2026-08-07 10 min

Right Triangle Calculator

Solve right triangles: side lengths, angles, area, perimeter, and altitude from any two known values with Pythagorean verification.

Calculator

Side a Help

Leg opposite α

Side b Help

Leg opposite β

Side c (hypotenuse) Help

Longest side

Angle α Help

Degrees

Angle β Help

Degrees

Angle Unit

Provide any 2 values — γ is always 90°

Results update automatically

Results update live as you type

Status:a=3.0000 b=4.0000 c=5.0000α=36.87° β=53.13°A=6.0000 P=12.0000

Right Triangle Summary

Sidesa=3.0000 b=4.0000 c=5.0000

Anglesα=36.87° β=53.13° γ=90°

Altitude h2.4000

Area & Perimeter6.0000 · 12.0000

3.0000² + 4.0000² = 5.0000² => 9.0000 + 16.0000 = 25.0000 ✓ Verified

Results

Optimal

Side a

3.0000

Leg opposite α

Side b

4.0000

Leg opposite β

Hypotenuse c

5.0000

Opposite 90°

Angle α

36.87°

At base between b and c

Angle β

53.13°

α + β = 90°

Altitude h

2.4000

(a×b)/c to hypotenuse

Area

6.0000sq units

(a×b)/2

Perimeter

12.0000

a + b + c

Solves any two-known right triangle (≥1 side) via Pythagoras and SOH-CAH-TOA. Legs must be shorter than the hypotenuse; acute angles under 90°.

Visualizations

Sides Compared

2.4h

Legs Split (a vs b)

a43%

b57%

α Share of 90°

α37°

Area vs Perimeter (numeric)

Area6

Formula

Side–side pairs use Pythagoras; side–angle pairs use sine/cosine/tangent; angles complete to 90°.

a² + b² = c²; sin α = a/c; cos α = b/c; tan α = a/b

Alternative Form

h = ab/c; A = ab/2; P = a+b+c; β = 90° − α

Variables

Symbol	Name	Unit	Description
a,b	Legs	units	Perpendicular sides
c	Hypotenuse	units	Longest, opposite 90°
α,β	Acute angles	°/rad	Sum to 90°
h	Altitude	units	ab/c to hypotenuse
A,P	Area / perimeter	sq units / units	ab/2 and a+b+c

Explanation

Special 30-60-90 (x:x√3:2x) and 45-45-90 (x:x:x√2) ratios shortcut angle-known cases.

Derivation

Pythagoras for sides; SOH-CAH-TOA for angles; geometric mean for altitude segments.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Identify the two knowns and pick Pythagoras vs trig.

Side–side → Pythagoras; side–angle → trig

a:3.0000

b:4.0000

c:5.0000

Result: Solved triple a=3.0000, b=4.0000, c=5.0000

Step 2

Solve the third side.

c=√(a²+b²) or leg=√(c²−leg²) or trig

Intermediate: 3.0000² + 4.0000² = 5.0000² => 9.0000 + 16.0000 = 25.0000 ✓ Verified

Result: c = 5.0000

Step 3

Solve acute angles via inverse trig or atan2.

α = atan2(a,b); β = 90°−α

α:36.87°

β:53.13°

Result: α = 36.87°, β = 53.13°

Step 4

Enforce α + β = 90° with γ = 90°.

α + β + 90° = 180°

Intermediate: 36.87 + 53.13 = 90.00

Result: Angle sum verified ✓

Step 5

Compute altitude to the hypotenuse.

h = (a×b)/c

h:2.4000

Intermediate: (3.0000×4.0000)/5.0000

Result: h = 2.4000

Step 6

Finish area, perimeter, and Pythagoras check.

A=ab/2; P=a+b+c

Area:6.0000

Perimeter:12.0000

Intermediate: 3.0000² + 4.0000² = 5.0000² => 9.0000 + 16.0000 = 25.0000 ✓ Verified

Result: A = 6.0000, P = 12.0000 ✓

Final Answer

3.0000

Worked Examples

See how the calculator works with different scenarios.

Scenario

a=3, b=4 — classic triple.

Inputs

a:3

b:4

Calculation

c = √(9+16) = 5 α = atan2(3,4) ≈ 36.87° β ≈ 53.13°

Result

c=5, α≈36.87°, β≈53.13°, A=6, P=12

Interpretation

Default view; 3-4-5 verified instantly.

Understanding Your Result

Your Result: Side a: 3.0000

Typical range: Leg < hyp; α,β in (0,90°); h < min(a,b); 30-60-90 and 45-45-90 as special checks.

What This Means

All sides, both acute angles, altitude, area, and perimeter consistent under Pythagoras and angle sum.

Why It Matters

Construction squaring, navigation bearings, surveying, force decomposition, and graphics all reduce to this solve.

Professional Interpretation

Confirm leg<hyp and α+β=90°; use radian mode consistently when scripting.

Recommendations

•Give any 2 valid values
•Respect leg < hyp
•Keep angles under 90°
•Use SOH-CAH-TOA deliberately
•Check 30-60-90/45-45-90 patterns
•Verify via Pythagoras line

Educational Content

Introduction

A right triangle is fixed by two data (with at least one side): Pythagoras gives sides, trig gives angles.

Concept Explanation

a²+b²=c² plus sin/cos/tan (SOH-CAH-TOA) plus α+β=90° solve all nine side/angle combos.

Applications

▸3-4-5 squaring on site

▸Roof, stair, ramp slopes

▸Bearings and triangulation

▸Vector components

▸Parallax distances

▸Camera and ray math

Benefits

•Nine-combo solver
•Degree/radian modes
•Altitude + area + perimeter
•Live Pythagoras proof line
•Special-triangle intuition

Limitations

•Right triangles only
•Needs ≥2 values incl. a side
•Two angles alone underdetermine size
•Ideal geometry

Best Practices

•Name hyp first
•Pick sin vs cos by opposite/adjacent
•Complete to 90°
•Check h = ab/c
•Round angles to 0.01°

Common Misconceptions

Myth: Two angles fix the triangle.

Truth: Angles give shape but not scale — a side is required.

Myth: Hypotenuse can be a leg.

Truth: Hypotenuse is always longest and opposite 90°.

Myth: 45-45-90 legs differ.

Truth: They are equal; hyp is leg×√2.

Myth: Altitude equals a leg.

Truth: h = ab/c is shorter than either leg (except degenerate).

Practical Tips

•Provide any 2 values
•Leg must be < hypotenuse
•Angles must be < 90°
•Use SOH-CAH-TOA for side+angle
•α+β always 90°
•Check h = ab/c
•Spot 30-60-90 and 45-45-90

Common Mistakes

Two angles only

Add at least one side — angles alone leave scale free.

Leg ≥ hypotenuse

Relabel: longest input must be c.

Angle ≥ 90° as α/β

Acute angles only; γ is the 90° one.

Wrong trig ratio

Opposite/hyp = sin, adjacent/hyp = cos, opp/adj = tan.

Degree/radian mix

Set the unit selector before typing angles.

Assumptions

This calculator makes the following assumptions to provide estimates:

Right angle at γ: γ fixed at 90° (π/2 rad).

Two knowns incl. side: Underdetermined otherwise.

Leg < hyp: Enforced with error messages.

Acute α/β: Each in (0, 90°) exclusive.

Flat geometry: Euclidean plane, consistent units.

Limitations

Be aware of these limitations when using this calculator:

Right-only: Oblique needs law of sines/cosines.

Scale needs a side: Angles alone give similarity only.

Ideal shape: No tolerance or thickness.

Single triangle: One solve at a time.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a right triangle?

A triangle with one exact 90° angle. The hypotenuse opposes it and is longest; the legs form the right angle. Basis of Pythagoras and trig ratios.

What is the Pythagorean theorem?

a² + b² = c² for legs a,b and hypotenuse c. The workhorse for missing sides, distance, and verification.

How do I find the hypotenuse of a right triangle?

c = √(a²+b²). For 3 and 4: √(9+16) = 5. Enter any two sides and the calculator roots the third.

How do I find a missing leg?

a = √(c²−b²) or b = √(c²−a²). For c=13, a=5: b = √(169−25) = 12. Leg must stay shorter than c.

What is a 30-60-90 triangle?

Sides in ratio x : x√3 : 2x (opposite 30°, 60°, hypotenuse). Short leg 5 → long 5√3 ≈ 8.66, hyp 10.

What is a 45-45-90 triangle?

Isosceles right triangle with legs x,x and hypotenuse x√2. Legs 7 → hyp 7√2 ≈ 9.899.

How do I find the area of a right triangle?

A = (a×b)/2 — legs are base and height since perpendicular. 3-4-5 gives 6 sq units.

How do I find the perimeter?

P = a+b+c. Solve the third side first via Pythagoras or trig, then sum.

What is the altitude of a right triangle?

From the right angle to the hypotenuse: h = (a×b)/c. It splits the triangle into two smaller similar right triangles.

What does SOH-CAH-TOA stand for?

Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent — which ratio to use for sides vs angles.

How do I find an angle from two sides?

Use inverses: angle = sin⁻¹(opp/hyp), cos⁻¹(adj/hyp), or tan⁻¹(opp/adj). The other acute angle is 90° minus it.

Can I enter values in radians?

Yes — switch Angle Unit to radian. A right angle is π/2 ≈ 1.5708 rad. You can type pi/2 style decimals.

What are Pythagorean triples?

Integer solutions like (3,4,5), (5,12,13), (7,24,25), (8,15,17). They give exact integer sides and quick layout checks.

How do I solve a triangle with one side and one angle?

Use trig: e.g., a known with α gives c = a/sin(α), b = a/tan(α); then β = 90°−α. The calculator automates all nine combos.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

SOH-CAH-TOA fluency

Picking the right ratio.

Special triangles

30-60-90 and 45-45-90 shortcuts.

Solving step by step

Sides, angles, checks.

Right triangles on site

3-4-5 and beyond.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Hypotenuse: Longest side opposite 90°.
Leg: Side forming the right angle.
SOH-CAH-TOA: Sin/cos/tan ratio mnemonic.
Altitude: h = ab/c to the hypotenuse.
30-60-90: Sides x : x√3 : 2x.
45-45-90: Isosceles right; hyp x√2.
Circumcircle: Center at hyp midpoint (Thales).
Inradius: (a+b−c)/2 for right triangles.
Geometric mean: h² = p·q on hyp segments.
3-4-5 rule: Site method for 90° corners.

Authoritative References

Trusted sources and standards referenced by this calculator.

Right Triangle — Wikipedia Pythagorean Theorem — Wikipedia Special Right Triangles — Wikipedia Trigonometry — Wikipedia Khan Academy: Right triangles Wolfram MathWorld: Right triangle Desmos Geometry

Applicable Standards

Common Core G-SRT — Similarity & trigNCTM Geometry expectations

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Page Information

Last Updated:

2026-08-07

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.