Rectangles and triangles can be measured directly because their edges are straight. This unit measures shapes that are not so obliging. A trapezium has two sides that stay the same distance apart and two that do not. A regular polygon has too many sides to count squares. A circle has no straight edge at all.
Each shape is turned into one you already know. Turn a copy of a trapezium and it joins to make a parallelogram. Join the centre of a regular polygon to its corners and it falls into triangles. Cut a circle into thin sectors and lay them side by side and they make a rectangle. Then the unit goes into three dimensions: a prism is any shape carried along a length, so its surface is two ends and a rectangle, and its volume is the area of one end times the length.
Lessons 2 to 4 use a scientific calculator, because the numbers have decimals. Pi is 3.14 throughout, so every answer on the page can be checked exactly.
The lessons, in order
- Lesson 1 · Area of a trapezium Area of a trapezium by copying and turning, the middle line as the average of the parallel sides, the height at a right angle, and finding a missing height or parallel side. Turn a copy of a trapezium to make a parallelogram, then lean the top edge and watch the area stay put. No calculator
- Lesson 2 · Regular polygons: perimeter and area Perimeter of a regular polygon, the apothem, area as n triangles (n × ½ × s × a), the angle at the centre, and finding a missing side or apothem. Cut a regular polygon into triangles at its centre, then lay the triangles in a row. Scientific calculator
- Lesson 3 · Circumference and π Circumference as a little over three diameters, π as the ratio C ÷ d, C = πd = 2πr, answers in terms of π, and finding a missing diameter, turns of a wheel and part-circle outlines. Lay a circle's edge along a line and count diameters, then roll a wheel and count turns. Scientific calculator
- Lesson 4 · Area of a circle Area of a circle as πr² by rearranging sectors, radius against diameter, answers in terms of π, area of a ring and of a circle with a rectangle, and how area scales with the radius. Cut a circle into sectors and build a strip, then compare two circles to the same scale. Scientific calculator
- Lesson 5 · Nets and surface area of prisms A prism as a cross-section carried along a length, nets of triangular and trapezoidal prisms, surface area as two ends plus perimeter times length, and finding a missing length. Build a net face by face, then unwrap the sides of a prism into one rectangle. No calculator
- Lesson 6 · Volume of prisms Volume of a prism as cross-section area times length, cross-sections that are triangles, trapeziums, regular polygons and L shapes, cubic units and litres, finding a missing length, and scaling. Slice a prism into layers and count them, then multiply every length and watch area and volume grow differently. No calculator
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.