How do I find a missing leg of a right triangle?

Rearrange the Pythagorean theorem. If the hypotenuse is c and one leg is a, the missing leg is b = sqrt(c squared minus a squared). For example, c = 13, a = 5, so b = sqrt(169 - 25) = sqrt(144) = 12. Always check that the leg is shorter than the hypotenuse before computing.

How do I find a missing angle in a right triangle?

If you know two sides, use the inverse trig functions. To find angle A: arctan(opposite leg / adjacent leg), arcsin(opposite leg / hypotenuse), or arccos(adjacent leg / hypotenuse). All three give the same angle. For example, with legs 3 and 4, angle A opposite the leg of 3 = arctan(3/4) = 36.87 degrees.

What are the three sides of a right triangle called?

The two shorter sides that form the right angle are called legs (also called the base and perpendicular, or side a and side b). The longest side opposite the right angle is called the hypotenuse (side c). The Pythagorean theorem relates them: a squared plus b squared equals c squared.

What is the Pythagorean theorem?

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a squared plus b squared equals c squared. It was known to Babylonian mathematicians and systematically proven by Euclid. It applies only to right triangles (those with a 90-degree angle).

What is a Pythagorean triple?

A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a squared plus b squared equals c squared. Common triples are 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 9-40-41. Any multiple of a triple also works: 6-8-10, 9-12-15, and so on. Recognising triples lets you find sides and angles quickly without a calculator.

How many decimal places should I use for my answer?

For side lengths, give your answer to 2 decimal places. For angles in degrees, also give 2 decimal places. The quiz checker accepts answers within a tolerance of plus or minus 0.05 for sides and plus or minus 0.5 degrees for angles, so minor rounding differences do not count as wrong.

What is the difference between arcsin, arccos, and arctan?

All three are inverse trigonometric functions that convert a ratio back into an angle. arcsin(x) gives the angle whose sine is x (range -90 to 90 degrees). arccos(x) gives the angle whose cosine is x (range 0 to 180 degrees). arctan(x) gives the angle whose tangent is x (range -90 to 90 degrees). In a right triangle where all angles are between 0 and 90 degrees, all three give equivalent results when applied to the correct ratio.

How do I know which trig function to use?

Label the three sides relative to the angle you want to find: the side opposite the angle, the side adjacent to the angle, and the hypotenuse. Then: sin(A) = opposite / hypotenuse, cos(A) = adjacent / hypotenuse, tan(A) = opposite / adjacent. A helpful mnemonic is SOH-CAH-TOA. To find the angle, take the inverse (arcsin, arccos, or arctan) of the ratio you can compute.

Can this quiz help with 30-60-90 and 45-45-90 special triangles?

Yes. Special right triangles are common in the random problems. In a 30-60-90 triangle the sides are in ratio 1 : sqrt(3) : 2. In a 45-45-90 triangle the sides are in ratio 1 : 1 : sqrt(2). Knowing these ratios lets you answer certain problems instantly. The quiz will show the full solution step after you check your answer.

Quiz: Right Triangle Side and Angle Calculator

Q: How do I find the hypotenuse of a right triangle?

Use the Pythagorean theorem: c = sqrt(a squared plus b squared), where a and b are the two legs and c is the hypotenuse. For example, if a = 6 and b = 8, then c = sqrt(36 + 64) = sqrt(100) = 10. The hypotenuse is always opposite the 90-degree angle and is always the longest side.

Check your right triangle skills. Solve the problem, enter your answer, and get instant feedback with a full worked solution.

📐 What is a Right Triangle Side and Angle Quiz?

A right triangle quiz is a structured practice tool that presents randomised right triangle problems and gives you instant feedback on your answer. Instead of passively reading formulas, you actively solve problems, which accelerates learning and builds the automatic recall needed in exams, standardised tests, and engineering courses.

Right triangles appear in an enormous range of real-world contexts. Architects use them when calculating roof pitch. Surveyors use them when measuring horizontal distances from slope measurements. Pilots and navigators use them to resolve velocity vectors. Electricians use them when calculating the length of conduit runs in three dimensions. Carpenters use the 3-4-5 rule to check whether corners are square. Every profession that works with spatial relationships relies on these same two formulas: the Pythagorean theorem for sides, and inverse trigonometric functions for angles.

The quiz covers two core skills: finding a missing side (using the Pythagorean theorem c = sqrt(a squared plus b squared), or b = sqrt(c squared minus a squared)) and finding a missing angle (using arctan for two known legs, or arcsin/arccos when the hypotenuse is known). These skills are the foundation of all further trigonometry, from the unit circle to the law of sines and cosines in oblique triangles.

The built-in random problem generator produces an unlimited variety of right triangles with different side lengths and angle magnitudes, preventing the rote memorisation of a fixed problem set. After each attempt the full worked solution is displayed, so whether you answered correctly or not, you can follow every step of the arithmetic and build genuine understanding rather than just answer-checking.

📐 Formulas Used

c = √(a² + b²)

a, b = the two legs (shorter sides that form the 90° angle)

c = hypotenuse (side opposite the 90° angle, always the longest)

Example: a = 5, b = 12 → c = √(25 + 144) = √169 = 13

A = arctan(a ÷ b)

A = angle at vertex A (in degrees)

a = leg opposite to angle A

b = leg adjacent to angle A

Also: A = arcsin(a / c) or A = arccos(b / c)

Example: a = 3, b = 4 → A = arctan(3/4) = 36.87°

📖 How to Use This Quiz

Steps to practise right triangle problems

Select question type - Choose Find the Side to practise the Pythagorean theorem, or Find the Angle to practise inverse trig (arctan, arcsin).

Read the given values - The quiz shows you which two values are known. Write them down and identify what you need to find.

Solve it yourself - Use pen and paper or mental arithmetic. Apply the correct formula based on what is given.

Enter and check - Type your numeric answer and click Check Answer. The quiz shows correct or incorrect and displays the complete worked solution.

Continue practising - Click New Question to get a fresh random problem. Aim for at least 10 consecutive correct answers to build solid fluency.

💡 Example Problems

Example 1 — Classic 3-4-5 Triple (Find Hypotenuse)

Leg a = 3 units, Leg b = 4 units. Find the hypotenuse.

Apply the Pythagorean theorem: c = √(a² + b²) = √(3² + 4²)

Compute: √(9 + 16) = √25 = 5

Hypotenuse c = 5.00 units

Try this example →

Example 2 — Find Missing Leg (5-12-13 Triple)

Leg a = 5 units, Hypotenuse c = 13 units. Find leg b.

Rearrange: b = √(c² - a²) = √(13² - 5²) = √(169 - 25)

Compute: √144 = 12

Leg b = 12.00 units

Try this example →

Example 3 — Find an Angle (arctan)

Leg a (opposite) = 5 units, Leg b (adjacent) = 5 units. Find angle A.

Apply: A = arctan(a / b) = arctan(5 / 5) = arctan(1)

arctan(1) = 45° exactly (this is the 45-45-90 special triangle).

Angle A = 45.00°

Try this example →

Example 4 — Find an Angle (arcsin)

Leg a (opposite) = 10 units, Hypotenuse c = 20 units. Find angle A.

Apply: A = arcsin(a / c) = arcsin(10 / 20) = arcsin(0.5)

arcsin(0.5) = 30° exactly (half of the 30-60-90 pair).

Angle A = 30.00°

Try this example →

❓ Frequently Asked Questions

How do I find the hypotenuse of a right triangle?+

Use the Pythagorean theorem: c = √(a² + b²), where a and b are the two legs. Square each leg, add the results, then take the square root. For example, if the legs are 6 and 8, then c = √(36 + 64) = √100 = 10. The hypotenuse is always opposite the 90-degree angle and always the longest side.

How do I find a missing leg using the Pythagorean theorem?+

Rearrange the formula. If c is the hypotenuse and a is the known leg, then b = √(c² - a²). For example, with c = 13 and a = 5: b = √(169 - 25) = √144 = 12. Always confirm that the leg you are solving for is shorter than the hypotenuse before computing, otherwise the value under the square root would be negative.

What is the SOH-CAH-TOA mnemonic?+

SOH-CAH-TOA is a memory device for the three basic trig ratios: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. It tells you which two sides go with each trig function. To find an angle, take the inverse: arcsin, arccos, or arctan of the corresponding ratio.

What are Pythagorean triples and why are they useful?+

Pythagorean triples are integer sets (a, b, c) where a² + b² = c². The most common are 3-4-5, 5-12-13, 8-15-17, 7-24-25, and 9-40-41. Multiples also work: 6-8-10, 10-24-26. Recognising these lets you solve certain quiz problems instantly without a calculator, which is useful in timed exams.

How do I find an angle when I know two legs?+

Use arctan(opposite / adjacent). If you know leg a is opposite to angle A and leg b is adjacent to angle A, then A = arctan(a / b). For example, legs 3 and 4 give angle A = arctan(3/4) = arctan(0.75) = 36.87 degrees. The other acute angle B = 90 - 36.87 = 53.13 degrees.

How do I find an angle when I know one leg and the hypotenuse?+

Use arcsin if you know the opposite leg, or arccos if you know the adjacent leg. For example, if the opposite leg is 7 and the hypotenuse is 25, then A = arcsin(7/25) = arcsin(0.28) = 16.26 degrees. If the adjacent leg is 24 and the hypotenuse is 25, then A = arccos(24/25) = arccos(0.96) = 16.26 degrees. Both methods give the same answer.

What are the special 30-60-90 and 45-45-90 triangles?+

In a 30-60-90 triangle the side ratios are 1 : √3 : 2 (approximately 1 : 1.732 : 2). If the shortest side is s, the other leg is s√3 and the hypotenuse is 2s. In a 45-45-90 (isosceles right) triangle the ratios are 1 : 1 : √2. If each leg is s, the hypotenuse is s√2. These special triangles appear often in quiz problems and are worth memorising.

How much tolerance does the quiz allow?+

For side lengths the quiz accepts any answer within plus or minus 0.05 of the correct answer. For angles the tolerance is plus or minus 0.5 degrees. This means rounding 7.071 to 7.07 or 7.08 is accepted. Rounding errors at the second decimal place will not be marked wrong. Larger errors (such as missing a decimal point) will be flagged as incorrect.

Can I use this quiz to prepare for the SAT, ACT, or GCSE exams?+

Yes. Right triangle problems involving the Pythagorean theorem and basic trig (arctan, arcsin, arccos) appear regularly in SAT Math, ACT Math, GCSE Mathematics, and most high school geometry and trigonometry courses. Practising with randomised problems builds the speed and confidence needed to solve these questions quickly under exam conditions.

What is the difference between the Pythagorean theorem and trigonometry?+

The Pythagorean theorem (a² + b² = c²) relates the three side lengths of a right triangle to each other. It says nothing about angles. Trigonometry (sin, cos, tan and their inverses) connects angles to side ratios. Both are needed for right triangles: the Pythagorean theorem finds sides, and trig finds angles or a side when an angle is known. This quiz covers both.

Why does the quiz use random numbers instead of fixed problems?+

Fixed problem sets can be memorised without understanding. Random problems force you to apply the formula fresh each time, which is how real exams and real-world problems work. After 20 to 30 random problems you will find that the Pythagorean theorem and arctan formula become automatic, which is the goal of this practice tool.

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📌 Quick Tips

💡Always identify the right angle first. In a right triangle the side opposite the right angle is always the hypotenuse and it is always the longest side.

💡The Pythagorean theorem (a squared plus b squared equals c squared) only works when you already know it is a right triangle. It does not apply to obtuse or acute triangles.

💡When finding an angle from two legs, use arctan(opposite / adjacent). When finding an angle from a leg and the hypotenuse, use arcsin(opposite / hypotenuse) or arccos(adjacent / hypotenuse).

💡Round your answer to 2 decimal places for sides, and to 2 decimal places for angles in degrees. The checker accepts answers within a small tolerance.

💡The 3-4-5, 5-12-13, and 8-15-17 Pythagorean triples give whole-number answers. Recognising them saves calculation time.