Geometry & Measures

A gardener is fencing a triangular flower bed with sides measuring 8 m, 9 m and 11 m. What length of fencing is needed to go around the whole bed?

• A170 m
• B28 m✓
• C200 m
• D190 m

Answer: The perimeter of any triangle is simply the total of its three side lengths added together. Adding 8 m, 9 m and 11 m gives 28 m of fencing. Each distractor takes the total found by missing out one side and shifts the decimal point one place to the right, making it ten times too large: 170 m comes from missing the 11 m side, 190 m comes from missing the 9 m side, and 200 m comes from missing the 8 m side.

A triangular pennant flag has a base of 12 cm and a perpendicular height of 9 cm. What is the area of the flag?

• A144 cm²
• B54 cm²✓
• C108 cm²
• D81 cm²

Answer: The area of a triangle is half of its base multiplied by its perpendicular height. Half of 12 cm times 9 cm gives 54 cm². 108 cm² comes from forgetting to halve the product, 144 cm² comes from mistakenly squaring the base length instead of using the height, and 81 cm² comes from mistakenly squaring the height length instead of using the base.

A parallelogram-shaped tile has a base of 15 cm and a perpendicular height of 6 cm. What is its area?

• A180 cm²
• B21 cm²
• C90 cm²✓
• D45 cm²

Answer: The area of a parallelogram is its base multiplied by its perpendicular height, with no halving needed. 15 cm times 6 cm gives 90 cm². 21 cm² comes from adding the base and height instead of multiplying, 45 cm² comes from wrongly halving the product as if it were a triangle, and 180 cm² comes from doubling the correct area.

A trapezium-shaped garden bed has parallel sides of 8 m and 14 m, with a perpendicular height of 5 m between them. What is its area?

• A40 m²
• B110 m²
• C27 m²
• D55 m²✓

Answer: The area of a trapezium is found by averaging its two parallel sides and multiplying by the perpendicular height between them. Averaging 8 m and 14 m gives 11 m, and 11 m times 5 m gives 55 m². 110 m² comes from forgetting to average the parallel sides first, 40 m² comes from using only the shorter parallel side, and 27 m² comes from simply adding all three measurements together.

A storage box is a cuboid measuring 6 cm by 5 cm by 4 cm. What is its volume?

• A120 cm³✓
• B30 cm³
• C240 cm³
• D15 cm³

Answer: The volume of a cuboid is found by multiplying its length, width and height together. Multiplying 6 cm, 5 cm and 4 cm gives 120 cm³. 15 cm³ comes from adding the three dimensions instead of multiplying them, 30 cm³ comes from multiplying only two of the three dimensions, and 240 cm³ comes from doubling the correct volume.

A tent has a triangular cross-section with a base of 6 cm and a height of 4 cm, and the tent is 10 cm long. What is the volume of the tent shape?

• A60 cm³
• B120 cm³✓
• C240 cm³
• D12 cm³

Answer: The volume of a prism is the area of its cross-section multiplied by its length. The triangular cross-section has area 12 cm² (half of 6 cm times 4 cm), and multiplying by the length of 10 cm gives 120 cm³. 240 cm³ comes from forgetting to halve the triangle's area, 12 cm³ comes from forgetting to multiply by the length at all, and 60 cm³ comes from halving the correct volume by mistake.

A cylindrical water tank has a radius of 3 cm and a height of 10 cm. What is its volume, to the nearest whole cm³? (Use π ≈ 3.14)

• A94 cm³
• B283 cm³✓
• C188 cm³
• D1130 cm³

Answer: The volume of a cylinder is π times the radius squared times the height. Squaring the radius of 3 cm gives 9, and 9 times 3.14 times 10 cm gives about 283 cm³. 1130 cm³ comes from mistakenly using the radius as if it were 6 cm, 94 cm³ comes from forgetting to square the radius, and 188 cm³ comes from using the circumference formula instead of the volume formula.

A circular table top has a diameter of 14 cm. What is its circumference, to the nearest whole cm? (Use π ≈ 3.14)

• A88 cm
• B44 cm✓
• C220 cm
• D154 cm

Answer: The circumference of a circle is π multiplied by its diameter. Multiplying 14 cm by 3.14 gives about 44 cm. 220 cm comes from using the radius instead of the diameter and then misplacing a decimal point, 154 cm comes from calculating the area instead of the circumference, and 88 cm comes from doubling the correct circumference.

A circular pond has a radius of 8 cm on a scale model. What is its circumference, to the nearest whole cm? (Use π ≈ 3.14)

• A16 cm
• B25 cm
• C201 cm
• D50 cm✓

Answer: The circumference of a circle is π multiplied by its diameter, which is twice the radius. Doubling 8 cm gives a diameter of 16 cm, and 16 cm times 3.14 gives about 50 cm. 25 cm comes from forgetting to double the radius before multiplying by π, 201 cm comes from calculating the area instead of the circumference, and 16 cm comes from doubling the radius correctly but then forgetting to multiply by π at all.

A circular clock face has a radius of 9 cm. What is its area, to the nearest whole cm²? (Use π ≈ 3.14)

• A57 cm²
• B1017 cm²
• C254 cm²✓
• D28 cm²

Answer: The area of a circle is π multiplied by the radius squared. Squaring 9 cm gives 81, and 81 times 3.14 gives about 254 cm². 57 cm² comes from calculating the circumference instead of the area, 28 cm² comes from forgetting to square the radius, and 1017 cm² comes from mistakenly squaring the diameter of 18 cm instead of the radius.

A garden shed is shaped like a rectangle 12 m by 6 m with a triangular roof section on top, the triangle having a base of 12 m and a height of 4 m. What is the total area of the front face of the shed?

• A48 m²
• B120 m²
• C72 m²
• D96 m²✓

Answer: The total area of a composite shape is the sum of the areas of its separate parts. The rectangle has area 72 m² and the triangle has area 24 m² (half of 12 m times 4 m), giving a total of 96 m². 72 m² comes from counting the rectangle only and forgetting the roof, 120 m² comes from forgetting to halve the triangle's area before adding it, and 48 m² comes from using the triangle's un-halved area (12 × 4 = 48) and forgetting the rectangle entirely.

In a right-angled triangle, the angle at the base is 32° and the side adjacent to it is 10 cm long. What is the length of the side opposite the 32° angle, to 1 decimal place?

• A5.3 cm
• B6.2 cm✓
• C16.0 cm
• D8.5 cm

Answer: In a right-angled triangle, the tangent ratio links the opposite and adjacent sides: tan(angle) = opposite ÷ adjacent. Rearranging gives opposite = 10 cm × tan(32°), which is about 6.2 cm. 5.3 cm comes from using sine instead of tangent, 8.5 cm comes from using cosine instead of tangent, and 16.0 cm comes from dividing by tan(32°) instead of multiplying.

Sam measures an angle with a protractor. He lines up the base line correctly, but then misreads the outer scale as 85° instead of using the inner scale, which correctly shows the angle is obtuse. What is the actual size of the angle?

• A105°
• B85°
• C95°✓
• D265°

Answer: A protractor has two scales running in opposite directions, and reading the wrong one for this kind of angle gives a value that is 180° away from the correct reading, since 85° and 95° add to 180°. Because the angle is described as obtuse, the correct reading must be 95°, not the acute 85° from the wrong scale. 105° comes from a small misalignment of the base line, and 265° comes from mistakenly reading the angle as a reflex angle instead.

A map has a scale of 1 : 200 000. The distance between two villages on the map is 6 cm. What is the real distance between the villages?

• A12 km✓
• B1.2 km
• C120 km
• D1200 m

Answer: A scale of 1 : 200 000 means every 1 cm on the map represents 200 000 cm in real life. Multiplying 6 cm by 200 000 gives 1 200 000 cm, which converts to 12 km. 120 km comes from misplacing a decimal point by a factor of 10, 1200 m comes from using a scale factor of 20 000 instead of 200 000 (missing a zero), and 1.2 km comes from misplacing the decimal point the other way.

A wall is 8 m long in real life. A scale drawing of the wall uses a scale of 1 : 50. How long should the wall be drawn on the scale drawing?

• A400 cm
• B16 mm
• C16 cm✓
• D0.16 cm

Answer: A scale of 1 : 50 means every 50 cm in real life is drawn as 1 cm. Converting 8 m to 400 cm and dividing by 50 gives 16 cm for the drawing. 0.16 cm comes from dividing without first converting metres to centimetres, 400 cm comes from multiplying by 50 instead of dividing, and 16 mm comes from getting the right number but the wrong unit.

Chloe draws two arcs of equal radius, one centred at point A and one centred at point B, so that the arcs cross above and below the line segment AB. She joins the two crossing points with a straight line. What has she constructed?

• AThe perpendicular bisector of AB✓
• BA line parallel to AB through its midpoint
• CThe angle bisector of angle A
• DJust the midpoint of AB, with no particular angle

Answer: Drawing equal-radius arcs from both ends of a line segment and joining where they cross is the standard ruler-and-compass method for constructing a perpendicular bisector. The resulting line crosses AB at 90° at its exact midpoint, meaning every point on it is equally distant from A and B. An angle bisector needs an angle rather than a line segment to start from, a parallel line would never cross AB at all, and while the construction does pass through the midpoint, it also fixes a specific right-angle direction rather than having no particular angle.

A carpenter needs a line from a point on his workbench that meets a straight wall at exactly 90°, using only compasses and a straight edge. Once he has drawn arcs from the point to cross the wall twice, then drawn further arcs from those two crossing points to find a new point, and joined that new point back to his original point, what has he constructed?

• AThe perpendicular bisector of the line
• BThe angle bisector at the point
• CThe perpendicular from the point to the line✓
• DA line parallel to the wall

Answer: Constructing arcs from a point to mark two points on a line, then arcs from those two points to find a new point, and joining it back is the standard method for dropping a perpendicular from a point to a line. This guarantees the final line meets the wall at exactly 90°. A perpendicular bisector instead starts from a line segment's two endpoints rather than an external point, an angle bisector needs two lines meeting at the point rather than a single line, and a parallel line would never touch the wall at all.

Three angles meet at a point on a flat surface. Two of them measure 140° and 145°. What is the size of the third angle?

• A215°
• B285°
• C75°✓
• D220°

Answer: Angles that meet at a single point on a flat surface always add up to 360°. Adding 140° and 145° gives 285°, and subtracting this from 360° leaves 75° for the third angle. 220° comes from subtracting only the 140° angle, 215° comes from subtracting only the 145° angle, and 285° comes from adding the two given angles but forgetting to subtract the total from 360° at all.

Two angles sit next to each other on a straight line. One of them measures 63°. What is the size of the other angle?

• A27°
• B127°
• C63°
• D117°✓

Answer: Angles on a straight line always add up to 180°. Subtracting 63° from 180° gives 117° for the other angle. 63° comes from simply repeating the given angle instead of finding the missing one, 27° comes from confusing this with a pair of complementary angles that add to 90°, and 127° comes from a subtraction slip.

Two straight lines cross at a point, forming four angles. One of the angles measures 74°. What is the size of the angle vertically opposite to it?

• A106°
• B37°
• C148°
• D74°✓

Answer: Vertically opposite angles, formed where two straight lines cross, are always equal to each other. This means the angle vertically opposite the 74° angle is also 74°. 106° is the size of the two angles next to it on the straight line, not the one opposite, 37° comes from halving the given angle by mistake, and 148° comes from doubling it by mistake.

Two parallel lines are cut by a transversal. One of the angles formed measures 115°. What is the size of the alternate angle on the other parallel line?

• A115°✓
• B65°
• C155°
• D25°

Answer: Alternate angles, which form a Z-shape between two parallel lines cut by a transversal, are always equal to each other, so the alternate angle is also 115°. 65° is the angle that is supplementary to 115° rather than equal to it, and 155° and 25° come from other arithmetic slips.

Two parallel lines are cut by a transversal. One of the angles formed measures 47°. What is the size of the corresponding angle on the other parallel line?

• A47°✓
• B137°
• C133°
• D227°

Answer: Corresponding angles, which sit in matching positions between two parallel lines cut by a transversal, are always equal to each other, so the corresponding angle is also 47°. 133° comes from finding the supplementary angle instead of the corresponding one, 137° comes from confusing this with a complementary pair and adding 90° instead of recognising the angles are equal, and 227° comes from adding 180° instead of subtracting when trying to find a supplementary angle.

In a triangle, two of the angles measure 54° and 68°. What is the size of the third angle?

• A58°✓
• B126°
• C122°
• D112°

Answer: The three interior angles of any triangle always add up to 180°. Subtracting 54° and 68° from 180° leaves 58° for the third angle. 126° comes from subtracting only the 54° angle, 112° comes from subtracting only the 68° angle, and 122° comes from adding the two given angles together instead of subtracting them from 180°.

A quadrilateral has three angles measuring 85°, 92° and 110°. What is the size of the fourth angle?

• A158°
• B183°
• C73°✓
• D165°

Answer: The four interior angles of any quadrilateral always add up to 360°. Subtracting 85°, 92° and 110° from 360° leaves 73° for the fourth angle. Forgetting to subtract the 92° angle gives 360° − 85° − 110° = 165°, forgetting to subtract the 110° angle gives 360° − 85° − 92° = 183°, and forgetting to subtract the 85° angle gives 360° − 92° − 110° = 158°.

What is the sum of the interior angles of a pentagon?

• A360°
• B270°
• C540°✓
• D900°

Answer: The interior angles of a polygon with n sides add up to (n − 2) × 180°. For a pentagon, n is 5, so the sum is 3 × 180°, which is 540°. 900° comes from multiplying 5 by 180° without subtracting 2 first, 270° comes from using 90° instead of 180° in the formula, and 360° is the angle sum for the exterior angles of any polygon, not the interior angles of a pentagon.

What is the size of each interior angle of a regular hexagon?

• A120°✓
• B60°
• C720°
• D144°

Answer: The interior angles of a regular polygon with n sides add up to (n − 2) × 180°, and dividing by n gives the size of each one. For a hexagon, this is 720° shared between 6 equal angles, giving 120° each. 60° is the size of each exterior angle rather than each interior angle, 720° is the total sum before dividing by the number of sides, and 144° comes from dividing the total by the wrong number of sides.

A regular polygon has exterior angles of 40° each. How many sides does the polygon have?

• A18 sides
• B9 sides✓
• C36 sides
• D12 sides

Answer: The exterior angles of any regular polygon add up to 360°, so dividing 360° by the size of one exterior angle gives the number of sides. Dividing 360° by 40° gives 9 sides. 12 sides, 18 sides and 36 sides all come from simple division slips when working out 360° ÷ 40°.

Triangle PQR and triangle STU have PQ equal to ST, QR equal to TU, and angle Q equal to angle T. Which congruence criterion proves the two triangles are congruent?

• ASSS (three sides)
• BSAS (two sides and the included angle)✓
• CRHS (right angle, hypotenuse, side)
• DASA (two angles and the included side)

Answer: The SAS criterion proves two triangles are congruent when two pairs of sides are equal and the angle between them, the included angle, is also equal. Here PQ equals ST, QR equals TU, and the angle between them, angle Q and angle T, are equal, which matches SAS exactly. SSS would need all three sides matched rather than two sides and an angle, ASA would need two angles and the side between them rather than two sides, and RHS only applies to right-angled triangles, which are not mentioned here.

Point A is at (2, 3) on a coordinate grid. It is translated by the vector (4, -5). What are the coordinates of the image of A?

• A(6, 8)
• B(6, -2)✓
• C(-2, 6)
• D(6, 3)

Answer: A translation by a vector adds the vector's x-value to the point's x-coordinate and adds the vector's y-value to the point's y-coordinate. Adding 4 to 2 gives 6, and adding -5 to 3 gives -2, so the image is at (6, -2). (6, 8) comes from subtracting -5 instead of adding it, (-2, 6) comes from swapping the two coordinates around, and (6, 3) comes from translating the x-coordinate correctly but forgetting to change the y-coordinate at all.

Point B is at (3, 1) on a coordinate grid. It is rotated 90° clockwise about the origin. What are the coordinates of the image of B?

• A(1, -3)✓
• B(3, -1)
• C(-3, -1)
• D(-1, 3)

Answer: Rotating a point 90° clockwise about the origin swaps its coordinates and changes the sign of the new second coordinate, turning (x, y) into (y, -x). Applying this to (3, 1) gives (1, -3). (-1, 3) is the result of rotating 90° anticlockwise instead of clockwise, and (3, -1) and (-3, -1) both come from getting the swap or the sign change wrong.

Point C is at (2, 5) on a coordinate grid. It is reflected in the x-axis. What are the coordinates of the image of C?

• A(5, 2)
• B(-2, 5)
• C(-2, -5)
• D(2, -5)✓

Answer: Reflecting a point in the x-axis keeps the x-coordinate the same and changes the sign of the y-coordinate. Applying this to (2, 5) gives (2, -5). (-2, 5) comes from reflecting in the y-axis instead of the x-axis, (-2, -5) comes from reflecting in both axes at once, and (5, 2) comes from swapping the coordinates as if reflecting in the line y = x.

A shape is enlarged by a scale factor of 2.5. One of its sides originally measures 4.8 cm. What is the length of that side after the enlargement?

• A12 cm✓
• B7.3 cm
• C1.92 cm
• D30 cm

Answer: An enlargement by a scale factor multiplies every length on the original shape by that factor. Multiplying 4.8 cm by 2.5 gives 12 cm. 7.3 cm comes from adding the scale factor to the length instead of multiplying, 1.92 cm comes from dividing by the scale factor instead of multiplying, and 30 cm comes from using 6.25 (2.5 squared) as if the scale factor applied to area rather than length.

Two shapes are congruent to each other. What must be true about them?

• AThey have the same perimeter but a different shape
• BSame shape, different size
• CThey are exactly the same shape and size✓
• DThey have the same area but a different shape

Answer: Congruent shapes are identical in every way, meaning they are exactly the same shape and the same size, even if one has been rotated, reflected or moved. Shapes that are the same shape but a different size are described as similar, not congruent. Having the same area or the same perimeter does not guarantee two shapes are the same shape at all, so neither of those describes congruence.

Triangle A has sides of 4 cm, 6 cm and 8 cm. It is similar to triangle B, whose shortest side is 6 cm. What is the length of the side on triangle B that corresponds to the 8 cm side on triangle A?

• A9 cm
• B5.3 cm
• C10 cm
• D12 cm✓

Answer: Similar shapes have sides that are all in the same ratio, so finding the scale factor from one pair of corresponding sides lets you scale up the rest. The shortest sides give a scale factor of 6 cm ÷ 4 cm, which is 1.5, and multiplying the 8 cm side by 1.5 gives 12 cm. 9 cm comes from scaling the middle side instead of the longest side, 10 cm comes from simply adding the difference between the shortest sides rather than scaling, and 5.3 cm comes from dividing instead of multiplying by the scale factor.

A right-angled triangle has two shorter sides of 9 cm and 12 cm. What is the length of the hypotenuse?

• A15 cm✓
• B7.9 cm
• C225 cm
• D21 cm

Answer: Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Squaring 9 cm and 12 cm and adding them gives 225, and the square root of 225 is 15 cm. 21 cm comes from simply adding the two side lengths instead of using Pythagoras' theorem, 225 cm comes from forgetting to take the square root at the end, and 7.9 cm comes from subtracting the squares instead of adding them.

A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. What is the length of the other shorter side?

• A144 cm
• B13.9 cm
• C12 cm✓
• D8 cm

Answer: Pythagoras' theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides, so a missing shorter side is found by subtracting rather than adding. Subtracting the square of 5 cm from the square of 13 cm gives 144, and the square root of 144 is 12 cm. 13.9 cm comes from adding the squares instead of subtracting them, 144 cm comes from forgetting to take the square root at the end, and 8 cm comes from simply subtracting 5 cm from 13 cm instead of using Pythagoras' theorem.

A right-angled triangle has a hypotenuse of 20 cm and one angle of 40°. What is the length of the side opposite the 40° angle, to 1 decimal place?

• A16.8 cm
• B15.3 cm
• C12.9 cm✓
• D31.1 cm

Answer: In a right-angled triangle, the sine ratio links the opposite side and the hypotenuse: sin(angle) = opposite ÷ hypotenuse. Rearranging gives opposite = 20 cm × sin(40°), which is about 12.9 cm. 15.3 cm comes from using cosine instead of sine, 16.8 cm comes from using tangent instead of sine, and 31.1 cm comes from dividing by sin(40°) instead of multiplying.

How many faces does a triangular prism have?

• A9 faces
• B6 faces
• C3 faces
• D5 faces✓

Answer: A triangular prism has two triangular end faces plus three rectangular side faces, making 5 faces in total. 6 faces is actually the number of vertices on a triangular prism, not the number of faces, 9 faces is actually the number of edges, and 3 faces comes from counting only the rectangular sides and forgetting the two triangular ends.