How to Find Missing Angles in a Right Triangle: 3 Easy Methods

Have you ever stared at a right triangle on your homework, knowing two sides but completely stuck on finding that missing angle? You’re not alone! Learning how to find missing angles in a right triangle is one of the most practical skills you’ll develop in trigonometry—and once you understand the methods, it becomes almost second nature.

Whether you’re preparing for a test, working on a construction project, or just trying to understand how triangles work, this guide will walk you through three reliable methods that work every time. Let’s turn that confusion into confidence!

Understanding Right Triangle Basics

Right Triangle Anatomy

Before we dive into the methods, let’s make sure we’re on the same page about right triangle anatomy.

A right triangle has:

• One 90° angle (the right angle, marked with a small square)
• Two acute angles (each less than 90°)
• Three sides: the hypotenuse (longest side, opposite the right angle) and two legs

Here’s something crucial to remember:

Since one angle is always 90° in a right triangle, the other two angles must add up to 90°. This simple fact is the foundation of our first method!

Method 1: Using the Triangle Angle Sum Property

This is the easiest method when you already know one of the acute angles. No calculator needed—just basic arithmetic!

How It Works

Since all three angles must equal 180°, and one angle is 90°:

Or even simpler:

Step-by-Step Example

Problem: A right triangle has one acute angle measuring 35°. Find the other acute angle.

Solution:

1. We know: Angle A = 90° (right angle), Angle B = 35°
2. Apply the formula: Angle C = 90° - 35°
3. Answer: Angle C = 55°

Verification: 90° + 35° + 55° = 180° ✓

When to Use This Method

• When you already know one acute angle
• When you need a quick mental calculation
• When no side lengths are given (but one angle is)

Method 2: Using Inverse Trigonometric Functions

Now we’re getting to the real power of trigonometry! When you know two sides of a right triangle but no angles (except the 90°), inverse trig functions are your best friends.

The Big Three: SOH CAH TOA

First, let’s recall the basic trig ratios:

To find angles FROM side ratios, we use the inverse functions: arcsin (sin⁻¹), arccos (cos⁻¹), and arctan (tan⁻¹).

For a deeper dive into these functions, check out our guide on inverse trigonometric functions.

Step-by-Step Example: Using Tangent

Problem: A right triangle has legs measuring 6 cm (opposite) and 8 cm (adjacent) relative to angle θ. Find angle θ.

Solution:

1. Identify what we have: opposite = 6, adjacent = 8
2. Choose the right ratio: We have opposite and adjacent, so use tangent
3. Set up the equation: tan(θ) = 6/8 = 0.75
4. Apply inverse tangent: θ = tan⁻¹(0.75)
5. Calculate: θ ≈ 36.87°

Step-by-Step Example: Using Sine

Problem: The hypotenuse of a right triangle is 10 m, and one leg is 4 m. Find the angle opposite the 4 m leg.

Solution:

1. Identify: opposite = 4, hypotenuse = 10
2. Choose ratio: We have opposite and hypotenuse, so use sine
3. Set up: sin(θ) = 4/10 = 0.4
4. Apply inverse sine: θ = sin⁻¹(0.4)
5. Calculate: θ ≈ 23.58°

Try the Right Triangle Calculator

Choosing the Right Function

Known Sides	Use This Function
Opposite & Hypotenuse	sin⁻¹ (arcsin)
Adjacent & Hypotenuse	cos⁻¹ (arccos)
Opposite & Adjacent	tan⁻¹ (arctan)

Need to convert between degrees and radians? Our angle converter makes it simple.

Method 3: Using the Pythagorean Theorem + Trig Functions

Sometimes you only know two sides, but they’re not the most convenient pair for inverse trig. Here’s where combining the Pythagorean theorem with trigonometry becomes powerful.

The Strategy

1. Use the Pythagorean theorem to find the third side
2. Then use inverse trig with the most convenient pair of sides

Where c is always the hypotenuse.

Step-by-Step Example

Problem: A right triangle has a hypotenuse of 13 inches and one leg of 5 inches. Find both acute angles.

Solution:

Step 1: Find the missing side

• a² + b² = c²
• 5² + b² = 13²
• 25 + b² = 169
• b² = 144
• b = 12 inches

Step 2: Find the first acute angle (θ₁)

• Using the side opposite to θ₁ = 5, adjacent = 12
• tan(θ₁) = 5/12 ≈ 0.4167
• θ₁ = tan⁻¹(0.4167)
• θ₁ ≈ 22.62°

Step 3: Find the second acute angle (θ₂)

• Method A: θ₂ = 90° - 22.62° = 67.38°
• Method B (verification): tan(θ₂) = 12/5 = 2.4, so θ₂ = tan⁻¹(2.4) ≈ 67.38° ✓

Calculate Missing Sides with Our Pythagorean Calculator

Special Right Triangles: A Shortcut Worth Memorizing

Before we wrap up, there are two special right triangles where you don’t need to calculate anything—the angles and side ratios are fixed!

45-45-90 Triangle

If you see two equal legs, you immediately know both acute angles are 45°!

30-60-90 Triangle

If side ratios match this pattern:

• The smallest angle (30°) is opposite the shortest side
• The 60° angle is opposite the medium side (√3 times the shortest)
• The 90° angle is opposite the hypotenuse (2 times the shortest)

Explore these in detail with our special right triangles calculator or read our comprehensive guide on 30-60-90 and 45-45-90 triangles.

Practice Problems

Test your skills with these problems. Solutions are provided below!

Problem 1: A right triangle has one angle of 52°. Find the other acute angle.

Problem 2: A ladder leans against a wall. The base is 4 feet from the wall, and the ladder reaches 7 feet up the wall. What angle does the ladder make with the ground?

Problem 3: A ramp has a hypotenuse of 20 meters and rises 5 meters vertically. Find the angle of incline.

Solutions

Solution 1: 90° - 52° = 38°

Solution 2:

• tan(θ) = opposite/adjacent = 7/4 = 1.75
• θ = tan⁻¹(1.75) ≈ 60.26°

Solution 3:

• sin(θ) = opposite/hypotenuse = 5/20 = 0.25
• θ = sin⁻¹(0.25) ≈ 14.48°

Common Mistakes to Avoid

For more on avoiding errors, check out our article on common right triangle mistakes students make.

Real-World Applications

Finding missing angles isn’t just academic—it’s used everywhere:

• Construction: Determining roof pitch angles
• Safety: Calculating proper ladder placement angles
• Navigation: Determining directions and bearings
• Sports: Analyzing launch angles and trajectories
• Art & Design: Creating accurate perspective drawings

Quick Reference Summary

Situation	Method	Formula
Know one acute angle	Angle Sum	Missing = 90° - Known
Know opposite & hypotenuse	Inverse Sine	θ = sin⁻¹(O/H)
Know adjacent & hypotenuse	Inverse Cosine	θ = cos⁻¹(A/H)
Know opposite & adjacent	Inverse Tangent	θ = tan⁻¹(O/A)
Know any two sides	Pythagorean + Trig	Find 3rd side, then use inverse trig

Conclusion

Finding missing angles in a right triangle is a fundamental skill that opens doors to countless applications in math, science, and everyday life. Remember:

1. Method 1 (Angle Sum) is perfect when you know one acute angle
2. Method 2 (Inverse Trig) is your go-to when you have two sides
3. Method 3 (Pythagorean + Trig) combines both tools for maximum flexibility

The key is identifying what information you have and choosing the appropriate approach. With practice, you’ll instinctively know which method to use.

Ready to put your knowledge into action? Our calculators do the heavy lifting while you focus on understanding the concepts.

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Keep practicing, and remember—every expert was once a beginner who refused to give up. You’ve got this! 🎯