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← High School Geometry
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Practice deliberately

Work through High School Geometry unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

High School Geometry coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the High School Geometry syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use geometry practice questions with answers.

01

Transformations

Connect formulas, tables, and graphs; identify domain and parameters; then interpret how changing an input or coefficient changes the output.

Goal: recognise a transformations problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Triangle congruence

Draw and label the givens, establish every theorem condition, calculate with exact relationships, and check scale and units.

Goal: recognise a triangle congruence problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Similarity

Draw and label the givens, establish every theorem condition, calculate with exact relationships, and check scale and units.

Goal: recognise a similarity problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Right triangles

Draw and label the givens, establish every theorem condition, calculate with exact relationships, and check scale and units.

Goal: recognise a right triangles problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Circles

Draw and label the givens, establish every theorem condition, calculate with exact relationships, and check scale and units.

Goal: recognise a circles problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Area and volume

Draw and label the givens, establish every theorem condition, calculate with exact relationships, and check scale and units.

Goal: recognise a area and volume problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Coordinate proofs

State definitions and hypotheses precisely, construct a valid argument, and test it against a tempting counterexample.

Goal: recognise a coordinate proofs problem from its wording, carry out the governing method, and check that the result is reasonable.

High School Geometry questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: The point (3, -4) is rotated 90 degrees counterclockwise about the origin. What are the coordinates of the image?

High School Geometry · Transformations

  1. (4, 3)
  2. (-4, 3)
  3. (-3, 4)
  4. (4, -3)
Show the worked answer
  1. A 90 degree counterclockwise rotation sends the point (x, y) to (-y, x).
  2. Here x = 3 and y = -4.
  3. -y = -(-4) = 4, and x stays as the new second coordinate, which is 3.
  4. So the image is (4, 3).

Answer: (4, 3)

Question 2: Triangle ABC is congruent to triangle DEF. If AB = 7, BC = 9, and DF = 12, what is the length of AC?

High School Geometry · Triangle congruence

  1. 12
  2. 9
  3. 7
  4. 16
Show the worked answer
  1. In a congruence statement the letters match up in order: A with D, B with E, C with F.
  2. So side AC matches side DF.
  3. Matching sides of congruent triangles are equal in length.
  4. DF = 12, so AC = 12.

Answer: 12

Question 3: Two similar triangles have sides in the ratio 3 : 5. The smaller triangle has an area of 18 square cm. What is the area of the larger one?

High School Geometry · Similarity

  1. 50 square cm
  2. 30 square cm
  3. 45 square cm
  4. 90 square cm
Show the worked answer
  1. Areas of similar figures compare as the square of the side ratio.
  2. The side ratio 3 : 5 gives an area ratio of 9 : 25.
  3. Set up 18 / 9 = 2, so one part of area is 2 square cm.
  4. The larger area is 25 parts: 25 x 2 = 50 square cm.

Answer: 50 square cm

Question 4: A skate ramp rises 5 m over a horizontal run of 12 m. How long is the sloping surface?

High School Geometry · Right triangles

  1. 13 m
  2. 17 m
  3. 11 m
  4. 7 m
Show the worked answer
  1. The rise and the run meet at a right angle, so the ramp surface is the hypotenuse.
  2. 5^2 = 25 and 12^2 = 144.
  3. 25 + 144 = 169.
  4. The square root of 169 is 13, so the sloping surface is 13 m long.

Answer: 13 m

Question 5: A circle has a radius of 12 cm. What is the length of an arc cut off by a 60 degree central angle?

High School Geometry · Circles

  1. 4 pi cm
  2. 2 pi cm
  3. 12 pi cm
  4. 24 pi cm
Show the worked answer
  1. A 60 degree angle is 60 / 360 = 1/6 of the whole circle.
  2. The full circumference is 2 x pi x 12 = 24 pi cm.
  3. Take one sixth of that circumference.
  4. 24 pi / 6 = 4 pi cm.

Answer: 4 pi cm

Question 6: A cone has a radius of 3 cm and a height of 8 cm. What is its volume?

High School Geometry · Area and volume

  1. 24 pi cubic cm
  2. 72 pi cubic cm
  3. 48 pi cubic cm
  4. 8 pi cubic cm
Show the worked answer
  1. The volume of a cone is one third of pi times radius squared times height.
  2. Radius squared is 3^2 = 9.
  3. 9 x 8 = 72, so pi r^2 h = 72 pi.
  4. One third of 72 pi is 24 pi cubic cm.

Answer: 24 pi cubic cm

Question 7: What is the distance between the points (-3, 7) and (5, -1)?

High School Geometry · Coordinate proofs

  1. 8 times the square root of 2
  2. 8
  3. 16
  4. 4 times the square root of 2
Show the worked answer
  1. Find the horizontal change: 5 - (-3) = 8.
  2. Find the vertical change: -1 - 7 = -8.
  3. Square and add: 8^2 + (-8)^2 = 64 + 64 = 128.
  4. The square root of 128 is the square root of 64 times 2, which is 8 times the square root of 2, about 11.31.

Answer: 8 times the square root of 2

Question 8: An inscribed angle in a circle measures 35 degrees. What is the measure of the arc it intercepts?

High School Geometry · Circles

  1. 70 degrees
  2. 35 degrees
  3. 17.5 degrees
  4. 145 degrees
Show the worked answer
  1. An inscribed angle is always half of the arc it cuts off.
  2. That means the arc is twice the inscribed angle.
  3. 2 x 35 = 70.
  4. So the intercepted arc measures 70 degrees.

Answer: 70 degrees

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed High School Geometry questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1TransformationsThe point (3, -4) is rotated 90 degrees counterclockwise about the origin. What are the coordinates of the image?(4, 3)
2Triangle congruenceTriangle ABC is congruent to triangle DEF. If AB = 7, BC = 9, and DF = 12, what is the length of AC?12
3SimilarityTwo similar triangles have sides in the ratio 3 : 5. The smaller triangle has an area of 18 square cm. What is the area of the larger one?50 square cm
4Right trianglesA skate ramp rises 5 m over a horizontal run of 12 m. How long is the sloping surface?13 m
5CirclesA circle has a radius of 12 cm. What is the length of an arc cut off by a 60 degree central angle?4 pi cm
6Area and volumeA cone has a radius of 3 cm and a height of 8 cm. What is its volume?24 pi cubic cm
7Coordinate proofsWhat is the distance between the points (-3, 7) and (5, -1)?8 times the square root of 2
8CirclesAn inscribed angle in a circle measures 35 degrees. What is the measure of the arc it intercepts?70 degrees

How the generated High School Geometry sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this High School Geometry practice page

What does this page cover?

It covers all 7 High School Geometry units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.