Lessons · Teaching · volume and surface area
Volume fills, surface area wraps
Volume is the number of unit cubes that fill a solid, found for a box by multiplying length by width by height. Surface area is the total area of all its faces, found by adding them. One is a filling, the other a wrapping, and they answer different practical questions.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
In beta. This lesson was written for Hone and has not yet been checked by a state certification officer. Practice material, not professional advice. What that means.
What it is for
Which one you want is decided by the situation and never by the numbers: how much water fills the tank is volume, how much paint covers it is surface area. Naming the question in words before reaching for a formula is the habit that prevents the whole error.
How to think about it
Say what is being asked in ordinary words first. Filling, holding or capacity means volume, in cubic units. Covering, painting or wrapping means surface area, in square units. For a box, a surface area is three pairs of identical faces.
Worked example
A box 4 by 3 by 2: volume is 4 x 3 x 2 = 24 cubic unitsTwenty-four unit cubes fit inside it.
Its faces come in three pairs: 4x3, 4x2 and 3x2Twelve, eight and six, each appearing twice.
Surface area = 2(12 + 8 + 6) = 52 square unitsAdd one of each pair, then double. Fewer chances to miss a face.
Doubling every edge multiplies volume by eight, not by twoBecause all three dimensions doubled. This surprises almost everybody.
Your turn
Name the unit kind that volume is measured in.
Volume is measured in units
Solve one, graded on the server
The trap
Giving a volume in square units or an area in cubic ones. The unit is not decoration: it says which quantity was computed, and an examiner reading square centimetres for a capacity knows the wrong formula was used.