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.
Non-calculator
Geometry and measures worksheet — GCSE Foundation
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- 1.A bottle holds 2 litres of water. A glass holds 250 ml. Work out how many of these glasses can be filled from one full bottle.
- 2.A scale drawing shows a room that is 5 cm wide. The real room is 15 m wide. Write the scale of the drawing in the form 1 : n.
- 3.A straight line is drawn from the centre of a circle to a point on the circle. Write down the name of this line.
- 4.Triangle ABC has angle A = 40°, angle B = 65° and AB = 10 cm. Triangle DEF has angle D = 40°, angle E = 65° and DE = 10 cm. Which condition proves the two triangles are congruent?
- 5.A circle has a radius of 1 cm. Work out the circumference of the circle. Give your answer in terms of π.
- 6.The diagram shows the plan of the base layer of a solid built from identical cubes. A second layer of cubes is added on top, filling the entire base layer to a height of 2 cubes everywhere. Work out how many squares are visible in the plan view of this solid.
- 7.How many vertices does a cuboid have?
- 8.Two straight lines AB and CD cross at point O. Angle AOC = 74°. Work out angle AOD.
- 9.In triangle ABC and triangle DEF, AB = DE, angle ABC = angle DEF and BC = EF. Write down the congruence condition that proves the two triangles are congruent.
- 10.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?
- 11.The diagram shows a cuboid. Work out the area of its front elevation, in square centimetres.
- 12.A garden flower bed is in the shape of a trapezium. Its two parallel edges are 3.5 m and 6.5 m long, and the perpendicular distance between them is 4 m. Bedding plants cost £2.50 per square metre to cover the bed. Work out the total cost of the bedding plants.
- 13.Write down the exact value of tan 45°.
- 14.Two angles lie on a straight line. One angle is 118°. Which reason correctly explains why the other angle is 62°?
- 15.A translation is described by the column vector . Write down the column vector of the translation that undoes it.
Answer key
- (a) 8 — Method: two capacities can only be divided once they are written in the same unit. Working: there are 1000 ml in 1 litre, so the bottle holds 2 × 1000 = 2000 ml. Then 2000 ÷ 250 = 8. Answer: 8 glasses. Using 1 litre = 10 000 ml gives 20 000 ÷ 250 = 80. Dividing the two numbers as they stand, 250 ÷ 2 = 125, ignores the units altogether. Converting the glass into litres with the wrong factor, as though 250 ml were 2.5 litres, gives 2 ÷ 2.5 = 0.8.
- (c) 1 : 300 — First convert both lengths to the same unit: 15 m = 1500 cm. The scale compares 5 cm on the drawing to 1500 cm in real life, so dividing both parts by 5 gives a scale of 1 : 300. A candidate who compares 5 to 15 without converting units gets 1 : 3. A candidate who converts 15 m to 150 cm, using the wrong conversion factor, gets 1 : 30. A candidate who converts 15 m to 15000 cm, again using the wrong conversion factor, gets 1 : 3000. The scale of the drawing is 1 : 300.
- (b) The radius — Method: each of the four names describes a line in a fixed position relative to the circle, so match the description in the question against those positions. Working: the line described has one end at the centre and its other end on the circle. A chord joins two points that both lie on the circle, so it is ruled out. A diameter joins two points on the circle by passing through the centre, so it is twice as long as the line described. A tangent touches the circle at exactly one point and never reaches the centre. The line with one end at the centre and one end on the circle is a radius. Answer: The radius. The distractors: The diameter comes from remembering only that the line involves the centre, and not noticing that it stops there instead of carrying on to the far side; The chord comes from seeing one end on the circle and assuming both ends are; The tangent comes from matching the phrase 'a point on the circle' to the line that touches at exactly one point, ignoring that a tangent never reaches the centre.
- (d) ASA - two angles and the included side equal — Two angles (A and B) are given, and AB is the side between them, so this is ASA. SAS needs two sides and the angle between them, but only one side is given. AAS also uses two angles and a side, but the side must NOT be between the two angles — here AB is between angle A and angle B, so it is ASA, not AAS. RHS needs a right angle, and neither 40° nor 65° is 90°.
- (c) 2π cm — Method: the circumference of a circle is 2πr, where r is the radius, or equivalently πd, where d is the diameter. Working: r = 1, so the circumference is 2 × π × 1 = 2π cm. Answer: 2π cm. The distractors: π cm comes from using the formula πd but substituting the radius in place of the diameter; 4π cm comes from doubling twice — changing the radius into the diameter of 2 cm and then putting that diameter into 2πr as though it were a radius; π cm² is the area of this circle, π × 1², and comes from reaching for the area formula when a distance round the outside was asked for, which is why it carries a squared unit.
- (c) 6 squares — Method: the plan view shows the footprint of the solid; a second layer stacked on top of floor positions that are already covered does not create any new squares in the plan. Working: the row of 4 cubes and the row of 2 cubes attached at the end do not overlap, so the footprint has 4 + 2 = 6 distinct squares. Answer: 6 squares. The distractors: 12 squares comes from counting the total number of cubes used, including the second layer (6 floor positions × 2 layers = 12), instead of the footprint. 4 squares comes from counting only the row of four and forgetting the attached row of two. 5 squares comes from wrongly treating the corner square as shared between the two rows (4 + 1 instead of 4 + 2).
- (d) 8 — A cuboid is a 3D shape with 6 rectangular faces, 12 edges and 8 vertices (corners). Counting the corners of a cuboid gives 8. A candidate who mixes up vertices with faces answers 6. A candidate who mixes up vertices with edges answers 12. A candidate who counts only the 4 corners of the top face, forgetting the 4 corners of the bottom face, answers 4. The correct number of vertices is 8.
- (a) 106° — Angle AOC and angle AOD share the arm OA and together they make up the straight line CD, so they add up to 180°: angle AOD = 180° − 74° = 106°. 74° comes from treating AOC and AOD as vertically opposite angles, which are equal — but they are adjacent angles on a straight line, not vertically opposite. 16° comes from subtracting 74° from 90°, treating the angles as complementary. 148° comes from doubling 74° instead of subtracting it from 180°.
- (b) SAS — Two pairs of matching sides are equal (AB = DE and BC = EF) and the angle between them is equal (angle ABC = angle DEF), which is exactly the Side-Angle-Side congruence test: SAS. SSS is wrong because only two pairs of sides are given, not three. ASA is wrong because only one angle is given, and it sits between the two sides rather than the side sitting between two angles. RHS is wrong because no right angle is mentioned in either triangle.
- (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.
- (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.
- (d) £50 — Area = 1/2 × (3.5 + 6.5) × 4 = 1/2 × 10 × 4 = 20 m². Cost = 20 × £2.50 = £50. (£100 comes from forgetting to halve the trapezium area, giving 40 m² instead of 20 m²; £65 comes from using only the longer parallel side, 6.5 × 4 = 26 m², instead of the trapezium formula; £35 comes from adding all three given lengths, 3.5 + 6.5 + 4, and treating that total as the area in square metres.)
- (d) 1 — At 45° the right-angled isosceles triangle has both shorter sides equal, so tan 45° = opposite ÷ adjacent = 1 ÷ 1 = 1. 0 is the value of tan 0° (mixing up the two angles). √3 is the value of tan 60°. √3/3 (equivalent to 1/√3) is the value of tan 30°.
- (b) angles on a straight line add up to 180° — 118° and 62° add up to 180°, matching the fact that angles on a straight line always add up to 180° — that is the reason the second angle is 62°. Angles round a point add up to 360° is a different fact, used for angles surrounding a single point, not two angles on a line. Vertically opposite angles are equal would make the second angle 118° too, not 62°. Angles in a triangle add up to 180° is a true fact, but it applies to the three angles inside a triangle, not two angles on a straight line.
- (c) $\binom{-6}{5}$ — Method: to undo a translation, travel the same journey backwards. A move of 6 to the right is undone by a move of 6 to the left, and a move of 5 down is undone by a move of 5 up, so both numbers change sign. Working: the top number 6 becomes −6 and the bottom number −5 becomes 5. Check by combining the two: 6 − 6 = 0 across and −5 + 5 = 0 up, so the shape finishes where it started. Answer: $\binom{-6}{5}$. Reversing only the horizontal movement gives $\binom{-6}{-5}$ and reversing only the vertical movement gives $\binom{6}{5}$, and each of those leaves the shape displaced. Swapping the two entries instead of changing their signs gives $\binom{-5}{6}$.
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