Math · Unit 13

Geometry and Trigonometry

Pythagoras, area and volume, circles and trig ratios

Trigonometry is the discovery that the ratios of a right triangle's sides depend only on its angles, which turns measuring the inaccessible into arithmetic.

The unit covers triangle congruence, Pythagoras, circles, area and volume, and the trigonometric ratios themselves.

This unit breaks down into 21 short steps and 120 questions, starting at difficulty 3 and building to 4. Below you can see exactly what it covers, how the path is structured, and worked examples with explanations.

Steps: 21
Questions: 120
Difficulty: 3-4

What this unit covers

Triangles and Congruence
The Pythagorean Theorem
Trigonometric Ratios
Circles
Area and Volume

Where this fits

Needs Geometry and Algebra. Feeds into Precalculus.

Where people slip

Pythagoras only applies to right triangles. Applying it to any other triangle is the most common error in the unit, and the answer looks perfectly plausible.

How the unit is structured

Geometry and Trigonometry runs as 21 short steps that unlock in order. 15 are practice rounds and 6 are challenge rounds that pull together everything before them. Questions start at difficulty 3 and climb to 4 as you progress.

Step 1 · easierStep 21 · harder

Challenge rounds

Example questions

30 real questions from this unit, with the answer and the reason behind it, grouped by what they practise. There are 120 in the unit altogether.

Area and Volume

Choose all that applyLevel 3
1. Which of these are volume formulas? Select all that apply.
- side^3correct
- length x width x heightcorrect
- pi x r^2 x hcorrect
- pi x r^2
pi x r^2 is the area of a circle; the other three give volumes.
Fact or fibLevel 3
2. The area of a square with side 4 is 16.
Answer: True
Area of a square is side^2, so 4^2 = 16.
Fill the blankLevel 3
3. The volume of a rectangular box is length times width times ____.
- heightcorrect
- area
- radius
- diameter
Volume of a box is length x width x height.
Match the pairsLevel 3
4. Match each shape to its area formula.
Answer: Rectangle = length x width; Triangle = half x base x height; Circle = pi x r^2
Each shape has its own area formula shown here.
Multiple choiceLevel 3
5. What is the area of a rectangle that is 5 units long and 3 units wide?
- 15correct
- 8
- 16
- 30
Area of a rectangle is length x width, so 5 x 3 = 15.
Guess the numberLevel 4
6. A rectangular box is 4 by 5 by 6. What is its volume?
Answer: 120
Volume = 4 x 5 x 6 = 120.

Circles

Build the sentenceLevel 3
7. Build a true sentence about circles.
Answer: A circle has no straight sides
A circle is a smooth curve with no straight sides or corners.
Fill the blankLevel 3
8. The area of a circle is pi times the radius ____.
- squaredcorrect
- cubed
- doubled
- halved
Area of a circle is pi x r^2, so you square the radius.
Guess the numberLevel 3
9. A circle has radius 7. About what is its area? Use pi = 3.14 and round to the nearest whole number.
Answer: 154
Area = pi x r^2 = 3.14 x 49 = 153.86, which rounds to about 154.
Multiple choiceLevel 3
10. What is the circumference of a circle with radius 5? Use C = 2 x pi x r.
- 10 x picorrect
- 25 x pi
- 5 x pi
- 20 x pi
C = 2 x pi x 5 = 10 x pi.
Choose all that applyLevel 4
11. Which statements about circles are true? Select all that apply.
- diameter = 2 x radiuscorrect
- circumference = pi x diametercorrect
- area = pi x r^2correct
- area = 2 x pi x r
2 x pi x r is the circumference, not the area; the other three statements are correct.
Sort into groupsLevel 4
12. Sort each circle part as a Straight line or a Curved line.
Answer: Radius = Straight; Arc = Curved; Diameter = Straight; Circumference = Curved
Radius and diameter are straight segments; arcs and the circumference follow the curve.

The Pythagorean Theorem

Build the sentenceLevel 3
13. Build a true sentence about right triangles.
Answer: The hypotenuse is the longest side
The hypotenuse, opposite the right angle, is the longest side.
Fact or fibLevel 3
14. The sides 3, 4, and 5 form a right triangle because 3^2 + 4^2 = 5^2.
Answer: True
9 + 16 = 25 = 5^2, so 3, 4, 5 is a right triangle.
Fill the blankLevel 3
15. The Pythagorean theorem says a^2 + b^2 = ____, where c is the hypotenuse.
- c^2correct
- c
- 2c
- a + b
The theorem states a^2 + b^2 = c^2.
Multiple choiceLevel 3
16. In a right triangle the two legs measure 3 and 4. How long is the hypotenuse?
- 5correct
- 7
- 6
- 12
3^2 + 4^2 = 9 + 16 = 25, and sqrt(25) = 5, so the hypotenuse is 5.
Odd one outLevel 3
17. Which set of three numbers is NOT a Pythagorean triple?
- 4, 5, 6
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
4^2 + 5^2 = 41, not 36, so 4, 5, 6 is not a right triangle; the others all work.
Put in orderLevel 3
18. Put the steps for finding a hypotenuse with the Pythagorean theorem in the correct order.
Answer: Square each leg -> Add the two squares -> Take the square root
You square both legs, add them, then take the square root to get the hypotenuse.

Triangles and Congruence

Build the sentenceLevel 3
19. Build a true sentence about congruent triangles.
Answer: Congruent triangles have the same size
Congruent triangles are identical in both size and shape.
Choose all that applyLevel 3
20. Which statements about triangles are true? Select all that apply.
- The angles sum to 180 degreescorrect
- An equilateral triangle has equal sidescorrect
- A right triangle has a 90 degree anglecorrect
- A scalene triangle has all equal sides
A scalene triangle has all different sides; the other three statements are true.
Fact or fibLevel 3
21. An equilateral triangle has three equal angles of 60 degrees each.
Answer: True
All three angles are equal and 180 / 3 = 60, so each is 60 degrees.
Fill the blankLevel 3
22. A triangle with all three sides equal in length is called ____.
- equilateralcorrect
- isosceles
- scalene
- right
Equilateral means all three sides (and angles) are equal.
Match the pairsLevel 3
23. Match each shape to its number of sides.
Answer: Triangle = 3 sides; Square = 4 sides; Pentagon = 5 sides
A triangle has 3 sides, a square 4, and a pentagon 5.
Multiple choiceLevel 3
24. The three angles inside any triangle add up to how many degrees?
- 180 degreescorrect
- 90 degrees
- 360 degrees
- 270 degrees
The interior angles of every triangle always sum to 180 degrees.

Trigonometric Ratios

Build the sentenceLevel 3
25. Build the sentence that defines sine.
Answer: Sine equals opposite over hypotenuse
Sine of an angle equals the opposite side over the hypotenuse.
Choose all that applyLevel 3
26. Which of these trig values are correct? Select all that apply.
- sin(30 degrees) = 0.5correct
- cos(60 degrees) = 0.5correct
- tan(45 degrees) = 1correct
- sin(90 degrees) = 0
sin(90 degrees) is actually 1, not 0; the other three values are all correct.
Fill the blankLevel 3
27. The tangent of an angle equals the opposite side divided by the ____ side.
- adjacentcorrect
- hypotenuse
- opposite
- radius
TOA means Tangent = Opposite / Adjacent.
Match the pairsLevel 3
28. Match each trig ratio to its definition.
Answer: Sine = opposite / hypotenuse; Cosine = adjacent / hypotenuse; Tangent = opposite / adjacent
These are the three ratios in SOHCAHTOA.
Multiple choiceLevel 3
29. In SOHCAHTOA, the sine of an angle equals which ratio?
- opposite / hypotenusecorrect
- adjacent / hypotenuse
- opposite / adjacent
- hypotenuse / opposite
SOH means Sine = Opposite / Hypotenuse.
Fact or fibLevel 4
30. sin(60 degrees) is about 0.87.
Answer: True
sin(60 degrees) = sqrt(3)/2, which is about 0.866, close to 0.87.

Where these questions come from. Each unit starts as a plan of the concepts it should cover and the difficulty it should span. Questions are written against that plan with AI assistance, then checked by a validator that rejects anything without a single defensible answer, an explanation, or plausible wrong options. How we write questions sets out the whole process, and corrections are fixed in the bank and reach the site and the app the same day.

How you practise

This unit mixes 14 different question formats, so you are recalling and applying rather than recognising the same layout every time.

Build the sentence
Choose all that apply
Fact or fib
Fill the blank
Guess the number
Match the pairs
Multiple choice
Odd one out
Picture question
Put in order
Sort into groups
Spell it
True or false
Type the answer

Practise Geometry and Trigonometry

120 questions across 21 steps. Start with step one and crawl at your own pace.

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Read about Geometry and Trigonometry

Explainers from our blog on what this unit covers. Each one ends with real questions from the bank.

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