Volume & Surface Area of 3-D Figures

Volume fills the inside — base area times height. Surface area wraps the outside — add up every face.

Take a cereal box and cut it open along the edges, then lay it flat.

What you now have is six rectangles — the front, the back, two sides, a top and a base.

a cereal box, flattened out the box front side side back top base six faces, laid flat the card you can see is the surface; the cereal inside is the volume

Add up those six areas and you have the amount of card the box is made from. That is the surface area.

But the cereal inside was never measured by that. The cereal filled the space, and space needs three measurements: length, width and height.

One is about the skin, the other about the filling — and they even have different units.

Surface area is measured in cm2 because it is a flat thing made of lengths times lengths. Volume is cm3. If the units on your answer are wrong, you have computed the other quantity.

A box unfolded into its net, showing six faces 7×53×5 7×53×5 7×37×3 surface area = the area of the whole net 2(7×5) + 2(3×5) + 2(7×3) = 142 volume is different: 7 × 5 × 3 = 105

One formula for every prism and cylinder

V = Bh, where B is the area of the base and h is the height.

Think of it as stacking: one layer of the base is B cubic units thick, and you stack h of them.

Rectangular prism: B = lw, so V = lwh. Triangular prism: B = 12bh△. Cylinder: B = π r2, so V = π r2 h. Same idea each time — only B changes.

Find the base first

The base is the shape that repeats all the way through the solid. Find its area, then multiply by how far it repeats. That single habit handles every prism.

Surface area is just adding faces

Unfold the solid into its net, find each piece's area, and add.

Rectangular prism: three pairs of matching faces, so SA = 2lw + 2lh + 2wh.

Cylinder: two circles plus one rectangle. The rectangle is the label peeled off — its width is the circumference. So SA = 2π r2 + 2π r h.

Keep the units straight

Volume is in cubic units: cm3, m3, in3. Surface area is in square units: cm2, m2.

If your answer has the wrong kind of unit, you used the wrong formula. This is a free check on every problem.

Ask "filling or wrapping?"

Filling a fish tank with water, or a box with sand → volume. Wrapping a present, or painting a wall → surface area. Saying which one out loud before you start prevents most errors in this unit.

Pyramids and cones hold a third

A pyramid or cone with the same base and height as a prism or cylinder holds exactly one third as much: V = 13Bh.

You can see this by pouring: three cone-fulls of water fill a cylinder of matching base and height exactly.

A square pyramid inside a prism and a cone inside a cylinder, each with the same base and height: the pointed solid holds one third same base, same height — the pointed one holds a third h prism V = Bh pyramid V = ⅓Bh h r cylinder V = πr²h cone V = ⅓πr²h three pyramids fill the prism; three cones fill the cylinder

Working backwards

Given the volume and two dimensions, divide to find the third. If V = 240 cm3 with base 8 × 5, then 240 = 40h, so h = 6 cm.

Given surface area, substitute and solve — often an equation from Unit 11 appears inside a geometry problem.

Worked examples

Triangular prism

A prism has a triangular base with base 6 cm and height 4 cm, and the prism is 10 cm long. Find its volume.

  1. Find B, the triangle: 12(6)(4) = 12 cm2 The triangle is the shape that repeats.
  2. The prism length is the height of the solid: 10 cm. How far the triangle repeats.
  3. V = Bh = 12 × 10 = 120 cm3 Base area times height.

120 cm3.

Cylinder, both measures

A can has radius 3 cm and height 8 cm. Find its volume and surface area. Use π ≈ 3.14.

  1. Volume: π r2 h = π (9)(8) = 72π ≈ 226.1 cm3 Square the radius first, then multiply.
  2. Two circles: 2π r2 = 2π(9) = 18π Top and bottom.
  3. The label: 2π r h = 2π(3)(8) = 48π A rectangle, circumference wide and h tall.
  4. Total: 18π + 48π = 66π ≈ 207.2 cm2 Add all the faces.

V ≈ 226.1 cm3; SA ≈ 207.2 cm2.

Backwards from volume

A rectangular tank holds 1,680 cm3. Its base is 14 cm by 10 cm. How deep is the water when full?

  1. Base area: 14 × 10 = 140 cm2 That is B.
  2. 1680 = 140h Put the known volume into V = Bh.
  3. h = 1680 ÷ 140 = 12 Divide both sides by 140.

12 cm deep.

Words to know

Volume
Space inside, in cubic units.
Surface area
Total area of all the outside faces, in square units.
Prism
A solid with two identical parallel bases joined by rectangles.
Lateral area
The side faces only, not counting the two bases.

Mistakes to watch for

Mixing up volume and surface area. Volume fills, surface area wraps. The units tell you instantly which you computed: cm3 fills a space, cm2 covers one.

Using the diameter in π r2. It must be halved first — and because r is squared, forgetting makes the answer four times too big rather than twice.

Forgetting a face in surface area. A box has six, not four: the top and base are easy to lose because you cannot see both at once in a drawing. Unfold it and count.

Doubling the volume when you double a side. Doubling every edge multiplies the volume by eight, since all three dimensions grew. A box twice as tall only doubles it.

Hiero of Syracuse suspected his goldsmith had mixed silver into a gold crown. Archimedes is said to have solved it in the bath, noticing the water rise.

Silver is less dense than gold, so a cheated crown of the right weight would take up more space — and displace more water.

Whether he really ran through the streets shouting eureka is anyone's guess, but displacement is still how you measure the volume of an awkward shape.

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