Geometry & Trigonometry practice questions
Angles, area, volume, circles, similar figures, and right-triangle relationships. Sketching the figure and labelling what you know resolves a surprising share of these before any formula is needed.
Geometry & Trigonometry questions
- 57CirclesStandard
A circle has a circumference of 36 pi. Enter its radius.
Student-produced response: write your own answer rather than choosing one.
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Answer: 18
Circumference is 2 pi r, so 2 pi r = 36 pi gives r = 18. Using the area formula instead is the usual wrong turn here.
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A cylinder has radius 5 cm and height 12 cm. What is its volume, in terms of pi?
- A300 pi cubic cm
- B120 pi cubic cm
- C60 pi cubic cm
- D900 pi cubic cm
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Answer: 300 pi cubic cm
Volume is pi r^2 h = pi x 25 x 12 = 300 pi cubic cm. Using r rather than r^2 gives 60 pi, and doubling the radius before squaring gives 900 pi.
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A right triangle has legs of length 9 and 12. What is the length of its hypotenuse?
- A21
- B15
- C16
- D13
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Answer: 15
9^2 + 12^2 = 81 + 144 = 225, and the square root of 225 is 15. Adding the legs gives 21, which is what the theorem exists to correct: the hypotenuse is always shorter than the two legs combined.
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In right triangle ABC the right angle is at B, AB = 8 and BC = 15, so AC = 17. What is sin A?
- A8/17
- B15/8
- C15/17
- D8/15
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Answer: 15/17
Sine is opposite over hypotenuse. The side opposite angle A is BC = 15 and the hypotenuse is AC = 17, so sin A = 15/17. The choice 8/17 is cos A, taken from the side next to the angle rather than across from it.
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What is the distance between the points (-3, 2) and (5, 8)?
- A14
- Bthe square root of 68
- C8
- D10
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Answer: 10
The horizontal gap is 8 and the vertical gap is 6, so the distance is the square root of 64 + 36 = 100, which is 10. Adding the gaps gives 14, and subtracting the coordinates in the wrong order before squaring is what produces the square root of 68.
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Triangle ABC is similar to triangle DEF, with AB = 6 and DE = 15. If the area of triangle ABC is 20, what is the area of triangle DEF?
- A50
- B125
- C75
- D500
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Answer: 125
The sides scale by 15/6 = 5/2, so the areas scale by the square of that, 25/4. Then 20 x 25/4 = 125. Scaling the area by 5/2 instead gives 50, which is the error this question is built to catch.
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What is the measure of each interior angle of a regular decagon?
- A36 degrees
- B150 degrees
- C144 degrees
- D1,440 degrees
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Answer: 144 degrees
The interior angles of an n-sided polygon total (n - 2) x 180, so for ten sides that is 8 x 180 = 1,440 degrees. Divided among ten equal angles, each is 144 degrees. The choice 36 is the exterior angle, and 1,440 is the total rather than one angle.
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A trapezium has parallel sides of 14 cm and 22 cm and a height of 9 cm. What is its area?
- A324 square cm
- B126 square cm
- C198 square cm
- D162 square cm
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Answer: 162 square cm
The area is the average of the parallel sides times the height: ((14 + 22) / 2) x 9 = 18 x 9 = 162 square cm. Forgetting to halve gives 324, and using one parallel side alone gives 126 or 198.
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A 45-45-90 triangle has a hypotenuse of length 10 times the square root of 2. What is the length of each leg?
- A10
- B10 times the square root of 2
- C5 times the square root of 2
- D20
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Answer: 10
In a 45-45-90 triangle the hypotenuse is the leg times the square root of 2. Dividing the hypotenuse by the square root of 2 leaves 10. Multiplying by the square root of 2 instead goes the wrong way along the same relationship.
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Two angles are supplementary and one of them measures 47 degrees. What does the other measure?
- A43 degrees
- B133 degrees
- C53 degrees
- D313 degrees
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Answer: 133 degrees
Supplementary angles total 180 degrees, so the other is 180 - 47 = 133. The choice 43 comes from using 90 degrees, which is the rule for complementary angles.
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In triangle ABC, angle A measures 3x degrees, angle B measures 2x degrees and angle C measures x degrees. What is the measure of angle A?
- A30 degrees
- B60 degrees
- C90 degrees
- D120 degrees
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Answer: 90 degrees
The angles of a triangle total 180, so 3x + 2x + x = 6x = 180 and x = 30. Angle A is 3x = 90 degrees. The choice 30 is x itself, which is angle C.
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Two parallel lines are cut by a transversal, and one of the angles formed measures 118 degrees. What is the measure of the same-side interior angle paired with it?
- A118 degrees
- B236 degrees
- C32 degrees
- D62 degrees
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Answer: 62 degrees
Same-side interior angles between parallel lines are supplementary, so the pair totals 180 and the other is 62 degrees. The choice 118 is the measure of the angles that are equal to the given one rather than supplementary to it.
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Each interior angle of a regular polygon measures 156 degrees. How many sides does it have?
- A15
- B12
- C10
- D18
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Answer: 15
Each exterior angle is 180 - 156 = 24 degrees, and exterior angles of any polygon total 360, so there are 360 / 24 = 15 sides. Checking forwards: (15 - 2) x 180 / 15 = 156.
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A ladder 13 m long leans against a vertical wall with its foot 5 m from the base of the wall. How far up the wall does the ladder reach?
- A18 m
- B12 m
- C8 m
- Dthe square root of 194 m
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Answer: 12 m
The ladder is the hypotenuse, so the height is the square root of 13^2 - 5^2 = 169 - 25 = 144, which is 12 m. Adding the squares instead of subtracting them gives the square root of 194, the answer to a different arrangement.
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A circle has its centre at (3, -2) and passes through the point (7, 1). What is its radius?
- A25
- Bthe square root of 7
- C5
- D7
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Answer: 5
The radius is the distance from the centre to the point: the gaps are 4 and 3, so the radius is the square root of 16 + 9 = 25, which is 5. The choice 25 stops one step early, at the square of the radius.
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What is the midpoint of the segment joining (-5, 8) and (3, -2)?
- A(-4, 5)
- B(-2, 6)
- C(4, -5)
- D(-1, 3)
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Answer: (-1, 3)
A midpoint averages each coordinate separately: (-5 + 3) / 2 = -1 and (8 + (-2)) / 2 = 3. The choice (-2, 6) adds the coordinates without halving them, and (4, -5) halves the differences instead, which gives a displacement rather than a point.
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In a 30-60-90 triangle, the shorter leg has length 7. What is the length of the hypotenuse?
- A14
- B7 times the square root of 3
- C7 times the square root of 2
- D3.5
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Answer: 14
In a 30-60-90 triangle the hypotenuse is twice the shorter leg, so it is 14. The square root of 3 belongs to the longer leg, and the square root of 2 to the 45-45-90 triangle.
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Two angles are complementary, and one is twice the size of the other. What is the measure of the larger angle?
- A30 degrees
- B60 degrees
- C120 degrees
- D45 degrees
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Answer: 60 degrees
Complementary angles total 90, so x + 2x = 90 gives x = 30 and the larger angle is 60. The choice 120 uses 180, which is the rule for supplementary angles.
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An isosceles triangle has a vertex angle of 44 degrees. What is the measure of each base angle?
- A44 degrees
- B136 degrees
- C68 degrees
- D46 degrees
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Answer: 68 degrees
The two base angles are equal and share what is left of 180: (180 - 44) / 2 = 68 degrees. The choice 136 is the pair added together rather than one of them, and 46 comes from using 90 instead of 180.
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The four angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6. What is the measure of the largest angle?
- A120 degrees
- B96 degrees
- C72 degrees
- D144 degrees
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Answer: 144 degrees
The angles of a quadrilateral total 360, and the ratio has 2 + 3 + 4 + 6 = 15 parts, so each part is 24 degrees. The largest is 6 x 24 = 144. Using 180 for the total, as for a triangle, gives 72.
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How many diagonals does a hexagon have?
- A9
- B6
- C12
- D15
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Answer: 9
Each of the 6 vertices joins to 3 others that are not itself or its two neighbours, giving 6 x 3 = 18 endpoints, and each diagonal has two, so there are 9. The choice 15 counts every pair of vertices, which includes the six sides.
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A sector of a circle of radius 12 has a central angle of 60 degrees. What is the area of the sector, in terms of pi?
- A12 pi
- B24 pi
- C48 pi
- D144 pi
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Answer: 24 pi
A 60-degree sector is 60/360 = one sixth of the circle. The whole circle has area pi x 12^2 = 144 pi, so the sector has 144 pi / 6 = 24 pi. The choice 144 pi is the whole circle.
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A cube has a volume of 343 cubic centimetres. Enter the length of one edge, in centimetres.
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Answer: 7
The volume of a cube is the edge cubed, so the edge is the cube root of 343, which is 7 because 7 x 7 x 7 = 343.
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A triangle has a base of 18 cm and a height of 11 cm. Enter its area in square centimetres.
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Answer: 99
The area of a triangle is half the base times the height: (18 x 11) / 2 = 198 / 2 = 99 square cm. Forgetting to halve gives 198.
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A rectangle has a perimeter of 34 cm and a width of 6 cm. What is its area?
- A66 square cm
- B204 square cm
- C28 square cm
- D102 square cm
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Answer: 66 square cm
The perimeter is 2(length + width), so 2(length + 6) = 34 gives length = 11 cm. The area is then 11 x 6 = 66 square cm. Multiplying the perimeter by the width gives 204.
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A circle has an area of 49 pi. What is its circumference, in terms of pi?
- A7 pi
- B14 pi
- C49 pi
- D98 pi
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Answer: 14 pi
From pi r^2 = 49 pi comes r^2 = 49, so r = 7. The circumference is 2 pi r = 14 pi. Answering 7 pi uses the radius where the diameter belongs.
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A rectangular prism measures 4 cm by 5 cm by 9 cm. What is its volume?
- A18 cubic cm
- B90 cubic cm
- C180 cubic cm
- D360 cubic cm
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Answer: 180 cubic cm
Volume is the product of the three dimensions: 4 x 5 x 9 = 180 cubic cm. The choice 18 adds them instead, and 90 multiplies only two.
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In a right triangle, the tangent of angle theta is 3/4 and the side opposite theta has length 9. What is the length of the side adjacent to theta?
- A6.75
- B15
- C3
- D12
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Answer: 12
Tangent is opposite over adjacent, so 9 / adjacent = 3/4 and adjacent = 9 x 4/3 = 12. Multiplying by 3/4 instead of dividing gives 6.75, and 15 is the hypotenuse of the 9-12-15 triangle this turns out to be.
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Two similar triangles have corresponding sides of 8 cm and 20 cm. The smaller triangle has a perimeter of 24 cm. What is the perimeter of the larger?
- A150 cm
- B9.6 cm
- C60 cm
- D48 cm
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Answer: 60 cm
Perimeter is a length, so it scales by the same factor as the sides: 20 / 8 = 2.5, and 24 x 2.5 = 60 cm. Squaring the factor, which is right for areas, gives 150.
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A line passes through (2, 3) and is perpendicular to the line y = -(1/4)x + 7. What is its slope?
- A-1/4
- B-4
- C1/4
- D4
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Answer: 4
Perpendicular slopes are negative reciprocals: the negative reciprocal of -1/4 is 4. Taking only the reciprocal gives -4 and only the negative gives 1/4; the point (2, 3) fixes the intercept, not the slope.
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Two angles of a triangle measure 37 degrees and 58 degrees. Enter the measure of the third angle in degrees.
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Answer: 85
The three angles total 180, so the third is 180 - 37 - 58 = 85 degrees.
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A right triangle has a hypotenuse of 25 and one leg of 7. Enter the length of the other leg.
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Answer: 24
The other leg is the square root of 25^2 - 7^2 = 625 - 49 = 576, which is 24. Because the hypotenuse is given, the two squares are subtracted rather than added.
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A circle has a diameter of 18 cm. What is its area, in terms of pi?
- A81 pi square cm
- B324 pi square cm
- C36 pi square cm
- D18 pi square cm
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Answer: 81 pi square cm
The radius is half the diameter, so 9 cm, and the area is pi x 9^2 = 81 pi square cm. Using the diameter in place of the radius gives 324 pi, four times too large.
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A cone has a radius of 3 cm and a height of 7 cm. What is its volume, in terms of pi?
- A63 pi cubic cm
- B21 pi cubic cm
- C7 pi cubic cm
- D9 pi cubic cm
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Answer: 21 pi cubic cm
The volume of a cone is one third of pi r^2 h: (1/3) x pi x 9 x 7 = 21 pi cubic cm. Omitting the one third gives 63 pi, which is the cylinder the cone sits inside.
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In a 30-60-90 triangle the shorter leg is 5 times the square root of 3. What is the length of the longer leg?
- A5
- B10 times the square root of 3
- C15
- D5 times the square root of 3
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Answer: 15
The longer leg is the shorter leg times the square root of 3, so it is 5 times the square root of 3, times the square root of 3, which is 5 x 3 = 15. The two roots multiply to 3 rather than staying under the radical.
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What is the slope of the line through (-2, -5) and (4, 7)?
- A1/2
- B-2
- C6
- D2
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Answer: 2
The change in y is 7 - (-5) = 12 and the change in x is 4 - (-2) = 6, so the slope is 12/6 = 2. Both subtractions involve a double negative, which is where the sign errors come from.
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One exterior angle of a triangle measures 115 degrees. What is the sum of the two interior angles not adjacent to it?
- A115 degrees
- B65 degrees
- C245 degrees
- D57.5 degrees
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Answer: 115 degrees
An exterior angle equals the sum of the two opposite interior angles, so the sum is 115 degrees. The choice 65 is the adjacent interior angle, which is the one this rule does not involve.
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A parallelogram has a base of 14 cm and a perpendicular height of 9 cm. What is its area?
- A63 square cm
- B126 square cm
- C23 square cm
- D46 square cm
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Answer: 126 square cm
The area of a parallelogram is base times perpendicular height: 14 x 9 = 126 square cm. Halving it, as for a triangle, gives 63.
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A regular pentagon has how many degrees in each of its exterior angles? Enter the number.
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Answer: 72
The exterior angles of any polygon total 360 degrees, and a regular pentagon has five equal ones: 360 / 5 = 72 degrees.
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A right triangle has legs of 20 and 21. Enter the length of its hypotenuse.
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Answer: 29
20^2 + 21^2 = 400 + 441 = 841, and the square root of 841 is 29. The 20-21-29 triangle is one of the less familiar right triangles with whole-number sides.
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A circle has a circumference of 44 pi. What is its radius?
- A44
- B11
- C22
- D88
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Answer: 22
Circumference is 2 pi r, so 2r = 44 and r = 22. Halving twice gives 11, which is the radius of a circle with half this circumference.
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A right triangle has legs of 7 and 24. What is its hypotenuse?
- A31
- B23
- C26
- D25
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Answer: 25
7^2 + 24^2 = 49 + 576 = 625, and the square root of 625 is 25. Adding the legs gives 31; the hypotenuse is always less than that.
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What is the measure of each interior angle of a regular octagon?
- A135 degrees
- B45 degrees
- C120 degrees
- D1,080 degrees
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Answer: 135 degrees
The interior angles total (8 - 2) x 180 = 1,080 degrees, and dividing among eight equal angles gives 135. The choice 45 is the exterior angle and 1,080 is the total.
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What is the midpoint of the segment joining (3, -7) and (11, 1)?
- A(7, -3)
- B(8, -6)
- C(14, -6)
- D(4, -4)
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Answer: (7, -3)
Average each coordinate: (3 + 11) / 2 = 7 and (-7 + 1) / 2 = -3. Adding without halving gives (14, -6), and halving the differences instead gives a displacement rather than a point.
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A closed cylinder has a radius of 5 and a height of 9. What is its total surface area, in terms of pi?
- A90 pi
- B140 pi
- C115 pi
- D225 pi
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Answer: 140 pi
The curved surface is 2 pi r h = 90 pi, and the two circular ends add 2 pi r^2 = 50 pi, giving 140 pi. Leaving out one end gives 115 pi and leaving out both gives 90 pi.
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In a right triangle, the side adjacent to angle theta is 12 and the hypotenuse is 13. What is cos theta?
- A5/13
- B5/12
- C12/13
- D13/12
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Answer: 12/13
Cosine is adjacent over hypotenuse, so 12/13. The remaining side is 5, making 5/13 the sine and 5/12 the tangent — all three are available, and the question asks for one.
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A sector of a circle of radius 8 has a central angle of 90 degrees. What is its area, in terms of pi?
- A64 pi
- B32 pi
- C8 pi
- D16 pi
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Answer: 16 pi
A 90-degree sector is a quarter of the circle. The whole circle has area 64 pi, so the sector has 16 pi. The choice 64 pi is the whole circle and 32 pi is half of it.
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Two similar figures have corresponding sides in the ratio 3 : 7. What is the ratio of their areas?
- A9 : 49
- B3 : 7
- C6 : 14
- D27 : 343
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Answer: 9 : 49
Areas scale by the square of the length ratio, so 3^2 : 7^2 = 9 : 49. The last choice cubes the ratio, which is right for volumes and not for areas.
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A circle has an area of 121 pi. What is its radius?
- A60.5
- B11
- C121
- D22
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Answer: 11
From pi r^2 = 121 pi comes r^2 = 121, so r = 11. The choice 121 stops before taking the square root, and 22 is the diameter.
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A right triangle has legs of 15 and 20. What is its hypotenuse?
- A35
- B30
- C25
- D27
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Answer: 25
15^2 + 20^2 = 225 + 400 = 625, and the square root of 625 is 25. This is the 3-4-5 triangle scaled by 5, which is worth recognising on sight.
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What is the measure of each interior angle of a regular nonagon?
- A40 degrees
- B1,260 degrees
- C120 degrees
- D140 degrees
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Answer: 140 degrees
The interior angles total (9 - 2) x 180 = 1,260 degrees, and dividing among nine equal angles gives 140. The choice 40 is the exterior angle and 1,260 is the total.
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What is the midpoint of the segment joining (-6, 3) and (2, 11)?
- A(-2, 7)
- B(-4, 14)
- C(4, -4)
- D(-8, -8)
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Answer: (-2, 7)
Average each coordinate: (-6 + 2) / 2 = -2 and (3 + 11) / 2 = 7. Adding without halving gives (-4, 14).
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A cylinder has a radius of 6 and a height of 4. What is its volume, in terms of pi?
- A48 pi
- B144 pi
- C96 pi
- D576 pi
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Answer: 144 pi
Volume is pi r^2 h = pi x 36 x 4 = 144 pi. Using r rather than r^2 gives 24 pi, and doubling the radius before squaring gives 576 pi.
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In a right triangle, the side opposite angle theta is 9 and the hypotenuse is 15. What is sin theta?
- A4/5
- B3/4
- C3/5
- D5/3
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Answer: 3/5
Sine is opposite over hypotenuse: 9/15 = 3/5. The remaining side is 12, making 4/5 the cosine and 3/4 the tangent — all three are available and the question asks for one.
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A sector of a circle of radius 9 has a central angle of 120 degrees. Enter the area of the sector as a multiple of pi.
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Answer: 27
A 120-degree sector is one third of the circle. The whole circle has area 81 pi, so the sector has 27 pi.
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Two similar solids have corresponding edges in the ratio 5 : 8. What is the ratio of their volumes?
- A5 : 8
- B25 : 64
- C10 : 16
- D125 : 512
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Answer: 125 : 512
Volumes scale by the cube of the length ratio: 5^3 : 8^3 = 125 : 512. Squaring instead gives 25 : 64, which is the ratio of their surface areas.
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