Printable · GCSE Foundation · ages 14-16
Geometry and measures worksheet — GCSE Foundation
Fifteen questions across the geometry and measures statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
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Geometry and measures worksheet — GCSE Foundation
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- 1.The diagram shows a cuboid. Work out the area of its front elevation, in square centimetres.
- 2.Angle ABC is 130°. The line BF divides angle ABC into two equal parts. Work out the size of angle FBC.
- 3.The three angles of a triangle are 10°, 58° and 112°. Write down the name that describes this triangle.
- 4.A solid pyramid has a square base of side 6 cm. Its apex is directly above the centre of the base, at a vertical height of 8 cm. Work out the volume of the pyramid.
- 5.Triangle ABC has a right angle at B, hypotenuse AC = 13 cm and AB = 5 cm. Triangle DEF has a right angle at E, hypotenuse DF = 13 cm and DE = 5 cm. Which condition proves the two triangles are congruent?
- 6.A zip-wire is fixed at an angle of 60° to the horizontal ground. The zip-wire is 12 m long. Using the exact value of cos 60°, work out the exact horizontal distance it covers.
- 7.A cube has all of its edges the same length. How many planes of symmetry does it have?
- 8.A parallelogram has an area of 54 cm² and a base of 9 cm. Work out its perpendicular height.
- 9.A rectangle has vertices at (2, 1), (9, 1), (9, 5) and (2, 5). Work out the area of the rectangle.
- 10.A carpenter measures two triangular braces. Brace 1 has a 55° angle, a 65° angle, and, opposite the 55° angle, a side of 8 cm. Brace 2 has matching 55° and 65° angles and, opposite its 55° angle, a side of 8 cm. The carpenter says that because the equal 8 cm side is not between the two equal angles in either brace, he cannot be sure the two braces are the same size. Is the carpenter correct?
- 11.Two similar triangles have lengths in the ratio 2 : 5. The sides of the smaller triangle are 4 cm, 6 cm and 8 cm. Work out the length of the longest side of the larger triangle.
- 12.Every diameter of a circle is an example of a more general type of straight line joining two points on the circle. Which term names that more general type of line?
- 13.A robotic arm's tip starts at the point (12.5, −4.25) on a grid measured in centimetres. It moves by the vector to pick up a component, then by the vector to place it. Work out the coordinates of the tip after both moves.
- 14.The diagram shows a prism resting on a table. Which shape is its front elevation, the view looking at it from directly in front?
- 15.A stained-glass window panel is designed as a kite. Two adjacent sides are each 25 cm, and the other two adjacent sides are each 40 cm. A frame is fitted around the whole panel. The frame costs £2.50 per metre, sold only in whole metres. Work out the total cost of the frame.
Answer key
- (c) 24 cm² — Method: the front elevation of a cuboid is a rectangle formed by the cuboid's length and its height, so its area is length × height. Working: 6 cm × 4 cm = 24 cm². Answer: 24 cm². The distractors: 12 cm² comes from using width × height (3 × 4) instead of length × height, mistaking the side elevation's dimensions for the front's. 18 cm² comes from using length × width (6 × 3), which gives the area of the plan view instead of the front elevation. 20 cm² comes from finding the perimeter of the front face instead of its area: 2 × (6 + 4) = 20.
- (a) 65° — Method: a line that divides an angle into two equal parts gives each part half of the original angle, so halve 130°. Working: 130 ÷ 2 = 65. Answer: 65°. The distractors: 130° is the whole of angle ABC, written down without halving it; 50° comes from working out 180 − 130, using the angles on a straight line instead of dividing the angle in two; 32.5° comes from dividing by 4 instead of by 2, as though the line split the angle into four equal parts.
- (c) Scalene — Method: equal angles in a triangle sit opposite equal sides, so compare the three angles with each other. Working: 10°, 58° and 112° are all different, so no two sides of the triangle are equal either, and a triangle with no equal sides is scalene. Answer: scalene. The distractors: isosceles is chosen by candidates who pair up the two acute angles, 10° and 58°, as base angles without checking that they are actually equal; equilateral is chosen by candidates who check that the three angles add to 180° and take that as meaning the triangle is regular; right-angled is chosen by candidates who see that 112° is larger than 90° and classify the triangle as containing a right angle, when in fact none of the three angles is 90°.
- (b) 96 cm³ — Method: the volume of a pyramid is one third of the base area multiplied by the vertical height. Work out the area of the square base, multiply by the height, then divide by 3. Working: the base area is 6 × 6 = 36 cm², then 36 × 8 = 288, and 288 ÷ 3 = 96. Answer: 96 cm³. The distractors: 288 cm³ comes from multiplying the base area by the height and forgetting the one third, which is the volume of a cuboid with the same base and height; 144 cm³ comes from halving that 288 instead of taking a third of it; 16 cm³ comes from using the base edge of 6 cm in place of the base area, (6 × 8) ÷ 3.
- (b) RHS - right angle, hypotenuse and one side equal — A right angle, the hypotenuse (13 cm) and one other side (5 cm) are equal in both triangles, so this is RHS. SAS would need the equal angle to be the one INCLUDED between the two equal sides, but the right angle at B is not between AB and the hypotenuse AC — it is opposite the hypotenuse instead, so SAS does not apply directly here. SSS needs all three sides, but only two sides are stated. ASA needs two angles, but only one angle (the right angle) is given.
- (d) 6 m — The horizontal distance is adjacent to the 60° angle and the zip-wire is the hypotenuse, so horizontal distance = 12 × cos 60° = 12 × 1/2 = 6 m. 6√3 m comes from using sin 60° = √3/2 instead of cos 60°, which would give the vertical drop, not the horizontal distance. 4√3 m comes from treating 12 as the side adjacent to a tangent ratio and dividing by tan 60° = √3. 24 m comes from dividing 12 by cos 60° instead of multiplying by it.
- (a) 9 — A cube has 9 planes of symmetry in total: 3 that pass through the middles of pairs of opposite faces, and 6 more that pass through pairs of opposite edges diagonally. A candidate who counts only the 3 face-to-face planes — which is correct for a cuboid with three different edge lengths, but forgets that a cube's equal edges create 6 more diagonal planes — answers 3. A candidate who counts only the 6 diagonal planes and forgets the 3 face-to-face ones answers 6. A candidate who confuses the number of planes of symmetry with the number of edges on a cube answers 12. The correct total for a cube is 9.
- (c) 6 cm — For a parallelogram, area = base × height, so height = area ÷ base = 54 ÷ 9 = 6 cm. 12 cm comes from using the triangle's reverse formula, height = 2 × area ÷ base, which does not apply to a parallelogram. 45 cm comes from subtracting the base from the area, 54 − 9, instead of dividing. 486 cm comes from multiplying the area by the base, 54 × 9, instead of dividing.
- (b) 28 — The width of the rectangle is the difference in x-coordinates, 9 − 2 = 7, and the height is the difference in y-coordinates, 5 − 1 = 4. The area is width × height = 7 × 4 = 28. 22 comes from using the perimeter formula, 2 × (7 + 4), instead of the area formula. 35 comes from multiplying 7 by 5 instead of 4, misreading one of the y-coordinates. 63 comes from multiplying 9 by 7, using an x-coordinate instead of the height.
- (d) No — third angle is also fixed — Since both braces have angles of 55° and 65°, their third angles must both be 60°, because angles in a triangle sum to 180°. All three angles now match, so the braces have the same shape. Both 8 cm sides lie in the same position relative to those angles — opposite the 55° angle in each brace — so one matching pair of corresponding sides fixes the size as well, exactly as ASA or AAS would. The braces are therefore guaranteed to be congruent and the carpenter is incorrect: 'No — third angle is also fixed' is correct. 'Yes — side must be included' is wrong because the side does not have to lie physically between the two named angles; once the third angle is fixed, a corresponding equal side anywhere is enough. 'No — any two angles enough alone' is wrong because two equal angles with no side length at all would only show the triangles are similar, not congruent. 'Yes — third angle may differ' is wrong because the third angle is fixed at 60° by the angle sum and cannot vary.
- (b) 20 cm — Method: corresponding sides of similar triangles are in the same ratio, and the longest side of one triangle corresponds to the longest side of the other; a ratio of 2 : 5 means each length is multiplied by 5 ÷ 2 = 2.5 going from the smaller triangle to the larger one. Working: the longest side of the smaller triangle is 8 cm, so the matching side of the larger triangle is 8 × 2.5 = 20. Answer: 20 cm. The distractors: 10 cm comes from scaling the shortest side, 4 cm, instead of the longest; 40 cm comes from multiplying by 5 and forgetting to divide by 2; 3.2 cm comes from multiplying by 2 ÷ 5 instead of 5 ÷ 2, which scales from the larger triangle down to the smaller one.
- (c) chord — A diameter is a straight line joining two points on the circle that happens to pass through the centre. The general term for any straight line joining two points on a circle is a chord — every diameter is a chord, but not every chord is a diameter. Calling it a radius is wrong because a radius runs from the centre to just ONE point on the circle, not between two points on the circle. A tangent touches the circle at only one point, so it cannot be a diameter at all. An arc is a curved part of the circle's edge, not a straight line.
- (d) (11, 0.75) — Add the moves to the starting point one component at a time. x: 12.5 + (−3.75) + 2.25 = 11; y: −4.25 + 6.5 + (−1.5) = 0.75, giving (11, 0.75). (8.75, 2.25) stops after the first move only and never applies the second vector. (6.5, 3.75) comes from subtracting the second vector instead of adding it. (0.75, 11) comes from swapping the final x-coordinate and y-coordinate.
- (b) Triangle — Method: the front elevation is the outline you see looking straight at the solid's front face. Working: this solid's front face is one of its two identical triangular ends, so looking directly at the front shows exactly that triangular outline. Answer: triangle. The distractors: rectangle comes from picturing the side view instead of the front view — looking along the ridge of the prism, the sloping roof faces project as a rectangle. Trapezium comes from misreading the two sloping roof faces as if together they formed a single four-sided shape. Pentagon comes from tracing round the outline of the whole sketch as it is drawn on the page, which is a five-sided shape, instead of drawing only the view seen looking straight at the front of the solid.
- (b) £5.00 — Perimeter of the kite = 25 + 25 + 40 + 40 = 130 cm = 1.3 m. The frame is sold only in whole metres, so 2 m must be bought. Cost = 2 × £2.50 = £5.00. A student who buys the exact 1.3 m instead of rounding up to whole metres gets 1.3 × £2.50 = £3.25. A student who never converts the perimeter from centimetres to metres and costs 130 × £2.50 gets £325.00.9 m, which rounds up to 3 whole metres, costing 3 × £2.50 = £7.50.
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