Math 8 · Nets, surface area and volume
Unfold it for the surface, fill it for the volume
A net is a 3-D object unfolded flat, and the area of the net is the object's surface area: the cardboard in a box, the label on a can. Volume is how much the object holds, and for every prism and cylinder it comes from one idea — the area of the base times the height.
- 1. The words, first
- 2. Nets of 3-D objects
- 3. Surface area of prisms and cylinders
- 4. Volume of prisms and cylinders
- 5. What costs marks
The words, first
The idea: Base is the word that trips people up. For a prism it means one of the two matching ends, which is not always the face on the bottom.
| Word | What it means |
|---|---|
| Face, edge | A face is a flat surface of a 3-D object; an edge is where two faces meet. |
| Right prism | A 3-D object with two identical, parallel bases joined by rectangular faces at 90° to them. It is named for its base: a rectangular prism (a box), or a triangular prism. |
| Base of a prism | One of the two matching ends. A tent lying on its floor still has triangles for bases. |
| Right cylinder | Two identical, parallel circles joined by a curved side at 90° to them, like a can. |
| Height of a prism or cylinder | The distance between the two bases. For a prism lying on its side, like a tent, it is the length. |
| Net | A flat pattern that folds up into a 3-D object, showing every face once. |
| Surface area, volume | The total area of every face, in square units such as cm²; and the space inside, in cubic units such as cm³. 1 cm³ holds 1 mL. |
| π, circumference | The distance round a circle is its circumference, C = 2 × π × r. This page uses π ≈ 3.14. |
Nets of 3-D objects
The idea: A net shows every face exactly once, joined along edges so it folds up with no gaps and no overlaps. Count the faces before you draw: a box has 6, a triangular prism 5, a cylinder 3.
- Box: 6 rectangles in 3 matching pairs — top and bottom, front and back, and the two ends.
- Triangular prism: 2 matching triangles and 3 rectangles. Each rectangle is as wide as the side of the triangle it folds against: 5, 8 and 5 cm for the tent.
- Cylinder: 2 circles and 1 rectangle. The rectangle wraps once round the circle, so its length is the circumference, not the diameter.
Where the 5 cm comes from. The dashed height splits each end triangle into two right triangles with legs of 4 cm and 3 cm, so the sloping side is a hypotenuse: 4² + 3² = 16 + 9 = 25, and √25 = 5.
Not every arrangement folds. Six squares in a cross fold into a cube. Six in a straight row just roll into a tube with both ends open, the last two squares landing on top of the first two.
Surface area of prisms and cylinders
The idea: Find the area of every face in the net and add them up. Faces come in matching pairs, so work out one and double it.
Box: SA = 2lw + 2lh + 2wh · Cylinder: SA = 2πr² + 2πrh, two circles and a rectangle 2πr long
Worked example: the tent. Use the net in the figure.
- Two triangles: each is 8 × 3 ÷ 2 = 12 cm², so together 24 cm².
- Three rectangles: 5 × 20 + 8 × 20 + 5 × 20 = 100 + 160 + 100 = 360 cm².
- Total: 24 + 360 = 384 cm².
The slip to avoid: using the 5 cm sloping side as the triangle's height, which is the 3 cm dashed line.
Worked example: the can. Radius 4 cm, height 11 cm, with π ≈ 3.14.
- Two circles: 3.14 × 4 × 4 = 50.24 cm² each, so 100.48 cm².
- The curved side: its length is the circumference, 2 × 3.14 × 4 = 25.12 cm, so its area is 25.12 × 11 = 276.32 cm².
- Total: 100.48 + 276.32 = 376.8 cm².
Read what is covered. The can's paper label is the curved side only, about 276 cm². A box 30 cm by 20 cm by 10 cm has a surface area of 2 × 600 + 2 × 300 + 2 × 200 = 2200 cm², but with no lid it needs 2200 − 600 = 1600 cm².
Volume of prisms and cylinders
The idea: Volume = area of the base × height, for every right prism and every right cylinder. Find the base first — the face the object is named after — then multiply by the distance between the two bases.
V = area of base × h · box: V = l × w × h · cylinder: V = πr² × h
Why it works. A prism is a stack of thin slices, each a copy of the base. A 1 cm slice of the tent holds 12 cm³, because its triangle has an area of 12 cm². The tent is 20 slices long, so it holds 12 × 20 = 240 cm³. Multiplying 8 × 3 × 20 gives 480 cm³, twice too much: that is the box the triangle only half fills.
The can. The base is a circle of 50.24 cm² and the height is 11 cm, so V = 50.24 × 11 = 552.64 cm³. Since 1 cm³ holds 1 mL, the can holds about 553 mL.
Worked example: working backwards. An aquarium has a base 60 cm by 30 cm and holds 54 L when full. How deep is the water?
- Litres to cubic centimetres: 1 L is 1000 cm³, so 54 L is 54 × 1000 = 54 000 cm³.
- Base area: 60 × 30 = 1800 cm².
- Volume = base area × height, so the height is 54 000 ÷ 1800 = 30 cm.
What costs marks
The idea: Most of these are a length used in the wrong place.
- Using the diameter as the radius. A can 8 cm across has r = 4 cm.
- Forgetting to halve a triangle, or using its sloping side as its height.
- Making the cylinder's rectangle as long as the diameter. It wraps round, so it is as long as the circumference.
- Missing a face, or counting one twice. Count the faces in the net first: 6, 5 or 3.
- Mixing up the units. Surface area is in cm², volume in cm³.