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.
Answer key: Geometry and measures worksheet — GCSE Foundation
MathsUKwww.geekhero.co.uk
- (a) 8.1 m — Multiply the length on the plan by the scale factor: 4.5 × 1.8 = 8.1, so the real length is 8.1 m. Choosing 6.3 m adds the two numbers instead of multiplying them (4.5 + 1.8 = 6.3). Choosing 2.5 m divides the plan length by the scale factor the wrong way round (4.5 ÷ 1.8 = 2.5) instead of multiplying. Choosing 9.0 m rounds the scale factor up to 2 before multiplying (4.5 × 2 = 9.0), losing the accuracy the 1.8 was giving.
- (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) 3/4 — 1 litre = 1000 ml, so 750 ml is 750/1000 of a litre. Dividing both the numerator and denominator by 250 simplifies this to 3/4. Writing the fraction upside down, as the litre out of the 750 ml, gives 4/3. Finding the fraction of the litre that is NOT filled, 250/1000, gives 1/4. Dividing the numerator by 250 but the denominator by only 100, an inconsistent simplification, gives 3/10.
- (c) 15 cm — Method: going from the smaller tile to the larger one, every length is multiplied by the scale factor 5 ÷ 3. Working: 9 ÷ 3 = 3, and 3 × 5 = 15. Answer: 15 cm. The distractors: 11 cm comes from adding the difference between the parts of the ratio, 5 − 3 = 2, to the 9 cm side, treating a ratio as a gap rather than a multiplier; 45 cm comes from multiplying by 5 and forgetting to divide by 3; 27 cm comes from multiplying by 3, which is the part of the ratio belonging to the smaller tile.
- (a) (9, −5) — Method: multiply every part of q by 2, then subtract the matching part from p. Working: 2q = (−4, 6); p − 2q gives top 5 − (−4) = 9 and bottom 1 − 6 = −5. Answer: p − 2q = (9, −5). A candidate who forgets to double q first, working out p − q instead, gets (7, −2). A candidate who doubles p instead of q, working out 2p − q, gets (12, −1). A candidate who adds 2q instead of subtracting it gets (1, 7).
- (b) (1, 6) — Method: when a square is set square-on to the grid, so that its sides run parallel to the axes, each vertex shares its x-coordinate with one neighbour and its y-coordinate with the other, and the missing vertex then borrows one coordinate from each of the two vertices it is joined to; so the first job is to check from the given points that the sides really do run parallel to the axes. Working: A(1, 2) and B(5, 2) share y = 2, so AB is a horizontal side; B(5, 2) and C(5, 6) share x = 5, so BC is a vertical side, which confirms that this square lies square-on to the axes and that the rule may be used. In square ABCD the vertex D is joined to C and to A. DC must be horizontal like AB, so D takes the y-coordinate of C, which is 6; DA must be vertical like CB, so D takes the x-coordinate of A, which is 1. D is therefore (1, 6), and checking confirms every side is 4 long. Answer: (1, 6). The distractors: (1, 5) comes from lifting the first number out of each of A and C, pairing the x-coordinate of A with the x-coordinate of C; (6, 1) comes from finding the right two numbers but writing them the wrong way round, height before sideways position; (9, 6) comes from stepping a further 4 to the right from C instead of closing the square back to the column A stands in.
- (a) (−1, 1) — Translating by $\binom{−2}{3}$ subtracts 2 from every x-coordinate and adds 3 to every y-coordinate. This gives image vertices (−1, 4), (3, 4), (3, 7) and (−1, 7). The point (−1, 1) is not one of these: it has the correct new x-coordinate (1 − 2 = −1) but keeps the original y-coordinate (1) instead of adding 3, as if only the horizontal part of the vector had been applied.
- (d) 4 — Method: use the definition of a rectangle — a quadrilateral in which every interior angle is a right angle. Working: a quadrilateral has four interior angles, and in a rectangle all four of them measure 90°, which is consistent with the 360° angle sum because 4 × 90° = 360°. Answer: 4. The distractors: 2 comes from applying the rule that opposite angles are equal and concluding that only one pair of corners is square; 1 comes from thinking that a single square corner is enough to make a shape a rectangle; 0 comes from confusing a rectangle with a general parallelogram, which has no right angles unless it is a rectangle.
- (b) 12 cm — Method: the area of a rectangle is one side multiplied by the other, so when the area and one side are known the other side is found by reversing that multiplication — divide the area by the side that is known. Working: 96 ÷ 8 = 12. Answer: 12 cm. The distractors: 88 cm comes from 96 − 8, subtracting the known side as though the area had been made by adding the two sides together; 768 cm comes from 96 × 8, running the area rule forwards on the two numbers given instead of reversing it; 40 cm comes from reading the 96 as a perimeter — halving it to 48 and taking the 8 cm side away — which reverses the perimeter rule rather than the area rule.
- (a) A square's diagonals are equal in length — A square has two pairs of parallel sides, four equal sides, and diagonals that are equal in length and cross at right angles. This tile's diagonals are not equal, so it is a rhombus that is not a square.
- (a) 15 km — Multiply the length drawn on the map by the scale: 3 × 5 = 15 km. Choosing 8 km adds the two scale numbers together (3 + 5 = 8) instead of multiplying them. Choosing 30 km comes from misreading the scale as 1 cm : 10 km and doubling the correct answer. Choosing 5 km simply repeats the scale's distance figure and ignores that the road is drawn 3 cm long, not 1 cm.
- (d) 1 — sin 30° = 1/2, so (sin 30°)² = 1/4. cos 30° = √3/2, so (cos 30°)² = 3/4. Adding these gives 1/4 + 3/4 = 1. '1/4' only calculates (sin 30°)² and forgets to add the cos 30° term. '3/4' only calculates (cos 30°)² and forgets to add the sin 30° term. '−1/2' comes from subtracting the two squared values instead of adding them: 1/4 − 3/4 = −1/2.
- (b) 53.1° — The angle of elevation is opposite the height of the flagpole, 12 m, and adjacent to the distance from its base, 9 m, so tan θ = 12/9 = 1.333..., giving θ = tan⁻¹(1.333...) = 53.13...° ≈ 53.1°. "36.9°" finds the OTHER acute angle of the triangle, 90° − 53.1°, the angle at the top of the flagpole rather than the angle of elevation at Freya's position. "48.6°" comes from wrongly treating 9/12 as a sine ratio and finding sin⁻¹(0.75) = 48.6°, when neither side here is the hypotenuse. "41.4°" comes from wrongly treating 9/12 as a cosine ratio and finding cos⁻¹(0.75) = 41.4°, again without a hypotenuse in the ratio at all.
- (a) Every square is a rectangle — Method: test each statement against the definitions. A rectangle is a quadrilateral with four right angles and opposite sides equal; a square is a quadrilateral with four right angles and all four sides equal; a rhombus is a quadrilateral with all four sides equal. Working: a square has four right angles and its opposite sides are equal, so every square meets the definition of a rectangle and the statement that every square is a rectangle is true. Answer: every square is a rectangle. The distractors: the claim that every rectangle is a square reverses the inclusion, and fails for any rectangle whose length and width differ; the claim that every rhombus is a rectangle treats four equal sides as enough, and drops the right-angle condition — a tilted rhombus has no right angles; the claim that every rectangle is a rhombus reads 'opposite sides equal' as though it meant 'all four sides equal'.
- (d) 310 — Method: since both turns are clockwise, add both angles to the starting bearing. Working: 245° + 50° = 295°; 295° + 15° = 310°. A student who answers 295 has only added the first turn and forgotten the second one. A student who answers 180 has subtracted both turns instead of adding them. A student who answers 320 has added the two turns as 75° instead of 65° by misreading the second turn. Answer: 310°.
Build your own mix at the worksheet builder.