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 the plan view and the front elevation of a cuboid-shaped box, each drawn as a rectangle. Work out the area of the box's side elevation, in square centimetres.
- 2.In a right-angled triangle, the two shorter sides are a and b, and the hypotenuse is c. Write down the correct statement of Pythagoras' theorem.
- 3.The point A(3, 2) is translated by the vector (−5, 4) to give point B. Work out the coordinates of B.
- 4.A water tank holds 2.5 m³. Work out this volume in cm³.
- 5.The vector describes a translation. A student writes it as by mistake. Write down what error the student has made.
- 6.Which of the following is the correct definition of a regular polygon?
- 7.A straight transversal crosses a pair of parallel lines. At one crossing point, the angle between the transversal and a parallel line is 118°. Work out the size of the co-interior (allied) angle at the other crossing point.
- 8.In triangle ABC, point D lies on side AB and point E lies on side AC, so that DE is parallel to BC. AD = 3 cm, DB = 6 cm and DE = 4 cm. Work out the length of BC.
- 9.A polygon has 9 sides. What is the name of this polygon?
- 10.A path goes from A(1, 1) to B(1, 5), then from B to C(6, 5). Work out the total length of the path from A to C.
- 11.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.
- 12.Two similar shapes have lengths in the ratio 1 : 3. A side of the smaller shape is 4 cm long. Work out the length of the corresponding side of the larger shape.
- 13.A straight fence is 5 m long. Describe the shape of the region made up of every point on the ground within 2 m of any part of the fence.
- 14.In geometry, which of the following best describes a 'plane'?
- 15.A right-angled triangle has two sides of length 5 cm and 12 cm, and the angle between those two sides is 90°. Work out the length of the hypotenuse.
Answer key
- (d) 15 cm² — Method: identify the length that is common to both views, since it is the box's length; then read off the width from the plan view and the height from the front elevation, and multiply those two measurements to find the area of the side elevation. Working: both rectangles share a side of 8 cm, which is the box's length; the plan view's other side gives a width of 5 cm, and the front elevation's other side gives a height of 3 cm. The side elevation is bounded by the width and the height: 5 cm × 3 cm = 15 cm². Answer: 15 cm². The distractors: 24 cm² comes from giving the area of the front elevation shown (8 cm × 3 cm) instead of working out a new rectangle for the side elevation. 40 cm² comes from giving the area of the plan view shown (8 cm × 5 cm) instead of working out the side elevation. 120 cm² comes from multiplying all three measurements together (8 cm × 5 cm × 3 cm), finding the volume of the box instead of the area of one face.
- (d) a² + b² = c² — Pythagoras' theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides, so a² + b² = c². "a + b = c" adds the sides directly without squaring them at all. "a² − b² = c²" subtracts the squares instead of adding them. "a² + b² = c" adds the squares correctly but forgets to square the hypotenuse on the other side of the equation.
- (a) (−2, 6) — Translating by the vector (−5, 4) means adding −5 to the x-coordinate and adding 4 to the y-coordinate: (3 + (−5), 2 + 4) = (−2, 6). (8, 6) comes from treating −5 as +5, adding instead of subtracting on the x-coordinate. (−2, −2) keeps the x-coordinate correct but subtracts 4 from the y-coordinate instead of adding it. (7, −3) comes from swapping the two components of the vector, applying 4 to the x-coordinate and −5 to the y-coordinate.
- (b) 2 500 000 cm³ — Method: a volume conversion uses the length factor three times, once for each dimension. Working: 1 m = 100 cm, so a cube of side 1 m is a cube of side 100 cm, and 100 × 100 × 100 = 1 000 000, giving 1 m³ = 1 000 000 cm³. Then 2.5 × 1 000 000 = 2 500 000. Answer: 2 500 000 cm³. Using the plain length factor 100 gives 250 cm³. Using 1000, which is the factor that turns cubic metres into litres, gives 2500 cm³. Using 10 000, which is the factor that belongs to square metres and square centimetres, gives 25 000 cm³.
- (c) Swapped the x and y components — The student's vector has the same two numbers, 5 and −2, but in swapped positions, so the error is swapping the x and y components rather than an error with signs or size. 'Reversed both signs' is wrong because the numbers 5 and −2 have not changed sign, only position. 'Reversed only the y sign' is wrong for the same reason — no sign has actually changed. 'Doubled the x component' is wrong because neither number has changed in size.
- (c) Equal sides and equal interior angles — Method: recall the full definition of 'regular' as applied to a polygon. Working: a regular polygon must have both equal side lengths and equal interior angles at the same time. Options: 'all sides equal' alone describes an equilateral but not necessarily equiangular shape, such as a rhombus, which is not regular; 'all angles equal' alone describes an equiangular but not necessarily equilateral shape, such as a rectangle, which is not regular; 'at least one line of symmetry' is a much weaker condition that many irregular shapes also satisfy. Answer: equal sides and equal interior angles.
- (c) 62° — Co-interior (allied) angles on parallel lines sum to 180°, so the co-interior angle is 180° − 118° = 62°. 118° comes from treating the angles as corresponding angles, which are equal, instead of co-interior angles, which sum to 180°. 59° comes from halving the given angle. 236° comes from doubling the given angle.
- (d) 12 cm — Since DE is parallel to BC, triangle ADE is similar to triangle ABC (the two triangles share angle A, and the parallel lines make the angles at D and E equal to the angles at B and C). The whole side AB = AD + DB = 3 + 6 = 9 cm. The scale factor from the small triangle to the large triangle is AB ÷ AD = 9 ÷ 3 = 3, so BC = DE × 3 = 4 × 3 = 12 cm. Scaling by DB ÷ AD instead of AB ÷ AD gives 4 × (6 ÷ 3) = 8 cm. Adding DB directly onto DE, instead of scaling, gives 4 + 6 = 10 cm. Treating the triangles as congruent rather than similar, so assuming corresponding sides are simply equal, gives BC = DE = 4 cm.
- (b) Nonagon — Method: match the number of sides given to the correct polygon name. Working: a polygon with 9 sides is called a nonagon. An octagon has 8 sides, a decagon has 10 sides, and a heptagon has 7 sides, so each names a different number of sides from the 9 given. Answer: nonagon.
- (a) 9 — AB is a vertical segment, since A and B share the x-coordinate 1, and its length is the difference in y-coordinates: 5 − 1 = 4. BC is a horizontal segment, since B and C share the y-coordinate 5, and its length is the difference in x-coordinates: 6 − 1 = 5. The total path length is 4 + 5 = 9. 20 comes from multiplying the two lengths, 4 × 5, instead of adding them. 5 is only the length of BC, forgetting to include AB. 4 is only the length of AB, forgetting to include BC.
- (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.
- (b) 12 cm — Method: in similar shapes every length is multiplied by the same scale factor, and a ratio of 1 : 3 means that factor is 3 going from the smaller shape to the larger one. Working: 4 × 3 = 12. Answer: 12 cm. The distractors: 7 cm comes from adding 3 to the side instead of multiplying by it, which is what happens when a ratio is read as a difference; 36 cm comes from multiplying by 3² = 9, the factor that scales areas, and applying it to a length; 4 cm comes from treating the two shapes as congruent, so that corresponding sides stay equal — similar shapes have equal angles, but their sides are in proportion.
- (a) a rounded rectangle: 5 m by 4 m with semicircular ends — Points within 2 m of the straight part of the fence form a rectangle running the 5 m length of the fence and 4 m wide (2 m on each side); points within 2 m of each END of the fence, beyond that rectangle, form a semicircle of radius 2 m there, since the nearest point of the fence to them is just that one end. Together this gives a rounded, stadium-shaped region. "a rectangle, 9 m by 4 m" extends the rectangle by 2 m at each end instead of rounding it, wrongly including corner points that are actually more than 2 m from every part of the fence. "a circle of radius 2 m" treats the whole 5 m fence as a single point. "a rectangle, 5 m by 2 m" uses 2 m as the full width instead of the distance on EACH side, so it only covers one side of the fence.
- (a) A flat surface extending infinitely in two directions — Method: recall the precise geometric meaning of 'plane', versus 'line', 'point' and 'face'. Working: a plane is a flat, two-dimensional surface extending infinitely in every direction within it. Options: 'a straight line extending in one direction' describes a line, not a plane; 'a single fixed position with no size' describes a point; 'a flat, bounded face on a 3D shape' describes a face, a bounded piece of a plane, not the plane itself, which has no boundary. Answer: a flat surface extending infinitely in two directions.
- (c) 13 cm — Method: the two given sides meet at the right angle, so they are the shorter pair and the hypotenuse comes from Pythagoras' theorem, a² + b² = c². Working: c² = 5² + 12² = 25 + 144 = 169, so c = √169 = 13. Answer: 13 cm. The distractors: 17 cm comes from adding the two sides, 5 + 12, rather than adding their squares; 60 cm comes from multiplying them, 5 × 12, which is twice the area of the triangle and not a length; 7 cm comes from subtracting, 12 − 5, as though the hypotenuse were the difference of the two shorter sides.
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