Mathematics II · Calculator allowed

Surface Area and Volume of 3D Shapes

Volume tells you how much fits inside a shape; surface area tells you how much wrapping covers the outside. Here is how to find both, one formula at a time.

Picture a moving box, a soup can, an ice-cream cone, and a pyramid. Each one holds a certain amount of space inside it (its volume), and each one has an outer skin you could paint or gift-wrap (its surface area). On the CAEC calculator section these show up as real-world problems, filling a tank, wrapping a parcel, pouring concrete, so let's get comfortable with both.

Good news: the volume formulas you need are printed on the formula sheet you are given during the test, so you do not have to memorize them. Your job is to know which formula fits the shape and to plug the numbers in carefully. Grab your calculator and let's go.

Volume vs. surface area: the key difference

Volume is the space inside a solid, how much it can hold. Because you are measuring length, width, and height (three directions), volume is always in cubic units: cm³, m³, in³, and so on.
Surface area is the total area of all the outside faces added together, how much paper would wrap it. Because area covers a flat region (two directions), surface area is in square units: cm², m², in².

Quick check on units: if a question asks how much water a tank holds, that is volume (cubic). If it asks how much paint or wrapping you need, that is surface area (square). Reading the units the question expects is half the battle.

The rectangular prism (a box)

A rectangular prism is a box shape with a length, a width, and a height. It is the friendliest solid to start with.

Volume (on the formula sheet): V = length × width × height. Just multiply the three dimensions.
Surface area is the sum of all six faces. The faces come in matching pairs (top and bottom, front and back, two sides), which gives SA = 2(lw) + 2(lh) + 2(wh).

Worked example #1: volume and surface area of a box

Use the box above: length 8 cm, width 4 cm, height 5 cm. We will find the volume first, then the surface area.

Volume: multiply the three dimensions: 8 × 4 × 5 = 160 cm³.
Surface area: find the three different faces, double each, and add: 2(8×4) + 2(8×5) + 2(4×5).

Volume:
V = l × w × h
V = 8 × 4 × 5
V = 160 cm³        (cubic, space inside)

Surface area:
SA = 2(lw) + 2(lh) + 2(wh)
SA = 2(8×4) + 2(8×5) + 2(4×5)
SA = 2(32)  + 2(40)  + 2(20)
SA = 64 + 80 + 40
SA = 184 cm²       (square, outside skin)

Answers: V = 160 cm³ and SA = 184 cm². Notice the volume is cubic and the surface area is square, writing the correct unit is part of getting it right.

The cylinder (a can)

A cylinder is a tube with two circular ends, like a soup can or a water tank. Its volume formula uses the area of the circular base (πr²) and multiplies by the height.

Volume (on the formula sheet): V = π × r² × h. Find the area of the round base, then stack it up to the height.
The r is the radius, the distance from the centre of the circle to the edge, which is half the diameter. If a problem gives you the diameter, halve it first.
For π, use the π button on your calculator, or 3.14 if you are estimating.

Worked example #2: volume of a cylinder

Use the can above: radius 3 cm, height 10 cm. Square the radius first, then multiply by π and by the height.

Square the radius: r² = 3² = 9.
Multiply through: π × 9 × 10 = 90π.
Calculator step: 90 × π ≈ 282.7 cm³.

V = π × r² × h
V = π × 3² × 10
V = π × 9 × 10       (square the radius first)
V = 90π
V ≈ 282.7 cm³        (90 × π on the calculator)

Answer: about 282.7 cm³. The big idea: do the exponent (3²) before you multiply, then let the calculator handle π.

Cones and pyramids: the "one-third" shapes

A cone (think ice-cream cone) and a pyramid both come to a single point at the top. There is a neat pattern here: each one holds exactly one third of the matching full shape with the same base and height.

Cone volume (on the formula sheet): V = ⅓ × π × r² × h. That is one third of a cylinder with the same circular base and height.
Pyramid volume (on the formula sheet): V = ⅓ × (base area) × h. For a rectangular base, the base area is length × width. It is one third of the matching prism.

Memory hook: pointy shapes (cone, pyramid) get the "⅓"; flat-topped shapes (cylinder, prism) do not. The h in these formulas is the straight-up height, not the slanted edge.

Worked example #3: volume of a cone

A cone-shaped paper cup has a radius of 4 cm and a height of 9 cm. How much water does it hold? Use the cone formula from the sheet.

Square the radius: 4² = 16.
Multiply by π and h: π × 16 × 9 = 144π.
Take one third: 144 ÷ 3 = 48, so the volume is 48π.

V = ⅓ × π × r² × h
V = ⅓ × π × 4² × 9
V = ⅓ × π × 16 × 9      (square the radius)
V = ⅓ × 144π
V = 48π
V ≈ 150.8 cm³           (48 × π on the calculator)

Answer: about 150.8 cm³. A tip: doing 144 ÷ 3 = 48 first keeps the numbers tidy before you reach for π.

Worked example #4: volume of a pyramid

A pyramid has a rectangular base 6 m long and 6 m wide, with a height of 10 m. Find its volume. First we need the area of the base, then we take one third.

Base area: a 6 by 6 rectangle is 6 × 6 = 36 m².
Times the height: 36 × 10 = 360.
Take one third: 360 ÷ 3 = 120.

V = ⅓ × (base area) × h
base area = 6 × 6 = 36 m²
V = ⅓ × 36 × 10
V = ⅓ × 360
V = 120 m³

Answer: 120 m³. Because every length was in metres, the volume is in cubic metres.

Tips that make these problems easier

Match the shape to the formula. Box → l×w×h. Can → πr²h. Pointy shapes → the same thing times ⅓. The formulas are on your sheet, so point to the right one.
Square the radius before multiplying. r² means r × r, not r × 2. Mixing these up is the most common cylinder and cone mistake.
Watch radius vs. diameter. The formula wants the radius. If you are handed the diameter, cut it in half first.
Label your units. Volume is cubic, surface area is square. The right number with the wrong unit can still be marked wrong.

Your turn: practice problems

You have a calculator and the formula sheet for these. Write out each step, keep your units straight, and check yourself after.

Find the volume of a rectangular prism with length 7 cm, width 3 cm, and height 2 cm.
Find the surface area of a box with length 5 cm, width 5 cm, and height 2 cm.
Find the volume of a cylinder with radius 5 m and height 4 m. Leave it in terms of π, then round to one decimal place.
Find the volume of a cone with radius 6 cm and height 7 cm. Round to one decimal place.

Tap to reveal the answers

1. V = l × w × h = 7 × 3 × 2 = 42 cm³.
2. SA = 2(lw) + 2(lh) + 2(wh) = 2(5×5) + 2(5×2) + 2(5×2) = 2(25) + 2(10) + 2(10) = 50 + 20 + 20 = 90 cm².
3. V = π × r² × h = π × 5² × 4 = π × 25 × 4 = 100π ≈ 314.2 m³.
4. V = ⅓ × π × r² × h = ⅓ × π × 6² × 7 = ⅓ × π × 36 × 7 = ⅓ × 252π = 84π ≈ 263.9 cm³.

Why this matters for the CAEC

Surface area and volume sit in the Geometry & Measurement part of the calculator math section. The volume formulas are handed to you on the formula sheet, the test is checking that you can pick the right one, plug in carefully, and label cubic or square units. A little practice makes these some of the most reliable points on the whole test.

Want more practice like this? Our CAEC math guide and the CAEC Ready Workbook are packed with worked examples and practice questions, or start with a free math sample to test yourself.

Disclaimer

This article is a general math tutorial for study purposes. CAEC Ready is an independent study resource and is not affiliated with or endorsed by any government, ministry of education, or official CAEC testing provider.