Area & Volume: Scaling, Shared Fractions & Displacement

Area on a grid is a count: every whole square is one square unit, and a triangle that cuts a square from corner to corner is exactly half. Count the whole squares and tick them as you go, then count the half-squares and pair them up. Five whole squares and four halves make seven square centimetres. Shapes on the test are drawn to tempt you into skipping a square or counting a half as a whole, so the ticking matters. Keep area and perimeter apart in your head: area is a number of squares, perimeter is a distance around the edge.

When every side of a shape is doubled, the area is not doubled — it is multiplied by four, because the shape is twice as wide and twice as tall, and two times two is four. Triple the sides and the area is nine times as big; the rule is that sides scale once and area scales twice. You can see it on the grid: each single square of the small shape becomes a two-by-two block in the big one. The tempting wrong answer, area doubled, is always among the options. For solids the same idea goes one step further: double every edge of a box and it holds eight times as much.

A favourite item shows two identical squares: one cut into sixty-four small squares, the other into eight triangles, with some of each shaded, and asks how many more small squares to shade so the shaded areas match. The pieces are different sizes, so you cannot compare counts directly. Because the squares are identical, eight triangles equal sixty-four small squares, so one triangle is worth eight. Three shaded triangles are twenty-four small squares; fourteen are already shaded, so ten more are needed. The whole question is a conversion between two units of the same whole.

When an object sinks in a jug, the water rises by exactly the object's volume. Displacement questions give a starting level, the level after a few identical objects, and then add more. Find the rise per object first: two blocks took the water from one hundred to one hundred and fifty, so each block is twenty-five millilitres. Five more blocks add five times twenty-five, one hundred and twenty-five, to the current level of one hundred and fifty, giving two hundred and seventy-five. The common slips are stopping at the rise instead of the new level, and adding to the original level instead of the current one.

The last idea is invariance. If you cut a shape into pieces and move them about, the area cannot change, because every square is still there; but the perimeter can, because pieces that used to touch along a long edge might now touch at a corner, leaving more edge on the outside. A six-by-two rectangle has area twelve and perimeter sixteen; cut it in half and shift one half sideways and the area is still twelve while the perimeter grows to twenty. Questions exploit the feeling that a shape which looks bigger has more area. Count squares and it never fools you. Volume works the same way: rebuild a solid from its cubes any way you like and the volume is unchanged. Quiz and exercises now!