Geometry practice test: 12 free questions with worked answers

The daily Geometry, Algebra 2 and SAT Math paper is a short timed practice test of 8 original questions. About a third of its questions are geometry: circle theorems, similarity and transformations, the Pythagorean theorem and trigonometry, and vectors. The rest are algebra and statistics, so treat it as mixed practice rather than a pure geometry test.

What's on a high school geometry test

• Angles and lines: parallel lines cut by a transversal, angles in polygons.
• Triangles: congruence (SSS, SAS, ASA, AAS, HL), the Pythagorean theorem and triangle angle facts.
• Similarity: scale factors and proportions in similar figures.
• Right-triangle trigonometry: sine, cosine and tangent, angles of elevation and depression.
• Circles: arcs, sectors, inscribed angles and the equation of a circle.
• Area, surface area and volume: prisms, cylinders, pyramids, cones and spheres.
• Coordinate geometry: distance, midpoint and slope.
• Transformations: reflections, rotations, translations and dilations.
• Proof: giving a reason for every statement.

Free geometry practice test: 12 questions

We wrote these questions for this page. Give yourself 40 minutes with a calculator, sketch a diagram for every question, and leave answers in terms of π where the question says so.

1. Two parallel lines are cut by a transversal. A pair of alternate interior angles measure (3x + 10)° and (5x − 30)°. Find x and the size of each angle.
2. An exterior angle of a triangle is 128°. One of the two interior angles not next to it is 47°. Find the other.
3. What is the measure of one interior angle in a regular octagon?
4. A 17 ft ladder leans against a wall with its foot 8 ft from the wall. How high up the wall does it reach?
5. Find the distance between A(−3, 4) and B(5, −2), and the midpoint of AB.
6. A 6 ft person casts a 4 ft shadow at the same moment a tree casts an 18 ft shadow. How tall is the tree?
7. From a point 40 m from the base of a tower, the angle of elevation to the top is 35°. How tall is the tower, to the nearest tenth of a meter?
8. A circle has radius 9 cm. A central angle of 80° cuts off an arc. Find the arc length and the sector area in terms of π.
9. An inscribed angle intercepts an arc of 110°. What is the inscribed angle?
10. A cylinder and a cone both have radius 3 in and height 10 in. Find each volume in terms of π.
11. Reflect the point (2, −5) over the line y = x, then rotate the original point 90° counterclockwise about the origin. Give both images.
12. Write the equation of the circle with center (2, −3) and radius 5. Does the point (5, 1) lie on it?

Geometry practice test answers

Angles (questions 1 to 3)

1. Alternate interior angles are congruent when the lines are parallel, so 3x + 10 = 5x − 30. That gives 40 = 2x and x = 20, so each angle is 70°. Check: 5(20) − 30 = 70.

2. An exterior angle equals the sum of the two remote interior angles, so the other angle is 128 − 47 = 81°.

3. The interior angles of an n-sided polygon add up to (n − 2) × 180°. For an octagon that is 6 × 180 = 1,080°, so each angle of a regular octagon is 1,080 ÷ 8 = 135°.

Right triangles and trigonometry (questions 4 to 7)

4. The ladder is the hypotenuse. Height² + 8² = 17², so height² = 289 − 64 = 225 and the ladder reaches 15 ft. 8, 15, 17 is a Pythagorean triple worth knowing.

5. The changes are 8 across and 6 down, so the distance is √(8² + 6²) = √100 = 10. The midpoint averages the coordinates: ((−3 + 5) ÷ 2, (4 + (−2)) ÷ 2) = (1, 1). The Midpoint and Distance Calculator checks these.

6. The sun's rays hit both at the same angle, so the triangles are similar and the sides are in proportion: height ÷ 18 = 6 ÷ 4. The tree is 18 × 1.5 = 27 ft tall.

7. You know the side adjacent to the angle and want the opposite side, so use tangent.

The tower is 28.0 m tall. Make sure your calculator is in degree mode: in radian mode, tan 35 gives a very different answer. The Trigonometry Calculator solves right triangles step by step.

Try one yourself from our Trigonometry course. Answer it here and press Mark.

Circles (questions 8, 9 and 12)

8. 80° is 80/360 of the whole circle, so take that fraction of the circumference and of the area.

That is about 12.57 cm and 56.55 cm². Mixing up the circumference and area formulas is the usual slip.

9. An inscribed angle is half the arc it intercepts, so it is 110 ÷ 2 = 55°. A central angle on the same arc would be the full 110°.

12. A circle with center (h, k) and radius r has equation (x − h)² + (y − k)² = r², so this one is (x − 2)² + (y + 3)² = 25. For (5, 1): (5 − 2)² + (1 + 3)² = 9 + 16 = 25, so yes, it lies on the circle. Note the sign change: a center at y = −3 gives (y + 3).

Volume and transformations (questions 10 and 11)

10. The cylinder's volume is πr²h = π(3²)(10) = 90π in³, about 282.7 in³. A cone with the same base and height holds a third as much: 30π in³, about 94.2 in³.

11. Reflecting over y = x swaps the coordinates: (2, −5) becomes (−5, 2). Rotating 90° counterclockwise around the origin turns (x, y) into (−y, x), so (2, −5) becomes (5, 2). Plot both to check that each image is the same distance from the origin as the original.

What to practice next

Links: the Pre-Algebra and Algebra 1 paper, Trigonometry course, Trigonometry Calculator and Vectors Calculator.

Geometry end-of-course exams and official practice tests

There is no national geometry exam in the US. Some states give a geometry end-of-course exam, others test geometry inside a broader math test, and the format and weight differ. Ask your teacher which test you will take, then use your state education department's own practice tests.

New York publishes the most complete set. Its past Regents Examinations in Geometry come with scoring keys, rating guides, model responses and conversion charts. The June 2026 exam ran on Tuesday, June 23, 2026, from 1:15 to 4:15 p.m., with 35 questions in four parts. Students needed a graphing calculator, a straightedge and a compass, and the conversion chart turned a top raw score of 80 into a scale score of 100.

Use the daily paper as a short timed practice test

1. Open today's Geometry, Algebra 2 and SAT Math paper with a calculator, ruler and compass.
2. Work to the suggested time, 35 to 40 minutes.
3. Draw and label a diagram for every geometry question before you calculate.
4. Mark it with the mark scheme. For each lost mark, note whether you used the wrong fact or made a slip, because the fixes are different.

Common geometry mistakes

• Calculator in radian mode for a degrees question.
• Using a² + b² = c² with the wrong side as c: c is always the hypotenuse, opposite the right angle.
• Matching the wrong sides in similar figures: pair sides opposite equal angles.
• Confusing inscribed and central angles: the inscribed angle is half the arc.
• Sign errors in a circle's equation: center (2, −3) gives (x − 2)² + (y + 3)².
• Assuming a diagram is to scale: use only what is given or what you can prove.

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