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
- 1Triangle anglesFoundation
Two angles of a triangle measure 48 degrees and 67 degrees. What is the measure of the third angle?
- A60
- B55
- C65
- D115
Show the answer and explanationHide the explanation
Answer: 65
The angles of a triangle total 180 degrees. 180 - 48 - 67 = 65.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 2AreaFoundation
What is the area of a rectangle with length 8 and width 5?
- A40
- B26
- C80
- D13
Show the answer and explanationHide the explanation
Answer: 40
Area of a rectangle is length times width: 8 x 5 = 40.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 3CirclesStandard
A circle has radius 4. What is its circumference in terms of pi?
- A32pi
- B4pi
- C16pi
- D8pi
Show the answer and explanationHide the explanation
Answer: 8pi
Circumference is 2pi r. With r = 4, the circumference is 8pi.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 4Right trianglesStandard
A right triangle has legs of length 5 and 12. What is the length of its hypotenuse?
- A25
- B17
- C15
- D13
Show the answer and explanationHide the explanation
Answer: 13
Use the Pythagorean theorem: 5^2 + 12^2 = 25 + 144 = 169. The square root of 169 is 13.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 5AreaStandard
A trapezoid has bases of length 6 and 10 and height 4. What is its area?
- A64
- B16
- C24
- D32
Show the answer and explanationHide the explanation
Answer: 32
Area = one-half times the sum of the bases times the height: (6 + 10) x 4 / 2 = 32.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 6Similar trianglesStandard
Two similar triangles have corresponding side lengths in the ratio 3:5. If the longer corresponding side is 15, what is the shorter side?
- A9
- B12
- C10
- D6
Show the answer and explanationHide the explanation
Answer: 9
The scale factor from 5 to 15 is 3. Apply it to 3: 3 x 3 = 9.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 7VolumeFoundation
A cube has side length 3. Enter its volume.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 27
Volume of a cube is side cubed: 3 x 3 x 3 = 27.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 8Coordinate geometryFoundation
What is the distance between (-1, 2) and (3, 2) on a coordinate plane?
- A2
- B5
- C4
- D8
Show the answer and explanationHide the explanation
Answer: 4
The points have the same y-coordinate, so the horizontal distance is 3 - (-1) = 4.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 9AreaFoundation
A square has perimeter 36. What is its area?
- A36
- B18
- C72
- D81
Show the answer and explanationHide the explanation
Answer: 81
Each side is 36 / 4 = 9. Its area is 9 x 9 = 81.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 10AreaFoundation
A triangle has base 10 and height 7. What is its area?
- A17.5
- B35
- C140
- D70
Show the answer and explanationHide the explanation
Answer: 35
Triangle area is one-half times base times height: 10 x 7 / 2 = 35.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 11VolumeStandard
A cylinder has radius 2 and height 5. What is its volume in terms of pi?
- A25pi
- B10pi
- C20pi
- D40pi
Show the answer and explanationHide the explanation
Answer: 20pi
Cylinder volume is pi r^2 h. That is pi x 2^2 x 5 = 20pi.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 12CirclesFoundation
A circle has diameter 10. What is its area in terms of pi?
- A5pi
- B10pi
- C25pi
- D100pi
Show the answer and explanationHide the explanation
Answer: 25pi
The radius is half the diameter, so r = 5. Area is pi r^2 = 25pi.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 13Right trianglesFoundation
A right triangle has one leg of length 6 and a hypotenuse of length 10. Enter the length of the other leg.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 8
By the Pythagorean theorem, 6^2 + b^2 = 10^2, so b^2 = 100 - 36 = 64 and b = 8.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 14Trigonometric ratiosStandard
In a right triangle, the side opposite angle A has length 3 and the hypotenuse has length 5. What is sin A?
- A3/5
- B4/5
- C3/4
- D5/3
Show the answer and explanationHide the explanation
Answer: 3/5
Sine is the ratio of the opposite side to the hypotenuse, which is 3/5.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 15AnglesFoundation
Two parallel lines are cut by a transversal, and one angle measures 65 degrees. What is the measure of its corresponding angle?
- A130 degrees
- B115 degrees
- C65 degrees
- D25 degrees
Show the answer and explanationHide the explanation
Answer: 65 degrees
Corresponding angles formed by a transversal crossing parallel lines are congruent, so the angle also measures 65 degrees.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 16VolumeChallenge
A sphere has radius 3. What is its volume in terms of pi?
- A36pi
- B12pi
- C108pi
- D27pi
Show the answer and explanationHide the explanation
Answer: 36pi
The volume of a sphere is (4/3)pi r^3. With r = 3, r^3 = 27, so the volume is (4/3)(27)pi = 36pi.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 17Coordinate geometryStandard
What is the midpoint of the segment joining (2, -1) and (8, 5)?
- A(3, 3)
- B(5, 2)
- C(6, 6)
- D(10, 4)
Show the answer and explanationHide the explanation
Answer: (5, 2)
Average each coordinate: x is (2 + 8) / 2 = 5 and y is (-1 + 5) / 2 = 2.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 18Similar trianglesStandard
Triangle ABC is similar to triangle DEF, with AB = 4, DE = 10, and BC = 6. Enter the length of EF.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 15
The scale factor from ABC to DEF is 10 / 4 = 2.5. Applying it to BC gives EF = 6 x 2.5 = 15.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 19CirclesChallenge
If a circle's radius is doubled, by what factor does its area increase?
- A4
- B8
- C16
- D2
Show the answer and explanationHide the explanation
Answer: 4
Area is pi r^2. Replacing r with 2r gives pi(2r)^2 = 4pi r^2, so the area is 4 times as large.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 20Coordinate geometryFoundation
Enter the distance between the points (0, 0) and (6, 8).
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 10
The distance is the square root of 6^2 + 8^2 = 36 + 64 = 100, which is 10.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 21PolygonsStandard
What is the sum of the interior angles of a pentagon?
- A720 degrees
- B360 degrees
- C900 degrees
- D540 degrees
Show the answer and explanationHide the explanation
Answer: 540 degrees
The sum of the interior angles of a polygon with n sides is (n - 2) x 180 degrees. For n = 5, that is 3 x 180 = 540 degrees.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 22Trigonometric ratiosStandard
In a right triangle, the leg opposite angle B has length 7 and the leg adjacent to angle B has length 24. What is tan B?
- A7/24
- B24/7
- C7/25
- D24/25
Show the answer and explanationHide the explanation
Answer: 7/24
Tangent is the ratio of the opposite leg to the adjacent leg, which is 7/24. The hypotenuse is not used.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 23VolumeStandard
A cone has radius 3 and height 4. What is its volume in terms of pi?
- A9pi
- B16pi
- C12pi
- D36pi
Show the answer and explanationHide the explanation
Answer: 12pi
The volume of a cone is (1/3)pi r^2 h. With r = 3 and h = 4, that is (1/3)(9)(4)pi = 12pi, which is one third of the cylinder with the same base and height.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 24Special right trianglesStandard
In a 30-60-90 triangle, the shortest side has length 5. What is the length of the hypotenuse?
- A10
- B5 sqrt(3)
- C15
- D5 sqrt(2)
Show the answer and explanationHide the explanation
Answer: 10
The sides of a 30-60-90 triangle are in the ratio x, x sqrt(3), 2x, with the hypotenuse twice the shortest side. With x = 5, the hypotenuse is 10.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 25Trigonometric ratiosChallenge
In a right triangle, angle A and angle B are the two acute angles. If sin A = 0.6, what is cos B?
- A1.6
- B0.6
- C0.4
- D0.8
Show the answer and explanationHide the explanation
Answer: 0.6
The two acute angles in a right triangle are complementary, and sin(x) = cos(90 - x). Because B = 90 - A, cos B equals sin A, which is 0.6.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 26Angle measureFoundation
An angle measures 90 degrees. What is its measure in radians?
- Api/2
- Bpi/4
- C2pi
- Dpi
Show the answer and explanationHide the explanation
Answer: pi/2
Because 180 degrees equals pi radians, multiply by pi/180: 90 x pi/180 = pi/2.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 27CirclesChallenge
A circle has area 36pi. Enter its circumference divided by pi.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 12
From pi r^2 = 36pi, r^2 = 36 and r = 6. The circumference is 2pi r = 12pi, so dividing by pi gives 12.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 28Triangle anglesFoundation
In an isosceles triangle, the two equal angles each measure 52 degrees. What is the measure of the third angle?
- A76 degrees
- B104 degrees
- C128 degrees
- D52 degrees
Show the answer and explanationHide the explanation
Answer: 76 degrees
The three angles total 180 degrees. Two of them account for 104 degrees, so the third is 180 - 104 = 76 degrees.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 29VolumeStandard
A rectangular tank measures 4 by 3 by 2 metres. If its height is doubled and the other dimensions are unchanged, what happens to its volume?
- AIt increases by 8 times.
- BIt doubles.
- CIt quadruples.
- DIt is unchanged.
Show the answer and explanationHide the explanation
Answer: It doubles.
Volume is length times width times height. Scaling only one dimension by 2 multiplies the volume by 2. Doubling all three would multiply it by 8.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 30Coordinate geometryStandard
A circle has centre (3, -1) and passes through (3, 4). What is its radius?
- Asqrt(10)
- B5
- C3
- D4
Show the answer and explanationHide the explanation
Answer: 5
The two points share an x-coordinate, so the distance between them is the difference in y: 4 - (-1) = 5. No distance formula is needed.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 31Similar trianglesChallenge
A 1.8-metre person casts a 2.4-metre shadow at the same moment a tree casts a 20-metre shadow. Enter the height of the tree in metres.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 15
The person and the tree form similar right triangles with the sun's rays, so height over shadow is constant: 1.8 / 2.4 = 0.75. The tree's height is 0.75 x 20 = 15 metres.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 32Trigonometric ratiosStandard
In a right triangle, the legs measure 8 and 15. What is the cosine of the angle adjacent to the leg of length 8?
- A15/8
- B8/17
- C8/15
- D15/17
Show the answer and explanationHide the explanation
Answer: 8/17
The hypotenuse is sqrt(64 + 225) = sqrt(289) = 17. Cosine is the adjacent leg over the hypotenuse, and the leg adjacent to that angle is 8, giving 8/17.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 33CirclesStandard
A circle has centre (2, -3) and radius 5. Which equation describes it?
- A(x - 2)^2 + (y + 3)^2 = 25
- B(x - 2)^2 + (y + 3)^2 = 5
- C(x + 2)^2 + (y - 3)^2 = 25
- D(x - 2)^2 + (y - 3)^2 = 25
Show the answer and explanationHide the explanation
Answer: (x - 2)^2 + (y + 3)^2 = 25
The equation is (x - h)^2 + (y - k)^2 = r^2. With centre (2, -3), y - (-3) becomes y + 3, and the right side is 5^2 = 25.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 34Right trianglesChallenge
A ladder 13 metres long leans against a wall with its foot 5 metres from the base. Enter how far up the wall it reaches, in metres.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 12
The ladder is the hypotenuse, so 5^2 + h^2 = 13^2 gives h^2 = 169 - 25 = 144 and h = 12 metres.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 35PolygonsStandard
Each interior angle of a regular polygon measures 140 degrees. How many sides does it have?
- A9
- B7
- C8
- D10
Show the answer and explanationHide the explanation
Answer: 9
Each exterior angle is 180 - 140 = 40 degrees, and exterior angles total 360, so there are 360 / 40 = 9 sides. Working through the exterior angle is faster than the interior-sum formula.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 36AreaStandard
A rectangle has perimeter 34 and length 10. What is its area?
- A140
- B170
- C240
- D70
Show the answer and explanationHide the explanation
Answer: 70
Perimeter 34 with length 10 gives 2(10) + 2w = 34, so w = 7. The area is 10 x 7 = 70.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 37Trigonometric ratiosChallenge
In a right triangle, tan A = 3/4. What is sin A?
- A3/5
- B4/5
- C5/3
- D3/4
Show the answer and explanationHide the explanation
Answer: 3/5
Tangent 3/4 means the opposite and adjacent sides are in the ratio 3 to 4, which makes the hypotenuse 5 by the Pythagorean theorem. Sine is opposite over hypotenuse, so 3/5.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 38VolumeFoundation
A rectangular box measures 5 by 4 by 3 centimetres. Enter its volume in cubic centimetres.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 60
Volume is length times width times height: 5 x 4 x 3 = 60 cubic centimetres.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 39Coordinate geometryStandard
A line segment has one endpoint at (1, 2) and midpoint at (4, 6). What is the other endpoint?
- A(5, 8)
- B(3, 4)
- C(2.5, 4)
- D(7, 10)
Show the answer and explanationHide the explanation
Answer: (7, 10)
The midpoint is the average of the endpoints, so the other endpoint is twice the midpoint minus the known one: (8 - 1, 12 - 2) = (7, 10).
How did you do?Saved in this browser. Weak Skills will replay this skill. - 40AnglesFoundation
Two angles are supplementary and one measures 47 degrees. What is the other?
- A137 degrees
- B43 degrees
- C133 degrees
- D313 degrees
Show the answer and explanationHide the explanation
Answer: 133 degrees
Supplementary angles total 180 degrees, so the other is 180 - 47 = 133 degrees. Complementary angles, which total 90, would give 43.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 41Triangle anglesStandard
In a triangle, the second angle is twice the first and the third is three times the first. What is the largest angle?
- A60 degrees
- B30 degrees
- C120 degrees
- D90 degrees
Show the answer and explanationHide the explanation
Answer: 90 degrees
The angles are x, 2x and 3x, totalling 6x = 180, so x = 30. The largest is 3x = 90 degrees.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 42CirclesChallenge
A circle has equation (x - 3)^2 + (y + 1)^2 = 49. What is its radius?
- A24.5
- B14
- C49
- D7
Show the answer and explanationHide the explanation
Answer: 7
The standard form sets the right-hand side to r^2, so r^2 = 49 and r = 7. Reading 49 as the radius skips the square root.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 43AreaStandard
A circle of radius 3 is cut from a square of side 6. What area remains, in terms of pi?
- A36 - 6pi
- B36 - 3pi
- C36 - 9pi
- D18 - 9pi
Show the answer and explanationHide the explanation
Answer: 36 - 9pi
The square has area 36 and the circle pi(3)^2 = 9pi. The circle fits exactly inside, so the remaining area is 36 - 9pi.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 44VolumeChallenge
A cylinder and a cone have the same radius and the same height. Enter the number of cone-fulls needed to fill the cylinder.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 3
A cone's volume is one third of the cylinder with the same base and height, so three cone-fulls fill it.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 45Coordinate geometryStandard
What is the slope of a line perpendicular to the segment joining (1, 3) and (5, 11)?
- A-2
- B-1/2
- C2
- D1/2
Show the answer and explanationHide the explanation
Answer: -1/2
The segment's slope is (11 - 3) / (5 - 1) = 2. A perpendicular slope is the negative reciprocal, which is -1/2.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 46Trigonometric ratiosChallenge
In a right triangle, cos B = 5/13. What is tan B?
- A13/5
- B12/5
- C5/12
- D12/13
Show the answer and explanationHide the explanation
Answer: 12/5
Cosine 5/13 means the adjacent side is 5 and the hypotenuse 13, so the opposite side is 12 by the Pythagorean theorem. Tangent is opposite over adjacent, giving 12/5.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 47PolygonsFoundation
What is the measure of each exterior angle of a regular hexagon?
- A720 degrees
- B60 degrees
- C45 degrees
- D120 degrees
Show the answer and explanationHide the explanation
Answer: 60 degrees
Exterior angles of any polygon total 360 degrees. For a regular hexagon that is 360 / 6 = 60 degrees each. The 120-degree figure is the interior angle.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 48Similar trianglesStandard
Two similar rectangles have widths 4 and 10. If the smaller has area 24, what is the area of the larger?
- A240
- B96
- C150
- D60
Show the answer and explanationHide the explanation
Answer: 150
The scale factor is 10/4 = 2.5, and area scales with its square: 24 x 6.25 = 150. Multiplying the area by 2.5 gives 60, which is the trap.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 49AreaStandard
A triangle has vertices at (0, 0), (6, 0) and (0, 8). What is its area?
- A10
- B48
- C24
- D14
Show the answer and explanationHide the explanation
Answer: 24
The legs along the axes are the base and height, 6 and 8. Area is (1/2)(6)(8) = 24. The value 48 is the rectangle, not the triangle.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 50CirclesStandard
A sector of a circle of radius 6 has a central angle of 60 degrees. What is its area in terms of pi?
- A3pi
- B12pi
- C6pi
- D36pi
Show the answer and explanationHide the explanation
Answer: 6pi
A 60-degree sector is one sixth of the circle. The full area is 36pi, so the sector is 36pi / 6 = 6pi.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 51Right trianglesChallenge
A rectangle measures 9 by 12. Enter the length of its diagonal.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 15
The diagonal is the hypotenuse of a right triangle with legs 9 and 12: sqrt(81 + 144) = sqrt(225) = 15.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 52VolumeStandard
A cylinder has volume 100pi and height 4. What is its radius?
- A25
- B5
- C10
- D6.25
Show the answer and explanationHide the explanation
Answer: 5
From pi r^2 h = 100pi with h = 4: r^2 = 25, so r = 5. Stopping at 25 skips the square root.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 53AnglesStandard
Two angles are complementary, and one is twice the other. What is the larger angle?
- A30 degrees
- B45 degrees
- C120 degrees
- D60 degrees
Show the answer and explanationHide the explanation
Answer: 60 degrees
Complementary angles total 90 degrees, so x + 2x = 90 gives x = 30 and the larger is 60. Using 180 would make them supplementary instead.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 54Similar trianglesChallenge
A 2-metre pole casts a 3-metre shadow. At the same moment a building casts a 42-metre shadow. Enter the building's height in metres.
Student-produced response: write your own answer rather than choosing one.
Show the answer and explanationHide the explanation
Answer: 28
Height over shadow is constant in similar right triangles: 2/3 of 42 is 28 metres.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 55Trigonometric ratiosStandard
In a right triangle with the right angle at C, side a = 6 and side b = 8. What is sin A, where a is opposite angle A?
- A0.75
- B0.8
- C1.25
- D0.6
Show the answer and explanationHide the explanation
Answer: 0.6
The hypotenuse is sqrt(36 + 64) = 10. Sine is the opposite side over the hypotenuse: 6/10 = 0.6.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 56Coordinate geometryStandard
Which point lies on the line y = 2x - 5?
- A(2, 1)
- B(4, 3)
- C(3, 4)
- D(0, 5)
Show the answer and explanationHide the explanation
Answer: (4, 3)
Substituting x = 4 gives y = 8 - 5 = 3, so (4, 3) is on the line. Checking (2, 1) gives y = -1, not 1.
How did you do?Saved in this browser. Weak Skills will replay this skill.
SAT® is a registered trademark of the College Board, which does not endorse this study guide.