SAT Geometry & Trigonometry
Practice Questions
Geometry & Trigonometry questions cover area, volume, angle relationships, the Pythagorean theorem, and the basics of right-triangle trigonometry. A reference sheet with key formulas is provided during the test, but knowing when and how to apply each formula efficiently is the real skill being assessed.
About Geometry & Trigonometry
Geometry and Trigonometry make up roughly 15% of the Math section, and the built-in calculator is available throughout Math. These questions focus on applying formulas and relationships to find missing lengths, areas, and angle measures in real or described diagrams. The provided reference sheet lists major formulas, so the central task is choosing and applying the appropriate relationship.
The four core subtopics move from two-dimensional figures to three-dimensional solids, from basic angle arithmetic to the relationships in special triangles, and from the Pythagorean theorem to sine, cosine, and tangent. A question can combine concepts—for example, using the Pythagorean theorem to find a circle chord or using similar triangles inside a trigonometry setup.
For right-triangle trigonometry, be comfortable setting up sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, and tanθ = opposite/adjacent, as well as recognizing complementary angle relationships: sin(θ) = cos(90° − θ). Circle questions can involve arc length, sector area, and central versus inscribed angles. Start by naming the relevant relationship, then substitute the labeled quantities.
What You'll Practice
- Computing areas and perimeters of standard 2D shapes
- Finding surface areas and volumes of 3D solids
- Using angle relationships: supplementary, vertical, alternate interior, corresponding
- Applying the Pythagorean theorem and recognizing Pythagorean triples
- Setting up and solving right-triangle trigonometry (sin, cos, tan, and their complements)
- Working with circle properties: central angles, inscribed angles, arc length, and sector area
Why Geometry & Trigonometry Matters for Your Score
Geometry connects diagrams to equations and develops spatial reasoning used in architecture, engineering, design, and science. Because many questions reuse a compact set of angle, area, volume, circle, and right-triangle relationships, an error log can show exactly which relationship needs review.
Geometry & Trigonometry Subtopics
Each subtopic page combines its available practice questions with concept explanations, common mistakes, and strategy tips tailored to that specific skill.
Area and Volume
Questions asking you to compute or compare areas of 2D figures and surface areas and volumes of 3D solids.
Lines, Angles, and Triangles
Questions covering angle relationships formed by parallel lines, properties of triangles, congruence, similarity, and the triangle inequality.
Right Triangles and Trigonometry
Questions applying the Pythagorean theorem, SOH-CAH-TOA, and complementary angle trig identities to find side lengths and angle measures in right triangles.
Circles
Questions covering circle area, circumference, arc length, sector area, central and inscribed angles, and the equation of a circle in the coordinate plane.
Geometry Sample Questions
More questionsPick an answer and hit Check Answer to see the detailed explanation. Questions are from easy, medium, and hard difficulty levels.
A rectangle has a length of 12 cm and a width of 7 cm. What is the area of the rectangle in square centimeters?
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Correct answer: A. 84
Explanation
Area of rectangle = length × width = 12 × 7 = 84 cm².
In triangle ABC, the exterior angle at vertex C measures 130°. The interior angle at vertex A measures 55°. What is the measure of the interior angle at vertex B?
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Correct answer: D. 75°
Explanation
The exterior angle theorem states that an exterior angle equals the sum of the two non-adjacent interior angles: 130° = angle A + angle B. So angle B = 130° − 55° = 75°.
In right triangle ABC, angle B = 90°. tan(A) = 5/12. If AC (the hypotenuse) = 26, what is the length of AB (adjacent to angle A)?
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Correct answer: C. 24
Explanation
tan(A) = opposite/adjacent = BC/AB = 5/12. Let BC = 5k and AB = 12k. By Pythagorean theorem: (5k)² + (12k)² = 26² → 25k² + 144k² = 676 → 169k² = 676 → k² = 4 → k = 2. AB = 12(2) = 24.
A circle has a radius of 9 cm. What is its circumference in terms of π?
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Correct answer: A. 18π cm
Explanation
Circumference = 2πr = 2π(9) = 18π cm.
A rectangular swimming pool is 20 meters long, 10 meters wide, and 2 meters deep. What is the volume of the pool in cubic meters?
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Correct answer: D. 400
Explanation
Volume of rectangular prism = length × width × height = 20 × 10 × 2 = 400 m³.
In triangle XYZ, the measure of angle X is twice the measure of angle Y, and the measure of angle Z is 40° more than angle Y. What is the measure of angle X in degrees?
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Correct answer: D. 70°
Explanation
Let angle Y = y. Then angle X = 2y and angle Z = y + 40. Summing to 180°: 2y + y + (y + 40) = 180 → 4y + 40 = 180 → 4y = 140 → y = 35°. Angle X = 2(35) = 70°.
In right triangle ABC, angle B = 90°, leg AB = 6, and leg BC = 8. What is the length of hypotenuse AC?
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Correct answer: B. 10
Explanation
Pythagorean theorem: AC² = AB² + BC² = 36 + 64 = 100. AC = √100 = 10.
A circle with radius 10 cm has a central angle of 120°. What is the area of the sector formed by this central angle in terms of π?
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Correct answer: D. 100π/3 cm²
Explanation
Sector area = (central angle / 360°) × πr² = (120/360) × π(100) = (1/3)(100π) = 100π/3 cm².
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Strategy Tips for Geometry
Draw and label everything
Even when the test provides a figure, redraw key information directly onto it: label angle measures you calculate, mark equal sides or angles with tick marks, and write any intermediate lengths. The geometry questions that stump students most often do so because they try to hold too much in working memory instead of annotating the figure.
Use the reference sheet strategically, not habitually
The test provides a reference sheet with formulas for area, volume, and special triangles. Use it when you're uncertain, but try to know the most common formulas cold—area of a triangle, circle, rectangle—so you only need the sheet for less common items like the volume of a cone or sphere. Constant reference-sheet trips waste time.
For trig, always label sides relative to the angle
Before applying SOH-CAH-TOA, label which side is opposite, adjacent, and the hypotenuse relative to the specific angle named in the problem. The hypotenuse is opposite the right angle; opposite and adjacent change when the reference angle changes.
Convert angles to arc length with the proportion
For an arc length or sector area, use the fraction of a full turn: (central angle / 360°) = (arc length / circumference) = (sector area / total circle area). Write the relevant equality, substitute the known values, and solve for the missing quantity.
Frequently Asked Questions — Geometry
Does the test provide a geometry formula sheet?
Yes. A reference sheet with the most important formulas—including areas of triangles, circles, rectangles, and trapezoids, as well as volumes of rectangular prisms, cylinders, spheres, cones, and pyramids, plus the 30-60-90 and 45-45-90 special triangle ratios—is accessible during the Math section. You should still aim to know the most common formulas from memory so you spend less time looking them up.
How much trigonometry is on the test?
Trigonometry has grown on the current test format. You should be comfortable with right-triangle trig (SOH-CAH-TOA), the complementary angle identity (sin θ = cos(90° − θ)), and using the unit circle for basic angles. The Law of Sines and Law of Cosines are not tested. Arc length, sector area, and radian-degree conversion are tested as part of the Circles subtopic.
Will I see figures/diagrams, or are problems text-only?
Most geometry questions include a figure, though some describe the figure entirely in the problem text. Even when a figure is provided, it may not be drawn to scale—the problem will say 'Note: Figure not drawn to scale.' In those cases, don't rely on visual estimation; use only the labeled values and the relationships you know.
Are coordinate geometry questions in this domain?
Questions involving the coordinate plane—distance formula, midpoint, slope, equations of circles—appear primarily in the Algebra and Advanced Math domains, not in Geometry & Trigonometry. This domain focuses on Euclidean geometry, classical angle relationships, and trigonometry.
What are the most important things to review first?
A practical review order is the Pythagorean theorem, common Pythagorean triples, triangle and circle area, the arc/sector proportion, and the three basic trig ratios. Then review how the reference sheet expresses the volume formulas and practice choosing the correct dimensions.
Other Math Topics
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Continue in the question bank for filtered practice, then use the timed Math test to check pacing across a full section.