Label the sides relative to the angle you are using: opposite, adjacent, hypotenuse.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Choosing the ratio
Pick the ratio containing what you know and what you want. That is the whole decision.
Two sides, no angle
If you have two sides and want the third, Pythagoras is faster than any ratio.
A sanity check
The hypotenuse is always the longest side. An answer that breaks that is wrong.
Choose the ratio containing what you know
List which sides and angles you have and which you want. If you know the opposite and the hypotenuse, use sine. Choosing the ratio before doing any arithmetic is what keeps these problems mechanical.
Finding an angle needs the inverse
If you know two sides and want the angle, use sin⁻¹, cos⁻¹ or tan⁻¹. The inverse takes a ratio and returns the angle. It is not a reciprocal — sin⁻¹(x) is not 1/sin(x), despite the notation.
Check the calculator mode
Degrees or radians. A calculator in the wrong mode gives confidently wrong answers with no warning. Checking that sin(30) returns 0.5 takes two seconds and confirms the mode.
Check the answer against the picture
The longest side must be the hypotenuse, and the largest angle must be opposite the longest side. If your computed side is longer than the hypotenuse, something went wrong before the arithmetic.
Step 2: Try It Yourself
Set the two legs and read every other measurement.
- sin — opposite over hypotenuse0.60
- cos — adjacent over hypotenuse0.80
- tan — opposite over adjacent0.75
- The marked angle36.87°
The hypotenuse is not given. It comes from 4² + 3² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Amara Pick a Ratio
A ladder makes 60° with the ground and reaches 5 m up a wall.
- Step 1
The 5 m is opposite the 60° angle, and the ladder is the hypotenuse.
Step 4: Your Turn
Practice makes it stick.
The Third Side
Problem 1 of 2
Legs 6 and 8. What is the hypotenuse?
The Longest Side
Problem 2 of 2
Which side is always longest in a right triangle? 1 for a leg, 2 for the hypotenuse.
Which Ratio, Which Side
1 of 8
Legs 3 and 4. Hypotenuse?
2 of 8
Legs 5 and 12. Hypotenuse?
3 of 8
Hypotenuse 13, one leg 5. Other leg?
4 of 8
Match each pair of known parts to the ratio you would use.
Tap a card on the left to start.
5 of 8
Legs 9 and 12. Hypotenuse?
6 of 8
Can a leg be longer than the hypotenuse? 1 for yes, 0 for no.
7 of 8
Legs 8 and 15. Hypotenuse?
8 of 8
Two sides known, third wanted, no angles. Use Pythagoras? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
Legs 7 and 24. Hypotenuse?
What You Learned
- Label the sides relative to the angle in use.
- Choose the ratio containing what you know and what you want.
- With two sides and no angle, Pythagoras is the shorter route.