Solving Right Triangles: Finding Missing Sides
Given any combination of sides and angles, learn to solve a right triangle completely β finding all three sides using the Pythagorean Theorem and trig ratios.
A structural engineer checking if a diagonal brace can withstand a load, a contractor figuring out the exact length of a rafter, or a game developer placing objects in a 3D world β all of them need to solve right triangles. "Solving" means finding every unknown side given any two pieces of information. Once you have one method (SOHCAHTOA or Pythagorean), you can always find the rest. This lesson makes you fluent with both tools together.
- Solve for any missing side given two sides using the Pythagorean Theorem
- Solve for a missing side given one side and one acute angle using SOHCAHTOA
- Recognize which tool to use based on what information is given
- Complete multi-step problems where you find one side and then use it to find another
- Two sides known, no angle? β Use the Pythagorean Theorem to find the third side.
- One angle + hypotenuse? β Use sin for opposite, cos for adjacent.
- One angle + a leg? β Use tan (opp and adj) or sin/cos (involving hyp).
Choosing Your Tool
| Given | Find | Tool |
|---|---|---|
| Both legs | Hypotenuse | \(c = \sqrt{a^2+b^2}\) |
| Hyp + leg | Other leg | \(a = \sqrt{c^2-b^2}\) |
| Angle + hyp | Opp leg | \(\text{opp} = \text{hyp}\cdot\sin\theta\) |
| Angle + hyp | Adj leg | \(\text{adj} = \text{hyp}\cdot\cos\theta\) |
| Angle + adj | Opp leg | \(\text{opp} = \text{adj}\cdot\tan\theta\) |
A right triangle has legs a = 6 and b = 8. Find the hypotenuse.
\[ c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = \mathbf{10} \]A right triangle has angle ΞΈ = 32Β° and hypotenuse = 18. Find both legs.
- Opposite leg: \(\text{opp} = 18\sin32Β° = 18 \times 0.530 \approx \mathbf{9.54}\)
- Adjacent leg: \(\text{adj} = 18\cos32Β° = 18 \times 0.848 \approx \mathbf{15.26}\)
- Check: \(\sqrt{9.54^2 + 15.26^2} = \sqrt{91.0 + 232.9} = \sqrt{323.9} \approx 18.0\) β
A ship travels 80 nautical miles due east, then turns and travels 60 nautical miles due north. How far is the ship from its starting point? (This is the hypotenuse of a right triangle.)
\[ d = \sqrt{80^2 + 60^2} = \sqrt{6400 + 3600} = \sqrt{10000} = \mathbf{100 \text{ nautical miles}} \]- Legs a = 5, b = 12. Find the hypotenuse.
- Angle = 48Β°, adjacent side = 10. Find the opposite side and hypotenuse.
- Hypotenuse = 25, one leg = 7. Find the other leg.
βΆ Show Answers
- \(c = \sqrt{25+144} = \sqrt{169} = \mathbf{13}\).
- \(\text{opp} = 10\tan48Β° \approx 10\times1.111 \approx \mathbf{11.11}\); \(\text{hyp} = 10/\cos48Β° \approx 10/0.669 \approx \mathbf{14.94}\).
- \(b = \sqrt{625 - 49} = \sqrt{576} = \mathbf{24}\).
- Using Pythagorean Theorem when angles are involved: If you have an angle, trig is usually cleaner than squaring. Pythagorean Theorem gives sides from sides only.
- Squaring before adding vs. after: \(\sqrt{a^2 + b^2} \neq a + b\). Always square first, then add, then take the square root.
- Mixing up which side is adjacent vs. opposite: Re-label the sides freshly at the start of every problem based on the reference angle given.
- Two sides known β Pythagorean Theorem.
- One angle + one side β SOHCAHTOA.
- Always verify your answer: \(a^2 + b^2\) should equal \(c^2\).
- A right triangle is fully solved when all 3 sides and both acute angles are found.
Civil engineers solve right triangles constantly: computing the diagonal span of a truss, finding the slope of a drainage pipe from horizontal run and elevation drop, or determining the anchor placement for a retaining wall. The Pythagorean Theorem and SOHCAHTOA underlie every structural calculation involving angles or diagonal distances.
Calculator Connection
The Right Triangle Solver finds all missing sides and angles from any two given values. The Pythagorean Theorem Calculator computes the missing side when two sides are known.
Try it with the Calculator
Apply what you've learned with these tools.
Solving Right Triangles: Finding Missing Sides β Quiz
5 questions per attempt Β· Beginner Β· Pass at 70%
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