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 hexagonal prism has 8 faces and 12 vertices. Using F + V − E = 2, work out how many edges it has.
- 2.A conservatory has a roof panel shaped like a regular octagon. Work out the interior angle of the octagon, then use it to work out the size of the reflex angle at the same vertex, on the outside of the roof panel.
- 3.u is the column vector with top number 7 and bottom number 3. v is the column vector with top number 2 and bottom number 5. Work out u − v, giving your answer as a column vector in the form (top, bottom).
- 4.A hexagon has interior angles of 130°, 142°, 125°, 150°, 135° and x°. Work out the value of x.
- 5.In a phone game, a character starts at the point (−5, 2) on a grid. It moves by the vector to collect a coin. The player now wants the character's next single move to finish at the point (3, 0). Write down the column vector of that second move.
- 6.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.
- 7.Write down the exact value of cos 45°.
- 8.A trapezium has parallel sides of 5 cm and 11 cm, and a perpendicular height of 4 cm. Work out the area of the trapezium.
- 9.A perpendicular bisector is drawn through the midpoint of a line segment AB. Which set of points does this construction show?
- 10.Write down the number of lines of symmetry a rhombus has.
- 11.A crane's hook starts at the point (3.4, 9.5) on a construction site plan measured in metres. It moves along the vector to pick up a beam, then along the vector to place it. Work out the single column vector that describes the hook's overall movement from its start position to where it places the beam.
- 12.Triangles LMN and XYZ have LM = XY, MN = YZ and LN = XZ. Give a reason why triangle LMN is congruent to triangle XYZ.
- 13.In an isosceles triangle each of the two base angles is 46°. Work out the size of the angle at the apex.
- 14.Write down the name given to a quadrilateral that has exactly one pair of parallel sides.
- 15.A quadrilateral has all four sides equal in length and all four interior angles equal to 90°. Its diagonals are equal in length and bisect each other at right angles. Which quadrilateral matches this description exactly?
Answer key
- (c) 18 — Rearranging F + V − E = 2 gives E = F + V − 2. Substitute F = 8 and V = 12: 8 + 12 − 2 = 18 edges. Choosing 20 comes from adding the faces and vertices but forgetting to subtract the 2 (8 + 12 = 20). Choosing 22 comes from adding the 2 instead of subtracting it (8 + 12 + 2 = 22). Choosing 16 comes from subtracting 2 twice by mistake (8 + 12 − 2 − 2 = 16).
- (a) 225 — Method: find the interior angle of the regular octagon, then subtract it from 360° to find the reflex angle at the same vertex, since the interior angle and the reflex angle together make a full turn. Working: there are 8 − 2 = 6 triangles' worth of angle in the octagon, so the interior angle = 6 × 180 ÷ 8 = 135; reflex angle = 360 − 135 = 225. Answer: 225°. A candidate who stops after finding the interior angle gives 135. A candidate who works out the exterior angle instead, 360 ÷ 8 = 45, gives 45. A candidate who subtracts the exterior angle from 360° instead of the interior angle, working out 360 − 45, gets 315.
- (b) (5, −2) — Method: subtract column vectors by subtracting the bottom vector's top number from the top vector's top number, and doing the same for the bottom numbers. Working: top numbers 7 − 2 = 5; bottom numbers 3 − 5 = −2. Answer: u − v = (5, −2). A candidate who drops the negative sign gets (5, 2). A candidate who works out v − u instead of u − v gets (−5, 2). A candidate who adds instead of subtracts gets (9, 8).
- (c) 38° — The interior angles of a hexagon sum to (6 − 2) × 180° = 720°. 130° + 142° + 125° + 150° + 135° = 682°. x = 720° − 682° = 38°. A student who uses 6 × 180° instead of (6 − 2) × 180°, forgetting to subtract the 2, gets 1080° − 682° = 398°. A student who mis-adds the five given angles as 680° instead of 682° gets x = 720° − 680° = 40°. A student who answers with one of the given angles instead of solving for x might write 142°.
- (c) $\binom{2}{7}$ — First find the character's position after the first move: (−5 + 6, 2 + (−9)) = (1, −7). The second move takes it from (1, −7) to (3, 0), so subtract the current position from the target: (3 − 1, 0 − (−7)) = $\binom{2}{7}$. $\binom{8}{−2}$ works from the starting point (−5, 2) and ignores the first move — (3 − (−5), 0 − 2) = $\binom{8}{−2}$. $\binom{−2}{−7}$ subtracts the wrong way round, current position minus target, instead of target minus current position — (1 − 3, −7 − 0) = $\binom{−2}{−7}$. $\binom{4}{−7}$ adds the target's coordinates to the current position instead of subtracting — (1 + 3, −7 + 0) = $\binom{4}{−7}$.
- (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.
- (b) √2/2 — cos 45° comes from a right-angled isosceles triangle with both shorter sides 1 and hypotenuse √2, giving cos 45° = 1/√2 = √2/2. 1/2 is the value of cos 60° (mixing up the two angles). √2 is the hypotenuse length itself, not divided by it (forgetting to divide by the hypotenuse). √3/2 is the value of cos 30° (using the wrong special triangle).
- (a) 32 cm² — Area of a trapezium = 1/2 × (sum of parallel sides) × height = 1/2 × (5 + 11) × 4 = 1/2 × 16 × 4 = 32 cm². (64 cm² comes from forgetting to halve the product; 44 cm² comes from using only the longer parallel side, 11 × 4, instead of both parallel sides; 20 cm² comes from adding all three given lengths together, 5 + 11 + 4, instead of using the area formula.)
- (d) points the same distance from A as from B — The perpendicular bisector of AB is, by definition, the locus of every point equidistant from A and B — any point on it forms two congruent right-angled triangles with A and B, which is exactly the property the construction guarantees. "points as far from A as the length AB" describes a circle centred at A with radius AB, not a bisector. "the single point exactly halfway along AB" names only the midpoint, one point, not the whole locus the construction produces. "points twice as far from A as from B" describes a different curve entirely, not a straight line construction.
- (b) 2 — A rhombus has two lines of symmetry — along each of its two diagonals.
- (b) $\binom{-0.4}{-7}$ — The overall movement is the sum of the two vectors: (−1.2 + 0.8, −4.5 + (−2.5)) = (−0.4, −7). '$\binom{-2}{-2}$' comes from subtracting the second vector from the first instead of adding them. '$\binom{-0.4}{7}$' gets the top number right but drops the negative sign on the bottom number. '$\binom{2}{-7}$' comes from treating −1.2 + 0.8 as if the signs did not matter, giving +2 instead of −0.4.
- (a) SSS – all three corresponding sides are equal — All three pairs of corresponding sides are stated as equal — LM = XY, MN = YZ and LN = XZ — with no angle mentioned. This matches the SSS condition, so triangle LMN is congruent to triangle XYZ.
- (d) 88° — Method: the three angles add up to 180°, and here it is the two equal base angles that are known, so take both of them away from 180°. Working: the two base angles come to 2 × 46° = 92°, and 180° − 92° = 88°. Answer: 88°. The distractors: 134° comes from subtracting only one base angle, 180° − 46°, and forgetting that there are two of them; 92° comes from doubling the base angle and stopping there, which is the two base angles together rather than the apex; 46° comes from assuming that all three angles of the triangle are equal to the one that is given.
- (c) Trapezium — A trapezium is defined as a quadrilateral with exactly one pair of parallel sides.
- (b) Square — A square has all four sides equal, all four angles equal to 90°, and diagonals that are equal in length and bisect each other at right angles — every part of the description matches, so Square is correct. A rhombus has all four sides equal and diagonals bisecting at right angles, but its interior angles are not generally 90° (only a square, a special rhombus, has that), so it does not fully match. A rectangle has four 90° angles and equal diagonals, but its sides are not all equal in general, so it fails the equal-sides condition. A kite has two pairs of adjacent equal sides rather than all four sides equal, and its diagonals are not generally equal in length, so it fails both conditions.
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