and two rectangles 3\times2=6\text{ cm}^2.
3. Total =2\times21+2\times14+2\times6=42+28+12=82\text{ cm}^2.

Each face has a matching face opposite it, so count the top, bottom, front and back first.

1. Top and bottom: 2\times12=24\text{ cm}^2.
2. Front and back: 2\times20=40\text{ cm}^2.
3. The two end faces together: 94-24-40=30\text{ cm}^2.
4. One end face =30\div2=15\text{ cm}^2.

There are two triangles and three rectangles. Use the perpendicular height only for the area of the triangles.

1. Each triangle: \frac{1}{2}\times10\times12=60\text{ cm}^2, so the two triangles give 120\text{ cm}^2.
2. Bottom rectangle: 10\times20=200\text{ cm}^2.
3. Two sloping rectangles: 2\times13\times20=520\text{ cm}^2.
4. Total surface area =120+200+520=840\text{ cm}^2.

The top and the bottom of a cylinder are both circles with the same radius as the cylinder.

Part (a)

1. Area =\pi\times5^2=25\pi\text{ cm}^2.

Part (b)

1. The two ends: 2\times25\pi=50\pi\text{ cm}^2.
2. Total surface area =80\pi+50\pi=130\pi\text{ cm}^2.

Halve the diameter to get the radius before you use any formula.

1. Radius =10\div2=5 cm.
2. Curved surface area =2\times\pi\times5\times12=120\pi=376.99...\text{ cm}^2.
3. Two ends =2\times\pi\times5^2=50\pi=157.07...\text{ cm}^2.
4. Total =170\pi=534.07...\text{ cm}^2, which is 534.1\text{ cm}^2 to 1 decimal place.

An open box has only five faces: the base and four sides.

1. Base: 30\times20=600\text{ cm}^2.
2. Front and back: 2\times30\times15=900\text{ cm}^2.
3. Two ends: 2\times20\times15=600\text{ cm}^2.
4. Total =600+900+600=2100\text{ cm}^2.

Find the area of one face first, then the length of one edge.

1. Area of one face =486\div6=81\text{ cm}^2.
2. Each face is a square, and 9\times9=81, so each edge is 9 cm.
3. Volume =9\times9\times9=729\text{ cm}^3.

Unrolled, the label is a rectangle. One side is the distance round the tin.

1. The distance round the tin is the circumference, \pi\times\text{diameter}=\pi\times8 cm.
2. The label is a rectangle \pi\times8 cm long and 10 cm high.
3. Area of the label =\pi\times8\times10.

There is only one circle to add to the curved surface, because the pot has no top.

1. Curved surface area =2\times\pi\times6\times15=180\pi=565.48...\text{ cm}^2.
2. Base =\pi\times6^2=36\pi=113.09...\text{ cm}^2.
3. Total =216\pi=678.58...\text{ cm}^2, which is 678.6\text{ cm}^2 to 1 decimal place.

Work out the surface area in square centimetres first, then change the units.

Part (a)

1. Top and bottom: 2\times100\times50=10\,000\text{ cm}^2.
2. Front and back: 2\times100\times40=8000\text{ cm}^2.
3. Two ends: 2\times50\times40=4000\text{ cm}^2.
4. Total =10\,000+8000+4000=22\,000\text{ cm}^2.

Part (b)

1. 1\text{ m}^2=100\times100=10\,000\text{ cm}^2.
2. 22\,000\div10\,000=2.2\text{ m}^2.

Find the total surface area of one drum, including both circular ends, then double it.

1. Curved surface of one drum =2\times\pi\times0.4\times0.9=2.261...\text{ m}^2.
2. Two ends of one drum =2\times\pi\times0.4^2=1.005...\text{ m}^2.
3. One drum =3.267...\text{ m}^2, so two drums =6.534...\text{ m}^2.
4. 6.534...\div2.5=2.61..., so she needs 3 tins (2 tins would not be enough).
5. Cost =3\times8.75= £26.25.

Add the surface areas of the two cubes, then take away the faces that are glued together and can no longer be seen.

1. Large cube: 6\times4\times4=96\text{ cm}^2.
2. Small cube: 6\times2\times2=24\text{ cm}^2.
3. The bottom of the small cube (4\text{ cm}^2) is hidden, and so is the 4\text{ cm}^2 of the large cube's top that it covers.
4. Total surface area =96+24-4-4=112\text{ cm}^2.

The rectangles along the length of the prism are all 10 cm long, and their widths go all the way round the edge of the L shape.

1. First find the inner sides: the upper height is 5-2=3 cm, and the inner horizontal side is 6-2=4 cm.
2. Split the end horizontally. The bottom rectangle is 6\times2; the upper rectangle is 2\times3. They meet along the dashed line, so no area is counted twice.

3. One L-shaped end: 6\times2+2\times3=12+6=18\text{ cm}^2. The two matching ends have area 2\times18=36\text{ cm}^2.
4. The six outside edges of the L shape total 6+2+4+3+2+5=22 cm. The dashed dividing line is inside the end, so it is not part of the perimeter.
5. Each outside edge makes a rectangular face 10 cm long. Their combined area is 22\times10=220\text{ cm}^2.
6. Total surface area: 36+220=256\text{ cm}^2.

Curved surface area = circumference \times height, so find the circumference first.

1. Circumference =2\times\pi\times10=62.83... cm.
2. Height =1500\div62.83...=23.87... cm.
3. Correct to 1 decimal place: 23.9 cm.

Find the edge length of box A from its volume, then work out the surface area of each box.

1. 4\times4\times4=64, so each edge of box A is 4 cm.
2. Box A: 6\times4\times4=96\text{ cm}^2.
3. Box B: 2\times(8\times4+8\times2+4\times2)=2\times56=112\text{ cm}^2.
4. Difference =112-96=16\text{ cm}^2.