GCSE Maths: Geometry & Measures — Deep Revision Workbook

Inside GCSE Maths: Geometry & Measures — Deep Revision Workbook

work missing angles on lines, at points and between parallel lines, find interior and exterior angles of polygons, calculate area, perimeter, circumference and volume, use Pythagoras’ theorem and SOHCAHTOA on right-angled triangles, then tackle an extended exam-style question with bearings and similarity. About 45 minutes.

Learning objectives

• angle and parallel-line rules — on a line, at a point, in triangles and quadrilaterals, exterior angles, and alternate, corresponding and co-interior angles;
• interior and exterior angles of polygons, using (n − 2) × 180° and the 360° exterior-angle sum, and working back to the number of sides;
• area, perimeter and circles — rectangle, triangle, parallelogram and trapezium areas, perimeter, and circle area (πr²) and circumference (πd);
• volume and surface area of cuboids, prisms and cylinders;
• Pythagoras’ theorem (a² + b² = c²), right-angled trigonometry (SOHCAHTOA), similarity and bearings, ending in an extended exam-style question.

Activities in this book

1. Section 1 — Angle & Parallel-Line Rules

   angles on a straight line add to 180°, angles at a point add to 360°, the angles in a triangle add to 180° and the angles in a quadrilateral add to 360°. The exterior angle of a triangle equals the sum of the two opposite interior angles. Between parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles add to 180°. All answers are exact whole numbers of degrees — no calculator.

2. Section 2 — Polygons: Interior & Exterior Angles

   the interior angles of a polygon with n sides add up to (n − 2) × 180°. The exterior angles of any polygon always add up to 360°, so a regular polygon’s exterior angle is 360° ÷ n, and its interior angle is 180° − the exterior angle. To find the number of sides from an angle, work back through the exterior angle.

3. Section 3 — Area, Perimeter & Circles

   area of a rectangle = base × height; triangle = ½ × base × height; parallelogram = base × height; trapezium = ½(a + b) × height. Perimeter is the total distance around the edge. For a circle, area = πr² and circumference = πd. Each circle question tells you whether to use π = 3.14 or to leave the answer in terms of π (then just give the number in front of π).

4. Section 4 — Volume, Pythagoras & Trigonometry

   volume of a prism = area of cross-section × length; volume of a cylinder = πr²h. Surface area of a cube = 6 × (side)². Pythagoras’ theorem says a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle); to find a shorter side, subtract. For angles, use SOHCAHTOA: sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent. Cylinder answers here use π = 3.14; the triple 6-8-10, 5-12-13 and 8-15-17 give exact answers.

5. Section 5 — Extended Exam-Style Challenge

   exam-style questions reward clear method as much as the final number. Set out each step, name the rule or formula, keep π where asked and round only at the end. Work through the multi-part question below on paper or on screen, showing your reasoning line by line.

Suggestions for teaching

• Name the rule. Before finding an angle, ask learners to state which rule they are using — the marks in the exam come from the reasoning.
• The polygon formulas. Model (n − 2) × 180° for the interior-angle sum and 360° ÷ n for a regular exterior angle, then work backwards to find the number of sides.
• Read the π instruction. Be explicit about when to use π = 3.14 and when to leave an answer in terms of π — both appear in GCSE papers.
• Pythagorean triples. Teach 6-8-10, 5-12-13 and 8-15-17 so learners can spot exact answers, then confirm with a² + b² = c².
• Method marks. On the Section 5 exam question, insist on full working, correct units and rounding only at the end — that is where marks are won or lost.