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.A regular hexagon is divided into six identical triangles by joining its centre to each of the six vertices. Work out the size of the angle of one of these triangles at the centre of the hexagon.
- 2.a is the column vector with top number 3 and bottom number −2. Work out 4a, giving your answer as a column vector in the form (top, bottom).
- 3.A circle has circumference 31.4 cm. Using π = 3.14, work out the radius of the circle.
- 4.Write down how many right angles a right-angled triangle has.
- 5.In quadrilateral ABCD the two sides AB and BC are each 6 cm long and meet each other at B. The two sides CD and DA are each 9 cm long and meet each other at D. No side of ABCD is parallel to any other side. Write down the mathematical name of this quadrilateral.
- 6.A bearing is described as being exactly halfway between north and east, measured clockwise. What is the three-figure bearing?
- 7.A sign-maker is told a metal bracket outline must have two pairs of adjacent sides equal in length (not opposite pairs), exactly one pair of opposite angles equal, and exactly one line of symmetry. Which quadrilateral matches this description?
- 8.Work out the gradient of the line segment joining A(2, 3) and B(6, 11).
- 9.Write down the exact value of sin 45°.
- 10.A ship sails 12 km on a bearing of 070° from a harbour to a buoy. It then sails a further 9 km, still on a bearing of 070°, from the buoy to a lighthouse. How far is the lighthouse from the harbour, in kilometres?
- 11.A furniture catalogue drawing uses a scale of 1 cm to 1.5 m. A sofa in the drawing is measured as 7 cm long. What is the real length of the sofa, in metres?
- 12.The diagram shows the plan, front elevation and side elevation of a solid. Which of these solids matches all three views?
- 13.A designer creates a repeating tile pattern. Each tile is translated from the one before it by the column vector with top number 4.5 and bottom number −2.5 (in centimetres). The first tile has its bottom-left corner at (1.5, 3). Work out the coordinates of the bottom-left corner of the third tile.
- 14.A sailmaker cuts two triangular sail panels from a pattern. Panel X has a 10 m side, with angles of 50° and 75° at its two ends. Panel Y also has a 10 m side, with angles of 50° and 75° at its two ends, in the same arrangement. The sailmaker wants to check the panels will match exactly, without cutting a third measurement. Which condition proves the two panels are congruent?
- 15.A square tile has sides of length 6 cm. Work out the area of the tile.
Answer key
- (d) 60° — Method: the six angles at the centre together make one complete turn of 360°, and because the hexagon is regular they are all equal, so divide 360° by 6. Working: 360 ÷ 6 = 60. Answer: 60°. The distractors: 120° is the interior angle of a regular hexagon, 720 ÷ 6, which is the angle at a vertex and not the angle at the centre; 45° comes from dividing 360 by 8, treating the hexagon as though it had eight sides; 30° comes from halving the angle at the centre, as though each of the six triangles were split again by a line of symmetry.
- (b) (12, −8) — Method: to multiply a column vector by a number, multiply every part of the vector by that number. Working: top number 4 × 3 = 12; bottom number 4 × (−2) = −8. Answer: 4a = (12, −8). A candidate who adds 4 to each part instead of multiplying gets (7, 2). A candidate who multiplies only the top number by 4 and leaves the bottom number unchanged gets (12, −2). A candidate who multiplies only the bottom number by 4 and leaves the top number unchanged gets (3, −8).
- (c) 5 cm — Diameter = circumference ÷ π, so 31.4 ÷ 3.14 = 10 cm, and the radius is half the diameter, so 10 ÷ 2 = 5 cm. 10 cm is the diameter itself, given as the radius by forgetting the final halving step. 15.7 cm comes from halving the circumference, 31.4 ÷ 2 = 15.7, and stopping there, treating half the circumference as the radius without ever dividing by π. 2.5 cm comes from halving the correct radius again, effectively dividing by 2 twice instead of once.
- (c) 1 — Method: a right angle measures 90°, the three angles of any triangle add up to 180°, and a right-angled triangle is defined as a triangle that contains a right angle. Working: taking one right angle out of the total leaves 180° − 90° = 90° to be shared between the other two angles, so both of those must be acute; a second right angle would use the whole of that remaining 90° and leave nothing at all for the third angle, which is impossible. The definition therefore fixes the count at exactly one. Answer: 1. The distractors: 2 comes from counting the two sides that form the right angle instead of counting the angles themselves; 3 comes from reading the name as a description of the whole triangle, so that all three of its angles are taken to be right angles, which would need an angle sum of 3 × 90° = 270°; 0 comes from over-applying the angle sum — a candidate who works out that 90° + 90° = 180° leaves nothing for a third angle can conclude from that alone that no triangle may contain a right angle at all.
- (b) Kite — Method: name a quadrilateral by matching what is given — which sides are equal, whether those equal sides lie next to each other or opposite each other, and whether any sides are parallel — against the definitions of the special quadrilaterals. Working: the two 6 cm sides meet at B and the two 9 cm sides meet at D, so each pair of equal sides is a pair of neighbours rather than a pair of opposites, and the stem rules out any parallel sides. The quadrilateral with two pairs of equal adjacent sides and no parallel sides is a kite. Answer: kite. The distractors: a rhombus is chosen by candidates who see two pairs of equal sides and read that as all four sides being equal, which the two different lengths of 6 cm and 9 cm rule out; a parallelogram is chosen by candidates who remember that a parallelogram has two pairs of equal sides but not that in a parallelogram the equal sides are the opposite ones, and who pass over the statement that nothing is parallel; an isosceles trapezium is chosen by candidates who notice that the shape is symmetrical about the line BD and treat symmetry on its own as the mark of a trapezium, when a trapezium needs a pair of parallel sides.
- (b) 045° — North is 000° and east is 090°; halfway between them, measured clockwise, is 045°. Choosing 090° gives east itself, not the midpoint between north and east. Choosing 135° gives the midpoint between east (090°) and south (180°) instead of between north and east. Choosing 225° is the back bearing of 045° (045° + 180°), not the bearing itself.
- (a) Kite — Method: check each named quadrilateral's properties against the three facts given, one at a time. Working: a kite has two pairs of adjacent sides equal (not opposite pairs), one pair of opposite angles equal (the two angles where an unequal pair of sides meet), and exactly one line of symmetry — matching all three facts. Options: a rhombus does have equal adjacent sides, but all four of its sides are equal, both pairs of its opposite angles are equal, and it has two lines of symmetry rather than exactly one; a parallelogram has its opposite sides equal rather than adjacent pairs, both pairs of opposite angles equal, and no line of symmetry at all; a trapezium does not generally have any pair of equal adjacent sides or a line of symmetry. Answer: kite.
- (a) 2 — Gradient = (change in y) ÷ (change in x) = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2. "0.5" comes from dividing the change in x by the change in y the wrong way round: 4 ÷ 8. "8" is only the change in y, forgetting to divide by the change in x at all. "−2" comes from a sign error, as if the y-coordinate had decreased rather than increased.
- (b) √2/2 — sin 45° = √2/2 (the same value as 1/√2, written with a rational denominator) — one of the exact values you need to know. √3/2 is the exact value of sin 60° and of cos 30°, not sin 45°. 1/2 is the exact value of sin 30° and of cos 60°. 1 is the exact value of sin 90°.
- (b) 21 — Both legs of the journey are on the same bearing, 070°, so the ship travels in one straight line the whole way and the distances simply add: 12 + 9 = 21 km. A candidate who subtracts instead of adding gets 12 − 9 = 3 km. A candidate who multiplies the two distances gets 12 × 9 = 108. A candidate who assumes the ship changed direction and treats the two legs as the sides of a right-angled triangle works out √(12² + 9²) = √225 = 15 km — but the bearing does not change, so there is no triangle and no hypotenuse to find. Because the ship stays on one straight line, the total distance is 21 km.
- (b) 10.5 — Method: multiply the drawing length by the scale factor to get the real length. Working: 7 cm × 1.5 = 10.5 m. A student who answers 8.5 has added the scale factor to the drawing length instead of multiplying (7 + 1.5). A student who answers 14 has rounded the scale factor up to 2 before multiplying. A student who answers 3.5 has divided the drawing length by 2 instead of multiplying it by 1.5. Answer: 10.5 m.
- (b) A triangular prism standing on its triangular end — Method: a rectangular elevation with no sloping sides means the solid keeps the same cross-section all the way from the bottom to the top; work out which solid, standing the right way up, has a triangular cross-section that stays that shape as you go higher. Working: a triangular prism standing upright on its triangular end has a triangle as its plan view, and because the cross-section is constant all the way up, both the front and side elevations are plain rectangles. Answer: a triangular prism standing on its triangular end. The distractors: a triangle-based pyramid standing on its triangular base does give a triangle as its plan view, but its cross-section shrinks towards the apex, so its front and side elevations come to a point and are triangles, not rectangles. A cuboid standing on a rectangular face is wrong because its plan view is a rectangle, not a triangle. A triangular prism lying on one of its rectangular faces is wrong because it is then the triangular end that faces the side, so its plan view is a rectangle and one of its elevations is a triangle.
- (c) (10.5, −2) — Method: the vector from the first tile to the third tile is the pattern's vector doubled, since two translations happen between them. Working: doubling (4.5, −2.5) gives (9, −5); adding this to the starting corner (1.5, 3) gives x-coordinate 1.5 + 9 = 10.5 and y-coordinate 3 − 5 = −2. Answer: (10.5, −2). A candidate who only applies the vector once, translating to the second tile instead of the third, gets (6, 0.5). A candidate who adds 2.5 instead of subtracting it in the y-coordinate gets (10.5, 8). A candidate who doubles the x-part of the vector correctly but forgets to change the y-coordinate at all gets (10.5, 3).
- (c) ASA — Method: check which condition matches two angles and the side between them, since that is all the sailmaker has measured. Working: the 10 m side lies between the 50° and 75° angles in both panels, so this is two Angles and the included Side, ASA. Options: SAS would need two sides and the angle between them, but only one side has been measured here; SSS would need three sides, but only one is known; RHS needs a right angle and a hypotenuse, and neither panel has a stated right angle. Answer: ASA.
- (c) 36 cm² — Method: the area of a square is its side length multiplied by itself. Working: 6 × 6 = 36. Answer: 36 cm². The distractors: 24 cm comes from working out the perimeter, 4 × 6, which is a length and not an area; 12 cm comes from doubling the side, 6 × 2, instead of squaring it; 18 cm² comes from halving the product, (6 × 6) ÷ 2, using the rule for the area of a triangle.
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