Similarity

• Similar shapes have equal angles and all matching lengths in the same ratio.
• Find a missing length using the scale factor, after matching the corresponding sides.
• Show that two triangles are similar with geometric reasons (equal angles).
• Areas and surface areas of similar shapes are in the ratio k² : 1.
• Volumes of similar solids are in the ratio k³ : 1.
• Solve problems and give simple explanations involving similarity.

Find k, then use it

Pick a pair of corresponding sides where both lengths are known. Their ratio is the scale factor. Every other length on the larger shape is k times its partner on the smaller shape.

Showing they are similar, then using it

Two triangles are similar if their three angles are equal (two pairs equal is enough, because the third pair then must be equal too). Give a reason for each pair: a common angle, corresponding angles, alternate angles or vertically opposite angles.

Write the triangles with matching vertices in the same order (triangle ADE and triangle ABC), then read off the corresponding sides.

Square the scale factor

If the lengths are multiplied by k, an area is multiplied by k × k = k². This also applies to the surface area of similar solids.

Cube the scale factor

A volume has three lengths multiplied together, so it scales by k³. Mass and capacity behave like volume when the material is the same. To go back from a volume ratio to a length ratio, take the cube root.

Mass, and "explain why"

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