SAT Math Practice Test: Geometry and Trigonometry
- Geometry and Trigonometry
- 11 questions
- 15 minutes
These questions practice SAT Geometry and Trigonometry: area and volume, lines, angles and triangles, right-triangle trigonometry and circles. The test gives you a reference sheet of formulas, but you still need to recognize which one a problem calls for.
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Question 1
The length of a rectangle is 3 inches more than twice its width. If the perimeter of the rectangle is 54 inches, what is the area of the rectangle, in square inches?
Correct answer: B
With width w, the perimeter is 2w + 2(2w + 3) = 6w + 6 = 54, so w = 8 and the length is 19. The area is 8 × 19 = 152; treating 54 as the sum of one length and one width leads to 629.
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Question 2
A cylindrical water tank has a diameter of 10 feet and a height of 12 feet. The tank is filled with water to 34 of its height. What is the volume of the water in the tank, in cubic feet?
Correct answer: A
The radius is 5 feet and the water is 9 feet deep, so the volume is π(52)(9) = 225π. The choice 300π is the volume of the full tank, and 900π uses the diameter as the radius.
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Question 3
Two parallel lines are intersected by a transversal. Two same-side interior angles formed by the transversal have measures (3x + 16)° and (5x − 4)°. What is the measure of the larger of these two angles?
Correct answer: C
Same-side interior angles between parallel lines are supplementary, so 8x + 12 = 180 and x = 21. The angles are 79° and 101°. Setting the two expressions equal, as if the angles were congruent, gives x = 10 and the trap values 46° and 134°.
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Question 4
In a right triangle with angles measuring 30°, 60° and 90°, the side opposite the 60° angle has a length of 9. What is the length of the hypotenuse?
Correct answer: B
In a 30°-60°-90° triangle the sides are in the ratio 1 : 3 : 2. The shorter leg is 9 ÷ 3 = 33, so the hypotenuse is 2(33) = 63. The choice 18 treats 9 as the shorter leg.
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Question 5
In right triangle ABC, angle C is the right angle, and sin A = 513. What is the value of tan B?
Correct answer: D
Let BC = 5 and AB = 13; then AC = 132 − 52 = 12. For angle B, the opposite side is AC and the adjacent side is BC, so tan B = 125. The choice 512 is tan A.
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Question 6
In the equation sin(3k + 12)° = cos(5k − 2)°, both angle measures are between 0° and 90°. What is the value of k?
Correct answer: C
The sine of an acute angle equals the cosine of its complement, so (3k + 12) + (5k − 2) = 90. Then 8k + 10 = 90 and k = 10, which gives angles of 42° and 48°. Setting the two angles equal gives the trap answer 7.
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Question 7
x2 + y2 − 6x + 10y − 2 = 0
The equation shown defines a circle in the xy-plane. What are the center and the radius of the circle?
Correct answer: A
Completing the square gives (x − 3)2 + (y + 5)2 = 2 + 9 + 25 = 36, so the center is (3, −5) and the radius is 6. The coordinates of the center have the opposite signs of the numbers inside the parentheses, and 36 is the radius squared.
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Question 8
A circle has a radius of 9 centimeters. What is the length, in centimeters, of an arc of the circle intercepted by a central angle of 140°?
Correct answer: B
The arc is 140360 of the circumference: 140360 × 2π(9) = 7π. The choice 31.5π is the area of the sector, and 14π uses the diameter in place of the radius.
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Question 9
In a circle, a central angle of 5π6 radians intercepts an arc of length 15π. What is the radius of the circle?
Correct answer: D
For an angle in radians, arc length = radius × angle, so the radius is 15π ÷ 5π6 = 15 × 65 = 18. Multiplying 15 by 56 instead of dividing gives 12.5.
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Question 10
In the figure, point D lies on side AB and point E lies on side AC of triangle ABC. Segment DE is parallel to segment BC, AD = 6, DB = 4 and DE = 9.
What is the length of BC?
Correct answer: C
Because DE is parallel to BC, triangle ADE is similar to triangle ABC with ratio AD : AB = 6 : 10. So 9 ÷ BC = 6 ÷ 10 and BC = 15. The choice 13.5 uses the ratio 6 : 4, which compares AD with DB instead of with the whole side AB.
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Question 11
Two right circular cylinders are similar. The ratio of the radius of the smaller cylinder to the radius of the larger cylinder is 2 to 3. If the volume of the smaller cylinder is 64π cubic centimeters, what is the volume of the larger cylinder, in cubic centimeters?
Correct answer: D
For similar solids, volumes scale by the cube of the ratio of lengths: 64π × (32)3 = 64π × 278 = 216π. The choice 96π scales by 32 only, and 144π by (32)2, which is the ratio of areas.
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