🧊Similar Solids & Composite Figures

Two solids are similar when one is an exact enlargement of the other: same shape, every length multiplied by the same number. That number is the scale factor, written k. Surveyor Kit up at Polygon Peaks prices every job by it — perimeter is rope, area is paint, volume is stone to carry — and the three of them do not grow at the same rate.

Start with a cube of edge 1 and double it to edge 2.

• ◆Edges: 1 becomes 2. Multiplied by k = 2.
• ◆Faces: 1 by 1 becomes 2 by 2. Area is 1 becomes 4, multiplied by k².
• ◆Whole solid: 1 by 1 by 1 becomes 2 by 2 by 2. Volume is 1 becomes 8, multiplied by k³.

An area is a length times a length, so it picks up the factor twice. A volume is a length times a length times a length, so it picks up the factor three times. That is the whole of the square-cube law, and it is why doubling a model does not double the stone.

Keep that little table in your head:

• ◆any length — an edge, a height, a radius, the perimeter — scales by k
• ◆any area — one face, or the whole surface area — scales by k²
• ◆any volume or capacity scales by k³

So if the scale factor is 2/3, the surface areas are in the ratio 4/9 and the volumes are in the ratio 8/27. A ratio question is answered as a fraction in lowest terms, exactly like 8/27, and the smaller solid always goes on top when the question says "smaller to larger".

You will also be asked to run this backwards. If two similar solids have volumes in the ratio 27/125, take the cube root of the top and the cube root of the bottom separately: k = 3/5. If their surface areas are in the ratio 9/49, take square roots: k = 3/7. And to get from one ratio to the other — areas 9/25 to volumes — there is no shortcut. Go back to k first (k = 3/5), then cube it (27/125).

Once you have k, a number comes with it. If the smaller of two similar cones holds 40 cm³ and k = 3 from smaller to larger, the larger holds 40 x 3³ = 40 x 27 = 1080 cm³. Going the other way you divide instead: the larger holds 1080, so the smaller holds 1080 / 27 = 40.

Two things make this go wrong, and both are easy to check.

• ◆Which power? Ask what the quantity is made of. Grams of paint cover a surface, so k². Litres of water fill a space, so k³. A length of trim round an edge is just a length, so k.
• ◆Which direction? The larger solid must end up with the larger number. If your answer for the big one came out smaller than the number you started with, you divided when you should have multiplied.

The scale factor does not always arrive with a label on it. Often you get two matching lengths instead — "the heights are 4 cm and 6 cm" — and you build it yourself: k = 6/4 = 3/2 going up, and volumes then scale by (3/2)³ = 27/8. A fractional k is not a special case; it works the same way.

Modelling questions are the same idea in a bigger coat. "A model is built to a scale of 1 : 20" means every real length is 20 times the model's, so real capacity is 20³ = 8000 times the model's. A model that holds 3 litres stands for a tank that holds 24000 litres. These questions often hand you a measurement you do not need — the model's height, say, when the capacity is already given. Extra information is not a hint; read what is asked and leave the spare number alone.

How to answer: a scaled quantity is exact. Type the whole number if it is one, or the single decimal place if it has one; the question tells you which to expect.

A regular polygon has all sides equal and all angles equal. Its apothem is the perpendicular distance from the centre to the middle of a side — not to a corner, which is longer.

Join the centre to every corner and a regular polygon with n sides falls apart into n identical triangles. Each triangle has base one side, s, and height the apothem, a, so each has area (1/2) x a x s. Add all n of them:

Area = n x (1/2) x a x s = (1/2) x a x (n x s)

and n x s is just the perimeter. So

Area = (1/2) x apothem x perimeter

One formula covers pentagons, hexagons, octagons and everything beyond. A hexagon of side 11 cm with apothem 9.5 cm has perimeter 6 x 11 = 66 cm and area (1/2) x 9.5 x 66 = 313.5 cm².

Run it backwards and nothing new is needed. Given the area and the apothem, double the area, divide by the apothem to get the perimeter, then divide by the number of sides to get one side.

A regular hexagon is the one case where the apothem can be built from the side on its own: apothem = (side / 2) x sqrt(3). When a question asks for that, it will tell you to use sqrt(3) = 1.732, in the same spirit as pi = 3.14.

How to answer: areas here are rounded to 1 decimal place, and the prompt says so.

A composite figure is a shape built from pieces you already know. There is no formula for it and there does not need to be one. The method is always the same three steps: split it into familiar pieces, find each piece, then add or subtract.

• ◆A window that is a rectangle with a semicircle on top: rectangle plus half a circle.
• ◆A sign shaped like a house: rectangle plus triangle.
• ◆An L-shaped slab: the whole rectangle minus the corner that was cut out.
• ◆A square tile with a quarter circle cut away: square minus a quarter of a circle.
• ◆A hopper that is a cylinder with a cone on top, or a tank with a dome: the two volumes added.
• ◆A block with a slot cut out: the block minus the slot.

Three traps account for most lost marks. The first is the fraction of a circle: a semicircle is (1/2) x pi x r², a quarter circle is (1/4) x pi x r², a hemisphere is (2/3) x pi x r³ — half of the sphere's (4/3), not a third; the third belongs to cones. The second is the diameter wearing a radius costume: when a semicircle sits on a 12 cm edge, that 12 cm is the diameter and the radius is 6. The third is adding when you should subtract: cutting a piece away makes the figure smaller, so if your answer is bigger than the shape you started with, the sign is wrong.

Composite surface area has a trap of its own. Put a square pyramid on top of a cube and the cube's top face and the pyramid's base are pressed together, so neither of them is on the outside. Paint 5 squares and 4 triangles, not 6 and 5. Where two solids meet, both hidden faces drop out.

How to answer: anything with a circle in it says "use pi = 3.14" and asks for 1 decimal place; the checker also accepts the answer you get from the pi key on a calculator. Answers with no curve in them are whole numbers.

Slice a solid with a flat plane and the flat face you expose is a cross-section. Picture the plane as a sheet of glass pushed through, and read the outline where it meets the surface.

The one rule worth memorising: a slice parallel to the base is a copy of the base, possibly smaller. So a cone or a cylinder sliced parallel to its base gives a circle; a square pyramid gives a square; a triangular prism sliced parallel to its triangular ends gives a triangle. Every other plane you have to picture on its own:

• ◆a cylinder cut straight down through the centre: a rectangle;
• ◆a cone cut straight down through its vertex: a triangle;
• ◆a square pyramid cut straight down through the apex: a triangle — but the same cut missing the apex leaves a trapezoid, because the sloping edges never meet;
• ◆a cone or cylinder cut at a slant: an ellipse, because the plane travels further one way than the other;
• ◆a sphere, cut any way at all: a circle.

Density ties measurement to the real world: density = mass / volume, so mass = density x volume. Iron at 7.8 grams per cubic centimetre means every single cubic centimetre carries 7.8 g, so find the volume first and multiply at the end. Population density is the same sentence about people: people per square kilometre, found by dividing the population by the area — and the area, not the distance round the outside.

Packing looks like density and is not. To ask how many 4 cm by 7 cm by 6 cm boxes fit inside a 19 cm by 16 cm by 15 cm crate, work along each direction separately and round down every time: 19 / 4 gives 4 boxes, 16 / 7 gives 2, 15 / 6 gives 2, so 4 x 2 x 2 = 16. Dividing the volumes would say 27, and 27 is wrong: it quietly assumes the leftover strips at the ends can be poured together into more boxes, and rigid boxes cannot be poured.

How to answer: a cross-section is a multiple choice — pick the shape's name. A mass, a population density and a box count are typed in, whole numbers or one decimal place as the question says.