Trigonometry • Core Flagship Pillar

Right Triangle Calculator

Solve for missing sides ($a, b, c$), acute angles ($\alpha, \beta$), area, and altitude with SOH-CAH-TOA trigonometric ratios and interactive geometric SVG triangle models.

Reviewed by Sanjay Samanta & Academic Review Board
|
Last Updated: September 2026
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Euclidean Geometry & Trigonometric Standards Verified
Quick-Select Famous Right Triangles Standard Benchmarks

Given Known Values

Geometric Proportional Triangle (γ = 90°)
Hypotenuse (c)
5.000
c = √(a² + b²)
Angle α
36.87°
0.6435 rad
Angle β
53.13°
0.9273 rad
Area (K)
6.000
½ · a · b
Perimeter (P)
12.000
a + b + c
Altitude (h_c)
2.400
(a · b) / c
⊿

Step-by-Step Pythagorean & SOH-CAH-TOA Proof

Direct Answer & Overview
Verified Educational Guide

How to Solve a Right Triangle

To solve any right triangle with given legs a and b: 1. Hypotenuse: c = √(a² + b²). 2. Acute Angle α: α = arctan(b / a). 3. Acute Angle β: β = 90° − α. 4. Area: K = ½ab. 5. Perimeter: P = a + b + c.

Primary Mathematical Formula Standard Mathematical Model
Standard Equation
ƒ(x)
Q.E.D.
c=a2+b2,α=arctan⁡(ba),Area=12abc = \sqrt{a^2 + b^2}, \quad \alpha = \arctan\left(\frac{b}{a}\right), \quad \text{Area} = \frac{1}{2}ab
Evaluated with exact mathematical formulation • Rigorously verified
Exact Formula
Input Parameters
Required
1
Any two known values: two legs (a, b), leg and hypotenuse (a, c), or side and acute angle
Expected Outputs
Calculated
All 3 side lengths (a, b, c), all 3 angles (α, β, γ = 90°)
Area (K), perimeter (P), altitude to hypotenuse, and geometric SVG model
Worked Numerical Example
Instant Verification
Solve right triangle with legs a = 3 and b = 4
→ c = √(3² + 4²) = √25 = 5; α = arctan(4/3) = 36.87°; β = 90° − 36.87° = 53.13°; Area = ½(3)(4) = 6
Hypotenuse c = 5 | α = 36.87° | β = 53.13° | Area = 6

Right Triangle Trigonometry & SOH-CAH-TOA Foundations

In a right-angled triangle where $\gamma = 90^\circ$, trigonometric functions define invariant ratios between side lengths:

Sine (SOH) sin(θ) = Opp / Hyp

Opposite side over hypotenuse

Cosine (CAH) cos(θ) = Adj / Hyp

Adjacent side over hypotenuse

Tangent (TOA) tan(θ) = Opp / Adj

Opposite side over adjacent

Special Right Triangle Identities (30-60-90 & 45-45-90)

Special Triangle Side Ratio (Short : Long : Hyp) Exact Trig Values
30°-60°-90°1 : √3 : 2sin(30°) = ½, cos(30°) = √3/2, tan(30°) = 1/√3
45°-45°-90° (Isosceles)1 : 1 : √2sin(45°) = 1/√2, cos(45°) = 1/√2, tan(45°) = 1

Real-World Applications of Right Triangle Trigonometry

Carpentry & Roof Rafters

Builders use run (base) and rise (height) with Pythagorean theorem to cut rafter diagonal lengths and pitch bevel angles.

Land Surveying & Elevations

Surveyors use theodolites to measure angle of elevation θ and horizontal distance d to compute mountain height h = d·tan(θ).

Maritime & Aviation Navigation

Navigators resolve vector crosswinds and ship headings into orthogonal north-south and east-west components.

Step-by-Step Worked Numerical Solutions

Example 1: Given Leg and Acute Angle a = 10, α = 30°

Problem: Solve right triangle with adjacent leg a = 10 and acute angle α = 30°.

1. Opposite Leg b = a · tan(30°) = 10 · (1 / √3) = 5.7735.
2. Hypotenuse c = a / cos(30°) = 10 / (√3 / 2) = 11.5470.
3. Angle β = 90° − 30° = 60°.
4. Area = ½ · a · b = ½(10)(5.7735) = 28.8675.
Result: b = 5.774, c = 11.547, β = 60°, Area = 28.868

Common Pitfalls & Mistakes

Radian vs. Degree Mode

Ensure angles entered in degrees (e.g. 30°) are converted to radians (30π/180) before evaluating in standard programming trig functions.

Confusing Opposite and Adjacent

Opposite and adjacent are relative to the reference angle. For angle α, side b is opposite; for angle β, side a is opposite.

Assuming Hypotenuse is a Leg

The hypotenuse c is ALWAYS the longest side opposite the 90° angle. A leg can never exceed the hypotenuse.

Fact-Checked & Verified • Computational Accuracy Standards
Updated August 2026 • Editorial Policy
Authored By
Sanjay Samanta

Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.

Reviewed & Verified By
Academic Review Board

Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.

Found an error or have an improvement suggestion? Report a calculation issue

Frequently Asked Questions

How do you solve a right triangle given any two sides?
If two legs a and b are known, find hypotenuse c using the Pythagorean theorem: c = √(a² + b²). Find the acute angles using inverse tangent: α = arctan(b / a) and β = 90° − α.
What is the SOH-CAH-TOA mnemonic in right triangle trigonometry?
SOH: Sine = Opposite / Hypotenuse (sin θ = opp/hyp). CAH: Cosine = Adjacent / Hypotenuse (cos θ = adj/hyp). TOA: Tangent = Opposite / Adjacent (tan θ = opp/adj).
What are the side length ratios of special 30°-60°-90° and 45°-45°-90° triangles?
In a 30°-60°-90° triangle, the sides are in ratio 1 : √3 : 2 (short leg : long leg : hypotenuse). In a 45°-45°-90° isosceles right triangle, the sides are in ratio 1 : 1 : √2.
How is the altitude to the hypotenuse (h_c) calculated?
By equating triangle area formulas (½ab = ½c·h_c), the altitude to the hypotenuse is: h_c = (a × b) / c.
What is the sum of angles in any right triangle?
The sum of all three angles is always 180°. Since the right angle is exactly 90°, the two acute angles α and β are always complementary: α + β = 90°.

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