CALTRX

Right Triangle Calculator

Enter any two sides of a right triangle to calculate the third side, angles, area, and perimeter. See the complete solution with step-by-step working and formula explanations.

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Enter your values above to see the results.

Tips & Notes

  • ✓SOH-CAH-TOA mnemonic: Sin=Opposite/Hyp, Cos=Adjacent/Hyp, Tan=Opposite/Adjacent.
  • ✓Hypotenuse is always the longest side — opposite the 90° angle.
  • ✓Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17. Memorizing saves calculation time.
  • ✓45-45-90 triangle: both legs equal, hypotenuse = leg × √2. Diagonal of a square.
  • ✓30-60-90 triangle: sides 1:√3:2. Short leg × 2 = hypotenuse. Short leg × √3 = long leg.

Common Mistakes

  • ✗Applying Pythagorean theorem to non-right triangles — only valid when one angle = 90°.
  • ✗Labeling hypotenuse incorrectly — it is always opposite the right angle and always the longest side.
  • ✗Using sin/cos/tan with degrees when calculator is in radian mode — check your calculator setting.
  • ✗Finding arctan gives angle in (−90°,90°). For obtuse triangles, check quadrant carefully.
  • ✗Not verifying: a²+b² should equal c² exactly. Floating point may give 24.9999 instead of 25.

Right Triangle Calculator Overview

A right triangle contains one 90° angle, making it the most mathematically tractable triangle shape. The relationship between its sides — the Pythagorean theorem — is one of the oldest mathematical results in history, appearing in Babylonian clay tablets from 1800 BCE. The trigonometric ratios that define sine, cosine, and tangent were originally defined through right triangles. This calculator finds all missing sides, angles, area, and perimeter from any two given measurements.

The Pythagorean Theorem — the relationship between the three sides:

a² + b² = c² where c is the hypotenuse (side opposite the 90° angle)
EX: legs a=3, b=4 → c = √(9+16) = √25 = 5 | EX: hypotenuse=13, leg=5 → other leg = √(169−25) = √144 = 12
Trigonometric ratios — SOH-CAH-TOA, where θ is one of the acute angles:
sin θ = opposite/hypotenuse | cos θ = adjacent/hypotenuse | tan θ = opposite/adjacent
EX: θ=30°, hypotenuse=10 → opposite = 10×sin(30°) = 10×0.5 = 5, adjacent = 10×cos(30°) = 10×0.866 = 8.66
Finding angles from sides — use inverse trigonometric functions:
EX: legs 5 and 12 → tan(θ) = 5/12 → θ = arctan(5/12) ≈ 22.6° | other angle = 90°−22.6° = 67.4°
Two special right triangles are worth memorizing for their exact ratios. The 45-45-90 triangle has sides in ratio 1:1:√2 — both legs equal, hypotenuse is leg × √2. The 30-60-90 triangle has sides in ratio 1:√3:2 — the leg opposite 30° is half the hypotenuse, and the leg opposite 60° is half the hypotenuse times √3. These ratios appear constantly in geometry, physics, and engineering problems that involve symmetric shapes or 60° angles. The Pythagorean theorem has over 370 documented proofs, more than any other theorem in mathematics — from Euclid's geometric proof to algebraic and trigonometric proofs to President James Garfield's original trapezoid proof. In practical applications, the 3-4-5 right triangle (and its multiples: 6-8-10, 5-12-13, 8-15-17) appears in construction for squaring corners — if the three sides of a triangle measure in these exact ratios, the angle between the two shorter sides is exactly 90°.

Frequently Asked Questions

The Pythagorean theorem states a² + b² = c² for any right triangle, where c is the hypotenuse (side opposite the right angle) and a, b are the legs. To find hypotenuse: c = √(a²+b²). Example: legs 3 and 4 → c = √(9+16) = √25 = 5. To find a missing leg: a = √(c²−b²). Example: hypotenuse 13, one leg 5 → other leg = √(169−25) = √144 = 12. Always identify the hypotenuse first — it is the longest side, always opposite the 90° angle.

Pythagorean triples are sets of integers (a, b, c) satisfying a²+b²=c². Primitive triples (no common factor): 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41. Multiples of primitive triples also work: 6-8-10, 9-12-15, 12-16-20 (all multiples of 3-4-5). Generation formula: for positive integers m>n, a=m²−n², b=2mn, c=m²+n². With m=2, n=1: a=3, b=4, c=5. These appear in ancient Egyptian construction (knotted ropes at 3-4-5 to create right angles) and throughout Diophantine mathematics.

For a right triangle with legs a, b and hypotenuse c: sin(A) = a/c, cos(A) = b/c, tan(A) = a/b, where A is the angle opposite leg a. Example: legs 3 and 4, hypotenuse 5. Angle A opposite leg 3: sin(A) = 3/5 = 0.6 → A = arcsin(0.6) = 36.87°. Angle B opposite leg 4: sin(B) = 4/5 = 0.8 → B = arcsin(0.8) = 53.13°. Verify: 36.87 + 53.13 = 90° ✓ (angles in a triangle sum to 180°, and one angle is already 90°).

Special right triangles have fixed angle ratios and known side relationships. 45-45-90 triangle: legs are equal, hypotenuse = leg × √2. If leg = 1, hypotenuse = √2 ≈ 1.414. 30-60-90 triangle: short leg (opposite 30°) = 1, long leg (opposite 60°) = √3 ≈ 1.732, hypotenuse = 2. These ratios are exact and worth memorizing. A square cut along the diagonal creates two 45-45-90 triangles; equilateral triangles bisected create two 30-60-90 triangles.

Right triangles appear in surveying (triangulation), navigation (calculating distances and bearings), architecture (ensuring structural perpendicularity), and physics (vector decomposition). The angle of elevation from a point 50 m from a building to its top is 60°. Height = 50 × tan(60°) = 50 × √3 ≈ 86.6 m. GPS triangulation uses distance measurements from multiple satellites to pinpoint location — each distance defines a sphere, and three spheres intersect at (approximately) one point, with a fourth satellite resolving ambiguity.

The area of a right triangle = (1/2) × leg₁ × leg₂. For legs 3 and 4: area = (1/2)(3)(4) = 6. The altitude h from the right angle to the hypotenuse satisfies: h = (leg₁ × leg₂)/hypotenuse = 12/5 = 2.4. Each leg² = hypotenuse × (projection of that leg on hypotenuse). Leg₁² = hypotenuse × projection₁: 9 = 5 × 1.8. These geometric mean relationships (called the right triangle altitude theorem) connect all elements of a right triangle and are used in geometric construction proofs.