Free SAT Math Additional Topics Geometry and Trigonometry Practice Test — Questions and Answers

Question 1: What is the area of a circle with radius 5?

5π
10π
25π (Correct answer)
50π

Correct answer: 25π

Area = πr² = π(5)² = 25π.

Question 2: In a right triangle, the two legs have lengths 3 and 4. What is the hypotenuse?

5 (Correct answer)
6
7
√7

Correct answer: 5

By the Pythagorean theorem: c² = 3² + 4² = 9 + 16 = 25. So c = 5.

Question 3: What is the circumference of a circle with diameter 10?

5π
10π (Correct answer)
20π
100π

Correct answer: 10π

Circumference = πd = π(10) = 10π.

Question 4: A rectangle has length 8 and width 5. What is its perimeter?

26 (Correct answer)
40
13
80

Correct answer: 26

Perimeter = 2(l + w) = 2(8 + 5) = 2(13) = 26.

Question 5: In a right triangle, if sin(θ) = 3/5, what is cos(θ)?

4/5 (Correct answer)
3/4
5/3
1/2

Correct answer: 4/5

Using the Pythagorean identity: sin²θ + cos²θ = 1. (3/5)² + cos²θ = 1 → 9/25 + cos²θ = 1 → cos²θ = 16/25 → cosθ = 4/5.

Question 6: What is the volume of a rectangular prism with length 4, width 3, and height 5?

12
24
47
60 (Correct answer)

Correct answer: 60

Volume = l × w × h = 4 × 3 × 5 = 60.

Question 7: The sum of interior angles in a pentagon (5 sides) is:

360°
450°
540° (Correct answer)
720°

Correct answer: 540°

Sum of interior angles = (n - 2) × 180°. For n = 5: (5 - 2) × 180° = 3 × 180° = 540°.

Question 8: A right triangle has hypotenuse 13 and one leg of 5. What is the other leg?

8
10
12 (Correct answer)
14

Correct answer: 12

By Pythagorean theorem: 5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = 12.

Question 9: In a right triangle, if tan(θ) = 1, what is θ?

30°
45° (Correct answer)
60°
90°

Correct answer: 45°

tan(45°) = 1. When the opposite and adjacent sides are equal, the angle is 45°.

Question 10: What is the area of a triangle with base 10 and height 6?

30 (Correct answer)
60
16
56

Correct answer: 30

Area = (1/2) × base × height = (1/2) × 10 × 6 = 30.

Question 11: Two angles of a triangle measure 55° and 75°. What is the third angle?

40°
50° (Correct answer)
60°
70°

Correct answer: 50°

The sum of angles in a triangle is 180°. Third angle = 180° - 55° - 75° = 50°.

Question 12: What is the distance between the points (1, 2) and (4, 6)?

3
4
5 (Correct answer)
7

Correct answer: 5

Distance = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5.

Question 13: A sphere has radius 3. What is its volume? (V = (4/3)πr³)

9π
12π
36π (Correct answer)
108π

Correct answer: 36π

V = (4/3)π(3)³ = (4/3)π(27) = 36π.

Question 14: In a right triangle with hypotenuse 1 and angle θ = 30°, what is the length of the side opposite to θ?

√3/2
1/2 (Correct answer)
√2/2
1

Correct answer: 1/2

sin(30°) = opposite/hypotenuse = opposite/1 = 1/2. The opposite side is 1/2.

Question 15: The midpoint of segment with endpoints (2, 4) and (8, 10) is:

(4, 6)
(5, 7) (Correct answer)
(6, 8)
(3, 5)

Correct answer: (5, 7)

Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2) = ((2+8)/2, (4+10)/2) = (5, 7).

Question 16: A cylinder has radius 4 and height 7. What is its volume? (V = πr²h)

28π
56π
112π (Correct answer)
196π

Correct answer: 112π

V = π(4²)(7) = π(16)(7) = 112π.

Question 17: Parallel lines are cut by a transversal. Which pair of angles are always equal?

Co-interior angles
Supplementary angles
Alternate interior angles (Correct answer)
Adjacent angles

Correct answer: Alternate interior angles

When parallel lines are cut by a transversal, alternate interior angles are equal (congruent).

Question 18: A 45-45-90 triangle has legs of length 6. What is the hypotenuse?

6
6√2 (Correct answer)
6√3
12

Correct answer: 6√2

In a 45-45-90 triangle, the hypotenuse = leg × √2 = 6√2.

Question 19: What is the area of a trapezoid with parallel sides 8 and 12, and height 5?

25
50 (Correct answer)
100
60

Correct answer: 50

Area = (1/2)(b₁ + b₂)(h) = (1/2)(8 + 12)(5) = (1/2)(20)(5) = 50.

Question 20: A 30-60-90 triangle has a shorter leg of 5. What is the length of the longer leg?

5
5√2
5√3 (Correct answer)
10

Correct answer: 5√3

In a 30-60-90 triangle, the longer leg = shorter leg × √3 = 5√3.

Question 21: Two triangles are similar. If the sides of the first triangle are 3, 4, 5 and the shortest side of the second is 6, what is the longest side?

8
10 (Correct answer)
12
15

Correct answer: 10

The scale factor is 6/3 = 2. The longest side = 5 × 2 = 10.

Question 22: What is the equation of a circle with center (3, -2) and radius 5?

(x-3)² + (y+2)² = 5
(x-3)² + (y+2)² = 25 (Correct answer)
(x+3)² + (y-2)² = 25
(x-3)² - (y+2)² = 25

Correct answer: (x-3)² + (y+2)² = 25

Standard form: (x-h)² + (y-k)² = r². With center (3,-2) and r = 5: (x-3)² + (y+2)² = 25.

Question 23: If cos(θ) = 5/13, what is sin(θ) (assuming θ is in the first quadrant)?

5/13
12/13 (Correct answer)
13/5
8/13

Correct answer: 12/13

sin²θ + cos²θ = 1. sin²θ = 1 - (5/13)² = 1 - 25/169 = 144/169. sinθ = 12/13.

Question 24: An arc of a circle subtends a central angle of 90°. If the radius is 8, what is the arc length?

2π
4π (Correct answer)
8π
16π

Correct answer: 4π

Arc length = (θ/360°) × 2πr = (90/360) × 2π(8) = (1/4) × 16π = 4π.

Question 25: Which of the following sets of side lengths CANNOT form a right triangle?

6, 8, 10
5, 12, 13
7, 24, 25
4, 5, 7 (Correct answer)

Correct answer: 4, 5, 7

Check 4² + 5² = 16 + 25 = 41 ≠ 49 = 7². So 4, 5, 7 cannot form a right triangle.

Question 26: A sector of a circle has a central angle of 60° and radius 6. What is the area of the sector?

3π
6π (Correct answer)
12π
36π

Correct answer: 6π

Area of sector = (θ/360°) × πr² = (60/360) × π(36) = (1/6) × 36π = 6π.

Question 27: In the unit circle, what are the coordinates at 270°?

(0, 1)
(0, -1) (Correct answer)
(-1, 0)
(1, 0)

Correct answer: (0, -1)

At 270°, the point on the unit circle is at (0, -1) — the bottom of the circle.

Question 28: A regular hexagon has a perimeter of 48. What is the length of each side?

6
7
8 (Correct answer)
9

Correct answer: 8

A regular hexagon has 6 equal sides. Side length = 48/6 = 8.

Question 29: What is the surface area of a cube with side length 4?

16
64
96 (Correct answer)
192

Correct answer: 96

A cube has 6 faces, each with area 4² = 16. Total surface area = 6 × 16 = 96.

Question 30: In a right triangle, sin(A) = cos(B). What can you conclude?

A = B
A + B = 90° (Correct answer)
A + B = 180°
A = 90°

Correct answer: A + B = 90°

sin(A) = cos(B) is always true when A and B are complementary angles (A + B = 90°) in a right triangle.

Question 31: A rectangular garden is 15 meters long and 10 meters wide. What is its area in square meters?

25
50
150 (Correct answer)
300

Correct answer: 150

Area = length × width = 15 × 10 = 150 square meters.

Question 32: Two supplementary angles have a ratio of 2:1. What are their measures?

60° and 30°
90° and 90°
120° and 60° (Correct answer)
100° and 80°

Correct answer: 120° and 60°

Supplementary angles sum to 180°. With ratio 2:1: 2x + x = 180 → 3x = 180 → x = 60. Angles are 120° and 60°.

Question 33: What is the area of a circle with circumference 12π?

12π
36π (Correct answer)
144π
6π

Correct answer: 36π

Circumference = 2πr = 12π → r = 6. Area = πr² = π(36) = 36π.

Question 34: In a triangle, which theorem relates the sides to the angles?

The Pythagorean theorem
The law of sines
The law of cosines
Both B and C (Correct answer)

Correct answer: Both B and C

Both the law of sines (a/sinA = b/sinB = c/sinC) and the law of cosines (c² = a² + b² - 2ab·cosC) relate sides and angles in any triangle.

Question 35: A point P has coordinates (6, 8). What is its distance from the origin?

7
8
10 (Correct answer)
14

Correct answer: 10

Distance from origin = √(6² + 8²) = √(36 + 64) = √100 = 10.

Question 36: A cone has radius 3 and height 4. What is its volume? (V = (1/3)πr²h)

4π
12π (Correct answer)
36π
48π

Correct answer: 12π

V = (1/3)π(3²)(4) = (1/3)π(9)(4) = 12π.

Free SAT Math Additional Topics Geometry and Trigonometry Practice Test

What is the area of a circle with radius 5?