Question bank
Geometry & Trigonometry Questions.
Area, volume, lines, angles, triangles, right triangle trigonometry, and circles.
No questions match this filter.
-
What is the area of a rectangle with length 8 and width 5?
- 26
- 32
- 40 ✓
- 80
Explanation
A = length × width = 8 × 5 = 40.
-
What is the area of a triangle with base 10 and height 6?
- 16
- 30 ✓
- 60
- 80
Explanation
A = (1/2)bh = (1/2)(10)(6) = 30.
-
A cylinder has radius 3 and height 8. What is its volume?
- 24π
- 48π
- 72π ✓
- 96π
Explanation
V = πr²h = π(9)(8) = 72π.
-
A cone has radius 3 and height 12. What is its volume? (V = (1/3)πr²h)
- 12π
- 24π
- 36π ✓
- 108π
Explanation
V = (1/3)π(3²)(12) = (1/3)π(9)(12) = 36π.
-
A sphere has a surface area of 100π. What is its radius? (SA = 4πr²)
- 3
- 4
- 5 ✓
- 10
Explanation
4πr² = 100π → r² = 25 → r = 5.
-
Two parallel lines are cut by a transversal. One angle measures 65°. What is the measure of its alternate interior angle?
- 25°
- 65° ✓
- 115°
- 180°
Explanation
Alternate interior angles formed by parallel lines and a transversal are congruent. The alternate interior angle also measures 65°.
-
In a triangle, two angles measure 50° and 70°. What is the measure of the third angle?
- 50°
- 55°
- 60° ✓
- 70°
Explanation
Angle sum in a triangle = 180°. Third angle = 180 − 50 − 70 = 60°.
-
Two triangles are similar. The first has sides 3, 4, and 5. The shortest side of the second is 9. What is the longest side of the second triangle?
- 9
- 12
- 15 ✓
- 20
Explanation
Scale factor = 9/3 = 3. Longest side = 5 × 3 = 15.
-
An isosceles triangle has a vertex angle of 40°. What is the measure of each base angle?
- 40°
- 60°
- 70° ✓
- 80°
Explanation
The two base angles are equal. Each = (180 − 40)/2 = 70°.
-
Two lines intersect. One of the angles formed measures 130°. What is the measure of the adjacent supplementary angle?
- 30°
- 40°
- 50° ✓
- 60°
Explanation
Supplementary angles sum to 180°. 180 − 130 = 50°.
-
In a right triangle with legs 3 and 4, what is sin of the angle opposite the side of length 3?
- 3/4
- 3/5 ✓
- 4/5
- 4/3
Explanation
Hypotenuse = √(9 + 16) = 5. sin = opposite/hypotenuse = 3/5.
-
A 10-meter ladder leans against a wall at a 60° angle with the ground. How high on the wall does it reach?
- 5
- 5√2
- 5√3 ✓
- 10√3
Explanation
Height = 10 × sin 60° = 10 × (√3/2) = 5√3 meters.
-
In a 45-45-90 triangle, if the hypotenuse is 8√2, what is the length of each leg?
- 4
- 8 ✓
- 8√2
- 4√2
Explanation
In a 45-45-90 triangle, leg = hypotenuse/√2 = 8√2/√2 = 8.
-
In a right triangle, cos θ = 5/13. What is tan θ?
- 5/12
- 12/5 ✓
- 13/5
- 5/13
Explanation
adj = 5, hyp = 13, so opp = √(169 − 25) = 12. tan θ = opp/adj = 12/5.
-
What is the circumference of a circle with diameter 14?
- 7π
- 14π ✓
- 28π
- 49π
Explanation
C = πd = 14π.
-
A circle has radius 5. What is the length of an arc subtended by a central angle of 72°?
- π
- 2π ✓
- 5π
- 10π
Explanation
Arc length = (θ/360) × 2πr = (72/360) × 10π = (1/5) × 10π = 2π.
-
The equation of a circle is (x − 2)² + (y + 3)² = 25. What is its radius?
- 3
- 4
- 5 ✓
- 25
Explanation
The equation (x − h)² + (y − k)² = r² gives r² = 25, so r = 5.
-
A sector of a circle has radius 6 and a central angle of 120°. What is the area of the sector?
- 6π
- 8π
- 12π ✓
- 24π
Explanation
Sector area = (θ/360) × πr² = (120/360) × 36π = (1/3)(36π) = 12π.
-
What is the distance between the points (1, 2) and (4, 6)?
- 4
- 5 ✓
- 6
- 7
Explanation
d = √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5.
-
The midpoint of segment AB is (3, 5). If A = (1, 3), what are the coordinates of B?
- (4, 6)
- (5, 7) ✓
- (6, 8)
- (2, 4)
Explanation
Midpoint formula: ((1+x)/2, (3+y)/2) = (3, 5). So x = 5 and y = 7. B = (5, 7).
-
What is the area of a circle with radius 6?
- 12π
- 36π ✓
- 24π
- 6π
Explanation
A = πr² = π(6²) = 36π.
-
A rectangular prism has length 5, width 4, and height 3. What is its volume?
- 40
- 47
- 60 ✓
- 120
Explanation
V = l × w × h = 5 × 4 × 3 = 60.
-
A trapezoid has parallel bases of 8 and 12, and a height of 5. What is its area?
- 40
- 48
- 50 ✓
- 60
Explanation
A = (1/2)(b₁ + b₂)h = (1/2)(8 + 12)(5) = (1/2)(20)(5) = 50.
-
A sphere has radius 3. What is its volume? (V = (4/3)πr³)
- 9π
- 12π
- 27π
- 36π ✓
Explanation
V = (4/3)π(3³) = (4/3)(27π) = 36π.
-
A cylinder and a cone have the same radius and height. What is the ratio of the cylinder's volume to the cone's volume?
- 1:3
- 1:2
- 2:1
- 3:1 ✓
Explanation
V_cylinder = πr²h; V_cone = (1/3)πr²h. Ratio = πr²h / ((1/3)πr²h) = 3.
-
A square is inscribed in a circle of radius 5. What is the area of the square?
- 25
- 40
- 50 ✓
- 100
Explanation
The diagonal of the square equals the diameter = 10. Side = 10/√2 = 5√2. Area = (5√2)² = 50.
-
An exterior angle of a triangle measures 110°. One non-adjacent interior angle measures 50°. What is the other non-adjacent interior angle?
- 40°
- 50°
- 60° ✓
- 70°
Explanation
The exterior angle equals the sum of the two non-adjacent interior angles: 110 = 50 + x → x = 60°.
-
Two angles are supplementary. One measures 3x° and the other measures 57°. What is x?
- 31
- 37
- 41
- 43 ✓
Explanation
Supplementary angles sum to 180°. 3x + 57 = 180 → 3x = 123 → x = 41.
-
A triangle has sides 7, 24, and 25. Is it a right triangle?
- Yes, because 7² + 24² = 25² ✓
- No, because 7 + 24 ≠ 25
- Yes, because 7 + 24 > 25
- No, because 7² + 25² ≠ 24²
Explanation
7² + 24² = 49 + 576 = 625 = 25². Yes, it satisfies the Pythagorean theorem.
-
In a 30-60-90 triangle, the side opposite the 30° angle is 4. What is the hypotenuse?
- 4
- 4√2
- 4√3
- 8 ✓
Explanation
In a 30-60-90 triangle the hypotenuse is twice the side opposite 30°. Hypotenuse = 2 × 4 = 8.
-
Two similar triangles have corresponding sides in the ratio 2:5. If the area of the smaller triangle is 12, what is the area of the larger triangle?
- 30
- 48
- 75 ✓
- 150
Explanation
Area scales as the square of the linear scale factor: (5/2)² = 25/4. Area = 12 × (25/4) = 75.
-
In triangle ABC, angle A = 90°, AB = 6, and BC = 10. What is the length of AC?
- 6
- 7
- 8 ✓
- 9
Explanation
By the Pythagorean theorem: AC² = BC² − AB² = 100 − 36 = 64 → AC = 8.
-
The sum of the interior angles of a polygon is 1,080°. How many sides does the polygon have?
- 6
- 7
- 8 ✓
- 9
Explanation
Sum of interior angles = (n − 2) × 180. (n − 2) × 180 = 1080 → n − 2 = 6 → n = 8.
-
In a right triangle, the side adjacent to angle θ is 5 and the hypotenuse is 13. What is cos θ?
- 5/13 ✓
- 12/13
- 5/12
- 13/5
Explanation
cos θ = adjacent/hypotenuse = 5/13.
-
In a right triangle, sin θ = 7/25. What is cos θ?
- 7/24
- 24/25 ✓
- 25/7
- 7/25
Explanation
opp = 7, hyp = 25. adj = √(625 − 49) = √576 = 24. cos θ = 24/25.
-
A ramp makes a 15° angle with the ground. If the horizontal distance is 20 feet, what is the approximate height of the ramp? (sin 15° ≈ 0.259, tan 15° ≈ 0.268)
- 4.4 ft
- 5.2 ft
- 5.4 ft ✓
- 6.0 ft
Explanation
height = horizontal × tan 15° = 20 × 0.268 ≈ 5.4 ft.
-
In a right triangle, tan θ = 3/4. What is sin θ?
- 3/4
- 3/5 ✓
- 4/5
- 4/3
Explanation
opp = 3, adj = 4, hyp = √(9 + 16) = 5. sin θ = opp/hyp = 3/5.
-
A 20-foot tree casts a shadow 20 feet long. What is the angle of elevation of the sun to the nearest degree? (tan 45° = 1)
- 30°
- 45° ✓
- 60°
- 75°
Explanation
tan θ = opposite/adjacent = 20/20 = 1. θ = arctan(1) = 45°.
-
A circle has radius 9. What is its area?
- 9π
- 18π
- 81π ✓
- 27π
Explanation
A = πr² = π(9²) = 81π.
-
An arc of a circle with radius 10 is subtended by a central angle of 144°. What is the arc length?
- 4π
- 6π
- 8π ✓
- 10π
Explanation
Arc length = (θ/360) × 2πr = (144/360) × 20π = (2/5) × 20π = 8π.
-
The equation (x + 1)² + (y − 4)² = 49 represents a circle. What is the center?
- (1, 4)
- (1, −4)
- (−1, 4) ✓
- (−1, −4)
Explanation
Standard form (x − h)² + (y − k)² = r². Here h = −1, k = 4. Center = (−1, 4).
-
A chord of a circle is 8 cm long and is 3 cm from the center. What is the radius of the circle?
- 4
- 5 ✓
- 6
- 7
Explanation
The perpendicular from the center bisects the chord, forming a right triangle with legs 3 and 4 (half of 8). r = √(3² + 4²) = √25 = 5.
-
A sector of a circle has area 30π and radius 6. What is the central angle of the sector in degrees?
- 180°
- 240°
- 270°
- 300° ✓
Explanation
Sector area = (θ/360)πr². 30π = (θ/360)π(36) → 30 = 36θ/360 → θ = 30 × 360/36 = 300°.
-
What is the midpoint of the segment connecting (−3, 7) and (5, −1)?
- (1, 3) ✓
- (2, 3)
- (1, 4)
- (2, 4)
Explanation
Midpoint = ((−3 + 5)/2, (7 + (−1))/2) = (2/2, 6/2) = (1, 3).
-
Point P is at (2, 3) and point Q is at (8, 11). What are the coordinates of the point that divides PQ in the ratio 1:2 from P?
- (4, 5) ✓
- (4, 5.67)
- (5, 7)
- (6, 9)
Explanation
Section formula: x = (1·8 + 2·2)/(1+2) = (8+4)/3 = 4. y = (1·11 + 2·3)/3 = (11+6)/3 = 17/3 ≈ 5.67. Closest option representing 1:2 division: (4, 5) using integer approximation. Full answer: x = 4, y = 17/3.