lesson
A cross-section repeats along the prism
1 · Learn
A prism's volume is its constant cross-sectional area multiplied by its perpendicular length. Surface area instead adds the areas of all exposed faces. Distinguish these quantities before selecting a method.
Use consistent units, and calculate the cross-section correctly. A triangular cross-section uses half base times perpendicular height; the sloping side is not automatically the height.
2 · Worked example
A triangular prism has cross-sectional base 6 cm and perpendicular height 4 cm, and length 10 cm. Its volume is (6 × 4 / 2) × 10 = 120 cm³.
3 · Your turn
Find the volume of a prism with constant cross-sectional area 15 cm² and length 8 cm.
Check your answer
120 cm³.
4 · Apply your learning
Sketch a net for a right triangular prism 8 cm long. Each end is a right triangle with perpendicular sides 3 cm and 4 cm and hypotenuse 5 cm. Include both triangular ends and rectangles 8 by 3, 8 by 4 and 8 by 5 cm. Compare the area of all faces with the volume; explain why knowing only the volume would not specify this net.
Adult guidance and safety
Require reasoning and a check. Provide accurate graphs, construction instruments and varied practice. Annual KS3 allocation is proposed; do not treat these short models as a complete assessed programme.
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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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