How to Solve Right Triangles with Missing Information

Every right triangle has six measurements: three sides and three angles. But here's the remarkable thing—you only need to know two of these measurements (besides the right angle itself) to calculate all the others. This principle transforms incomplete information into complete understanding through the power of trigonometry.

Whether you're working on homework, designing a project, or solving a real-world problem, knowing how to systematically find missing information is an essential skill. This guide will walk you through every possible scenario, providing clear strategies for whatever information you start with.

The Foundation: What We Always Know

Before solving any right triangle, remember what you already know:

• One angle is always 90° (the right angle)
• The three angles sum to 180°, so the two acute angles sum to 90°
• The sides follow the Pythagorean relationship: a² + b² = c²
• Trigonometric ratios connect angles to sides

These four principles are your tools. Every solution uses at least one of them.

Understanding the Six Scenarios

Right triangles can be solved when you know:

1. Two sides
2. One side and one acute angle
3. The hypotenuse and one acute angle
4. Two sides (one being the hypotenuse)
5. One leg and one acute angle
6. The hypotenuse and one leg

Let's examine each scenario systematically.

Scenario 1: Two Sides Given (Both Legs)

Given: Side a and side b (the two legs)
Find: Side c (hypotenuse), angle A, angle B

Strategy: Use the Pythagorean theorem, then trigonometry for angles.

Step-by-Step Process

Example: a = 6, b = 8

Step 1: Find the hypotenuse using a² + b² = c²

6² + 8² = c²
36 + 64 = c²
100 = c²
c = √100 = 10

Step 2: Find one acute angle using tangent

tan(A) = opposite/adjacent = a/b = 6/8 = 0.75
A = tan⁻¹(0.75) ≈ 36.87°

Step 3: Find the other angle

Since angles in a triangle sum to 180°, and we have a 90° angle:

B = 180° - 90° - A
B = 90° - 36.87°
B ≈ 53.13°

Verification: 36.87° + 53.13° = 90°

Alternative for Step 2: You could also use sine or cosine:

sin(A) = a/c = 6/10 = 0.6 → A = sin⁻¹(0.6) ≈ 36.87°
cos(A) = b/c = 8/10 = 0.8 → A = cos⁻¹(0.8) ≈ 36.87°

All methods give the same answer—use whichever feels most natural.

Scenario 2: Two Sides Given (One Leg and Hypotenuse)

Given: Side a (one leg) and side c (hypotenuse)
Find: Side b (other leg), angle A, angle B

Strategy: Use the Pythagorean theorem, then trigonometry.

Step-by-Step Process

Example: a = 5, c = 13

Step 1: Find the missing leg using a² + b² = c²

Rearrange to: b² = c² - a²

b² = 13² - 5²
b² = 169 - 25
b² = 144
b = √144 = 12

Step 2: Find angle A

sin(A) = opposite/hypotenuse = a/c = 5/13 ≈ 0.385
A = sin⁻¹(0.385) ≈ 22.62°

Step 3: Find angle B

B = 90° - A = 90° - 22.62° = 67.38°

Key Point: When you have the hypotenuse and one leg, sine is often the most direct route to finding angles because you immediately have opposite/hypotenuse.

Scenario 3: One Leg and One Acute Angle

Given: Side a (one leg) and angle A (opposite to side a)
Find: Side b, side c, angle B

Strategy: Use the angle to find the other angle, then use trigonometry to find sides.

Step-by-Step Process

Example: a = 7, A = 30°

Step 1: Find the other acute angle

B = 90° - A = 90° - 30° = 60°

Step 2: Find the adjacent leg using tangent

tan(A) = a/b
tan(30°) = 7/b
0.577 = 7/b
b = 7/0.577 ≈ 12.12

Step 3: Find the hypotenuse using sine

sin(A) = a/c
sin(30°) = 7/c
0.5 = 7/c
c = 7/0.5 = 14

Alternative for Step 2: Use the Pythagorean theorem after finding c:

b² = c² - a² = 14² - 7² = 196 - 49 = 147
b = √147 ≈ 12.12

Note: This example creates a 30-60-90 triangle, a special right triangle with the ratio 1:√3:2. Here we see 7:12.12:14 ≈ 1:1.73:2, confirming the pattern.

Scenario 4: Hypotenuse and One Acute Angle

Given: Side c (hypotenuse) and angle A
Find: Side a, side b, angle B

Strategy: Find the other angle, then use trigonometry for both legs.

Step-by-Step Process

Example: c = 20, A = 40°

Step 1: Find angle B

B = 90° - A = 90° - 40° = 50°

Step 2: Find side a (opposite to angle A)

sin(A) = a/c
sin(40°) = a/20
0.643 = a/20
a = 20 × 0.643 ≈ 12.86

Step 3: Find side b (adjacent to angle A)

cos(A) = b/c
cos(40°) = b/20
0.766 = b/20
b = 20 × 0.766 ≈ 15.32

Verification using Pythagorean theorem:

12.86² + 15.32² ≈ 165.38 + 234.70 ≈ 400
√400 = 20

Scenario 5: Two Angles Given (Besides the Right Angle)

Given: Angle A and angle B
Find: This is a trick scenario!

Critical Insight: You cannot determine the size of a triangle from angles alone. You can only determine its shape. All triangles with angles 30°-60°-90° are similar (same shape, different sizes), but without at least one side length, you cannot solve for specific measurements.

What you CAN determine:

• The ratios between sides
• That it's a valid triangle (if A + B = 90°)
• The type of special triangle it might be

What you CANNOT determine:

• Actual side lengths

Example: If A = 35° and B = 55°, you know:

It's a valid right triangle (35° + 55° + 90° = 180°)
tan(35°) = a/b ≈ 0.7, so side a is about 70% of side b
But without knowing a, b, or c, you cannot solve further

Decision Tree: What to Calculate First

When solving a right triangle, follow this decision tree:

Start: What information do I have?

Two sides?

• Use Pythagorean theorem for third side
• Use any trig function for one angle
• Subtract from 90° for other angle

One side and one angle?

• Find other angle (subtract from 90°)
• Use trig functions to find remaining sides
• Verify with Pythagorean theorem

Hypotenuse and angle?

• Find other angle first
• Use sine for opposite side
• Use cosine for adjacent side

Two legs?

• Pythagorean theorem for hypotenuse
• Tangent for one angle
• Subtract from 90° for other angle

Common Solving Mistakes and How to Avoid Them

Mistake 1: Using the Wrong Trig Function

Problem: Confusing which side is opposite, adjacent, or hypotenuse relative to your angle.

Solution:

• Draw the triangle
• Mark your angle clearly
• Identify which side is which relative to THAT angle
• Remember: SOH-CAH-TOA
  • Sine = Opposite/Hypotenuse
  • Cosine = Adjacent/Hypotenuse
  • Tangent = Opposite/Adjacent

Mistake 2: Forgetting to Take the Square Root

Problem: Finding a² + b² = 100 and writing c = 100 instead of c = 10.

Solution: The Pythagorean theorem gives you c², not c. Always take the square root as your final step.

Mistake 3: Using Degrees vs. Radians

Problem: Calculator is in radian mode when you need degrees (or vice versa).

Solution:

• Check your calculator mode before starting
• If sin(30°) doesn't equal 0.5, your mode is wrong
• Most practical problems use degrees

Mistake 4: Rounding Too Early

Problem: Using tan(35°) = 0.7 in subsequent calculations instead of the full precision 0.7002...

Solution:

• Keep full calculator precision until the end
• Round only your final answer
• If working by hand, use at least 3-4 decimal places

Mistake 5: Not Checking Your Answer

Problem: Accepting an answer without verification.

Solution: Always verify using a different method:

• If you found c using Pythagorean theorem, check using c = a/sin(A)
• If angles don't sum to 90°, something is wrong
• Do the side lengths make sense? Is the hypotenuse longest?

Advanced Technique: Working Backwards

Sometimes you need to reverse-engineer a triangle from specific requirements.

Example: "I need a ramp that rises 2 feet over a 10-foot horizontal distance. What angle is this, and what's the actual ramp length?"

Analysis:

• a = 2 feet (rise)
• b = 10 feet (run)
• Find angle and hypotenuse

Solution:

tan(angle) = 2/10 = 0.2
angle = tan⁻¹(0.2) ≈ 11.31°
c = √(4 + 100) = √104 ≈ 10.2 feet

This working-backwards approach applies to real-world problems where you're designing something to meet specifications.

Using Technology: Calculator Strategies

Essential Calculator Functions

For solving triangles, you need:

• Square root (√)
• Squaring (x²)
• Inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹)
• Standard trig functions (sin, cos, tan)

Pro tips:

• Store intermediate values in memory rather than writing them down
• Use parentheses liberally to ensure correct order of operations
• Most calculators have a degrees/radians toggle—know where yours is

Online Calculators

When you want to verify your work or check complex calculations, a right triangle calculator can provide instant verification. Use it to:

• Check your manual calculations
• Verify you used the correct formulas
• Ensure your final answers are reasonable

But always solve by hand first—the calculator is a verification tool, not a replacement for understanding.

Complete Worked Examples

Example 1: Both Legs Known

Given: a = 15, b = 20
Find: Everything else

Solution:

c² = 15² + 20² = 225 + 400 = 625 → c = 25
tan(A) = 15/20 = 0.75 → A = 36.87°
B = 90° - 36.87° = 53.13°

Answer: c = 25, A ≈ 36.87°, B ≈ 53.13°

Note: This is the 3-4-5 triple scaled by 5!

Example 2: Hypotenuse and Angle

Given: c = 50, A = 28°
Find: Everything else

Solution:

B = 90° - 28° = 62°
a = c × sin(A) = 50 × sin(28°) = 50 × 0.469 = 23.47
b = c × cos(A) = 50 × cos(28°) = 50 × 0.883 = 44.15

Verification: 23.47² + 44.15² = 550.84 + 1949.22 ≈ 2500 = 50²

Answer: a ≈ 23.47, b ≈ 44.15, B = 62°

Example 3: One Leg and Angle

Given: a = 9, A = 60°
Find: Everything else

Solution:

B = 90° - 60° = 30°
tan(60°) = 9/b → b = 9/tan(60°) = 9/1.732 ≈ 5.20
sin(60°) = 9/c → c = 9/sin(60°) = 9/0.866 ≈ 10.39

Alternative for c: c² = 9² + 5.20² = 81 + 27.04 = 108.04 → c ≈ 10.39

Answer: b ≈ 5.20, c ≈ 10.39, B = 30°

Recognition: This is a 30-60-90 triangle! The sides should follow the 1:√3:2 ratio. Let's verify: 5.20:9:10.39 ≈ 1:1.73:2

Special Cases and Shortcuts

When You Recognize Special Triangles

If you identify a 30-60-90 or 45-45-90 triangle, you can solve it instantly using ratios:

30-60-90 ratios: 1 : √3 : 2
45-45-90 ratios: 1 : 1 : √2

Example: c = 10 in a 45-45-90 triangle

Both legs = c/√2 = 10/√2 = 7.07

Much faster than calculating each side individually!

When You Recognize Pythagorean Triples

If sides are 3-4-5 or 5-12-13 (or multiples), you know immediately:

• The third side without calculation
• It's definitely a right triangle
• Angles can be found, but the sides are instantly known

Practice Problems

Test your understanding with these problems:

Problem 1: a = 24, b = 7. Find c, A, and B.
Solution: c = √(24² + 7²) = √625 = 25; tan(A) = 24/7 → A ≈ 73.74°; B ≈ 16.26°

Problem 2: c = 15, A = 50°. Find a, b, and B.
Solution: B = 40°; a = 15×sin(50°) ≈ 11.49; b = 15×cos(50°) ≈ 9.64

Problem 3: a = 8, A = 45°. Find b, c, and B.
Solution: B = 45° (45-45-90 triangle!); b = 8; c = 8√2 ≈ 11.31

Problem 4: a = 5, c = 13. Find b, A, and B.
Solution: b = √(13² - 5²) = 12 (5-12-13 triple!); sin(A) = 5/13 → A ≈ 22.62°; B ≈ 67.38°

Systematic Approach Summary

For any right triangle problem:

Step 1: Draw it - Sketch the triangle and label what you know

Step 2: Identify what you have - Two sides? Side and angle? Hypotenuse?

Step 3: Choose your path

• Need a side? → Pythagorean theorem or trig function
• Need an angle? → Inverse trig function or subtract from 90°

Step 4: Calculate systematically - Do one step at a time, clearly

Step 5: Verify - Check using an alternative method

Step 6: Check reasonableness - Does your answer make sense?

Conclusion: The Power of Systematic Thinking

Solving right triangles isn't about memorizing formulas—it's about understanding relationships. The Pythagorean theorem connects the sides. Trigonometric functions connect angles to sides. The angle sum property connects the angles to each other.

With these three tools, any right triangle with two known measurements becomes completely solvable. The key is systematic thinking: identify what you know, determine what you need, choose the appropriate tool, and verify your answer.

Every right triangle problem is a puzzle with a clear solution path. Sometimes multiple paths lead to the answer, and that's part of the beauty—the mathematics is so interconnected that you can approach problems from different angles (literally!) and arrive at the same truth.

The triangle doesn't change based on how you solve it. 3-4-5 is always a right triangle. A 30-60-90 triangle always has sides in the ratio 1:√3:2. These are mathematical certainties. Your job is simply to apply the right tools in the right order to reveal the complete picture.

Master these techniques, and you'll find that right triangles transform from obstacles into opportunities—clear problems with clear solutions, waiting for you to apply the elegant tools of geometry and trigonometry to unlock their secrets.