ACT Math Practice Test: Geometry and Trigonometry
- Geometry and Trigonometry
- 12 questions
- 14 minutes
These 12 questions practice ACT geometry and trigonometry: angles, triangles, similar figures, circles, area and volume, coordinate geometry, right-triangle trigonometry and unit circle values. The ACT does not give you a formula sheet, so learn the common area and volume formulas; a less common formula such as the law of cosines is usually stated in the question.
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Question 1
What is the measure of each interior angle of a regular octagon?
Correct answer: C
The interior angles of any octagon add up to (8 − 2) × 180° = 1,080°, and the 8 angles of a regular octagon are equal: 1,080° ÷ 8 = 135°. The choice 45° is an exterior angle, and 120° is an interior angle of a regular hexagon.
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Question 2
Segment PA is tangent to a circle with center O at point A. The radius OA is 5 centimeters long, and PA is 12 centimeters long. What is the distance, in centimeters, from P to O?
Correct answer: C
A tangent is perpendicular to the radius at the point of tangency, so triangle OAP has a right angle at A and PO is its hypotenuse: 52 + 122 = 13. The choice 119 treats PA as the hypotenuse, and 17 simply adds the two lengths.
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Question 3
At the same time of day, a person who is 6 feet tall casts a shadow 4 feet long, and a tree casts a shadow 26 feet long. How tall, in feet, is the tree?
Correct answer: D
The sun's rays form similar right triangles, so height ÷ shadow is the same for both: 64 = h26, and h = 39. Setting up the ratio upside down gives 17⅓, and adding the 2-foot difference between the person's height and shadow gives 28.
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Question 4
A circle has a circumference of 18π inches. What is the area of the circle, in square inches?
Correct answer: B
From 2πr = 18π, the radius is 9, so the area is π(92) = 81π. The choice 324π uses 18 as the radius, and 162π multiplies the radius by the diameter instead of by itself.
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Question 5
A right circular cone has a base radius of 6 inches and a height of 10 inches. What is the volume of the cone, in cubic inches?
Correct answer: C
The volume of a cone is 13πr2h = 13π(36)(10) = 120π. The choice 360π leaves out the 13 (it is the volume of a cylinder with the same base and height), and 20π uses r instead of r2.
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Question 6
The diagonal of a square is 10 centimeters long. What is the area of the square, in square centimeters?
Correct answer: A
The diagonal of a square is 2 times its side, so the side is 102 and the area is 1002 = 50. The choice 100 squares the diagonal, and 200 multiplies the diagonal by 2 instead of dividing by it.
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Question 7
In the standard (x, y) coordinate plane, triangle ABC has vertices A(−2, 1), B(6, 1) and C(3, 7). What is the area of triangle ABC, in square units?
Correct answer: B
Side AB is horizontal with length 6 − (−2) = 8, and the height is the vertical distance from C to that side, 7 − 1 = 6. The area is 12(8)(6) = 24. The choice 48 forgets the 12, and 28 uses the y-coordinate 7 as the height.
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Question 8
In the standard (x, y) coordinate plane, the points (−3, 4) and (5, −2) are the endpoints of a diameter of a circle. Which of the following is an equation of the circle?
Correct answer: D
The center is the midpoint of the diameter, ((−3 + 5) ÷ 2, (4 − 2) ÷ 2) = (1, 1). The diameter has length 82 + 62 = 10, so the radius is 5 and r2 = 25. The equation ending in 100 uses the diameter as the radius, and the one ending in 5 forgets to square the radius.
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Question 9
In right triangle PQR, angle R is the right angle, the measure of angle P is 35°, and PQ = 20 feet. Which of the following expressions gives the length of QR, in feet?
Correct answer: A
PQ is the hypotenuse because it is opposite the right angle, and QR is the side opposite angle P. Since sin P = QR20, QR = 20 sin 35°. The expression 20 cos 35° gives PR, the leg adjacent to angle P.
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Question 10
If cos θ = −35 and π2 < θ < π, what is the value of tan θ?
Correct answer: A
Since sin2 θ = 1 − 925 = 1625 and sine is positive in Quadrant II, sin θ = 45. Then tan θ = 45 ÷ (−35) = −43. The choice −34 divides the cosine by the sine instead.
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Question 11
What is the value of sin 2π3?
Correct answer: D
The angle 2π3 (120°) is in Quadrant II with a reference angle of π3, and sine is positive there, so its sine equals sin π3 = 32. The value 12 is the cosine of π3, and −32 gives the sine the wrong sign for Quadrant II.
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Question 12
Two sides of a triangle have lengths 5 and 8, and the angle between them measures 60°. What is the length of the third side?
(Note: For a triangle with sides of lengths a, b and c, where C is the angle opposite side c, the law of cosines states c2 = a2 + b2 − 2ab cos C. Also, cos 60° = 12.)
Correct answer: B
c2 = 25 + 64 − 2(5)(8)(12) = 89 − 40 = 49, so c = 7. The choice 89 treats the triangle as a right triangle, and 3 forgets to multiply by cos 60°.
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