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SAT Geometry and Trigonometry Quizzes

Practice 10 SAT Geometry and Trigonometry questions involving area, volume, scale factors, lines, angles, similar triangles, special right triangles, trigonometric ratios, circles, arc length, and equations of circles. Use hints when needed and review the explanation after each answer.

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1 Question 1 of 10
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Two similar rectangles have corresponding side lengths in the ratio 3:5. If the area of the smaller rectangle is 54 square units, what is the area of the larger rectangle?

Hint: Area changes by the square of the linear scale factor.
Explanation: The linear scale factor from smaller to larger is 5/3. The area scale factor is (5/3)² = 25/9. Therefore, the larger area is 54 × 25/9 = 6 × 25 = 150 square units.
2 Question 2 of 10
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A right circular cylinder has radius 3 and height 8. What is its volume?

Hint: Use V = πr²h.
Explanation: V = π(3²)(8) = π(9)(8) = 72π cubic units.
3 Question 3 of 10
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Two parallel lines are cut by a transversal. One acute angle formed is 68°. What is the measure of each obtuse angle formed?

Hint: An acute angle and an adjacent obtuse angle form a supplementary pair.
Explanation: Supplementary angles sum to 180°. Therefore, each obtuse angle is 180° - 68° = 112°.
4 Question 4 of 10
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Triangle ABC is similar to triangle DEF. Side AB corresponds to DE, and side AC corresponds to DF. If AB = 12, AC = 18, and DE = 20, what is DF?

Hint: Use the same scale factor for corresponding sides.
Explanation: The scale factor from triangle ABC to triangle DEF is 20/12 = 5/3. Therefore, DF = 18 × 5/3 = 30.
5 Question 5 of 10
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In a 30°-60°-90° triangle, the shorter leg has length 7. What is the length of the hypotenuse?

Hint: In a 30°-60°-90° triangle, the side lengths are in the ratio 1:√3:2.
Explanation: The hypotenuse is twice the shorter leg. Therefore, the hypotenuse is 2 × 7 = 14.
6 Question 6 of 10
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In a right triangle, angle θ has opposite side length 9 and adjacent side length 12. What is tan θ?

Hint: Tangent is opposite divided by adjacent.
Explanation: tan θ = opposite/adjacent = 9/12 = 3/4.
7 Question 7 of 10
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Angles A and B are complementary acute angles. If sin A = 5/13, what is cos B?

Hint: For complementary angles, sin A = cos(90° - A).
Explanation: Since A and B are complementary, B = 90° - A. Therefore, cos B = cos(90° - A) = sin A = 5/13.
8 Question 8 of 10
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A line is tangent to a circle at point T. The radius OT is 10, and point P lies on the tangent line with PT = 24. What is OP?

Hint: A radius to a point of tangency is perpendicular to the tangent line, forming a right triangle.
Explanation: OT is perpendicular to PT, so triangle OPT is right. By the Pythagorean theorem, OP = √(10² + 24²) = √676 = 26.
9 Question 9 of 10
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A circle has radius 9. What is the length of an arc subtended by a central angle of 80°?

Hint: Arc length is the same fraction of the circumference as the central angle is of 360°.
Explanation: Arc length = (80/360)(2π × 9) = (2/9)(18π) = 4π.
10 Question 10 of 10
Not answered

The equation (x - 4)² + (y + 3)² = 49 represents a circle in the xy-plane. What is the radius of the circle?

Hint: Compare the equation with (x - h)² + (y - k)² = r².
Explanation: The right side is r² = 49, so r = 7. The circle has center (4, -3) and radius 7.

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SAT Geometry & Trig Studio

SAT geometry rewards diagrams, scale reasoning, and a small set of relationships used precisely

These 10 questions cover the four SAT Geometry and Trigonometry skill groups: area and volume; lines, angles, and triangles; right triangles and trigonometry; and circles. The fastest route is often to identify which geometric relationship is fixed before choosing a formula.

Area & volume
Lines, angles & triangles
Right triangles & trig
Circles
Scale-factor reasoning

Lengths scale once; areas scale by the square of the scale factor

If corresponding side lengths change from a 3:5 ratio, the area does not change by 5/3 — it changes by (5/3)².

Smaller figure

Area = 54

Larger figure

54 × 25/9 = 150
Parallel lines & angles

One angle can determine every angle in a transversal diagram

Once an acute angle is known, vertical and corresponding angles match it, while adjacent angles are supplementary.

Each obtuse angle = 180° − 68° = 112°.
Similar triangles

Corresponding sides must use one consistent scale factor

If 12 in the first triangle corresponds to 20 in the second, the scale factor is 5/3.

scale factor = 20/12 = 5/3
DF corresponds to 18
DF = 18 × 5/3 = 30
Right triangles & trigonometry

SAT trig questions often reduce to a side ratio or a special-triangle pattern

30°-60°-90° triangle

The side ratio is 1 : √3 : 2, so a shorter leg of 7 gives a hypotenuse of 14.

short : long : hyp = 1 : √3 : 2
Tangent ratio

With opposite side 9 and adjacent side 12, tangent is the simplest direct relationship.

tan θ = 9/12 = 3/4
Radius-to-tangent theorem

A radius drawn to the point of tangency creates a right angle

That converts a circle problem into a right-triangle problem.

OT = 10 and PT = 24
OT ⟂ PT
OP² = 10² + 24² = 676
OP = 26
Circle structure lab

Circle questions switch among angles, arcs, radii, and coordinate form

Arc length

An 80° central angle cuts off 80/360 of the circumference of a radius-9 circle.

(80/360)(18π) = 4π

Equation of a circle

Standard form exposes the center and radius directly.

(x − 4)² + (y + 3)² = 49 → r = 7
Skill diagnostic

Review missed questions by geometric relationship, not by the final formula

Area & volumescale factors, dimensions, cylinder volume
Angles & similaritytransversals, supplementary angles, corresponding sides
Right trianglesspecial triangles, trig ratios, complementary angles
Circlestangents, arcs, radii, coordinate equations
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