lesson
Count dimensions in the formula
1 · Learn
A triangle's area is half base times perpendicular height. A trapezium's area is half the sum of its parallel sides times their perpendicular separation. A prism's volume is cross-sectional area times length.
Use consistent units and identify the correct height. A sloping edge is not automatically a perpendicular height. Squared and cubed units tell the reader what quantity was measured.
Derive the formula rather than only memorising it. A triangle and a congruent rotated copy fit into a parallelogram with the same base and perpendicular height, so the triangle's area is half bh. A parallelogram has area bh because a triangular piece from one end can be moved to the other to make a rectangle without changing area.
For the supplied trapezium, the parallel sides are 6 cm and 10 cm and their perpendicular separation is 4 cm. A congruent rotated copy joins it to make a parallelogram with base 6+10 = 16 cm and the SAME height 4 cm. Combined area is 64 cm², so one copy has area 32 cm². In general, two copies have area (a+b)h, giving trapezium area (a+b)h/2. Do not use the sloping edge for h.
A cuboid is a stack of rectangular layers: base area l×w repeated through height h gives lwh. The same reasoning applies to a prism with constant cross-sectional area, multiplied by its perpendicular length. A pyramid's cross-sections shrink, so that prism formula cannot simply be copied to it.
2 · Worked example
A trapezium with parallel sides 6 cm and 10 cm and height 4 cm has area (6+10) × 4 ÷ 2 = 32 cm².
3 · Your turn
A prism has cross-sectional area 12 cm² and length 7 cm. Find its volume.
Check your answer
84 cm³.
4 · Apply your learning
Draw labelled diagrams showing which measurements each formula uses and estimate before calculating.
Adult guidance and safety
Require a reasoned method and an independent check. Use accurate number lines, geometric drawings and data displays where the task needs them. These text models require varied practice and diagnostic assessment before curriculum approval.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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