Right Triangle Calculator

Foundations of Right-Angled Triangle Geometry & Trigonometry

A Right Triangle is a three-sided geometric polygon containing exactly one 90-degree right angle. The two sides forming the perpendicular right angle are called legs (a and b), while the longest side directly opposite the right angle is the hypotenuse (c).

Fundamental Geometric & Trigonometric Equations

• Pythagorean Theorem: a^2 + b^2 = c^2 c = √(a^2 + b^2), a = √(c^2 - b^2)

• Basic Trigonometric Definitions (SOH-CAH-TOA):

α = (a) / (c), α = (b) / (c), α = (a) / (b)

• Complementary Acute Angles: α + β = 90^ β = 90^ - α

• Enclosed Triangle Area: Area = (a × b) / (2)

• Boundary Perimeter: P = a + b + c

• Altitude to Hypotenuse: h_c = (a × b) / (c)

• Incircle Radius (r) & Circumcircle Radius (R): r = (a + b - c) / (2), R = (c) / (2)

Famous Pythagorean Triples

• 3 : 4 : 5 Ratio: The classic integer triple and its multiples (6-8-10, 9-12-15, etc.).

• 5 : 12 : 13 Ratio:.

• 8 : 15 : 17 Ratio:.

How to Use This Calculator

Enter your Given Known Dimensions — choose from Two Legs (Side a & Side b), Leg (Side a) & Hypotenuse (Side c), Leg (Side a) & Acute Angle α, First Value (Leg a or Hypotenuse c) (e.g. 6), Second Value (Leg b, Hypotenuse c, or Angle α °) (e.g. 8), and Measurement Unit — choose from Centimeters (cm), Meters (m), Inches (in). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Right Triangle Calculator

Input variables:

solveModeSelect — Given Known Dimensions

firstDimensionValue — First Value (Leg a or Hypotenuse c)

secondDimensionValue — Second Value (Leg b, Hypotenuse c, or Angle α °)

unitSelect — Measurement Unit

Computed outputs:

hypotenuseSideC — Hypotenuse (Side c)

sideAOutput — Leg (Side a)

sideBOutput — Leg (Side b)

angleAlphaDegrees — Angle α (Opposite a)

angleBetaDegrees — Angle β (Opposite b)

Mathematical relationships extracted from calculation logic:

radToDeg = (rad) => rad * (180 / Math.PI)

degToRad = (deg) => deg * (Math.PI / 180)

c = Math.sqrt((a a) + (b b))

b = Math.sqrt(Math.max(0, (c c) - (a a)))

area = (a * b) / 2

perimeter = a + b + c

altitude = safeDiv(a * b, c)

Standard: NIST / ISO mathematical definitions

Worked Example

A roof rafter rising 6 feet vertically across an 8-foot horizontal run requires a 10-foot rafter beam with a pitch angle of 36.87°.

Understanding Your Result

Geometric calculations assume ideal shapes — real-world measurements should include a 5–15% tolerance for irregular shapes, cutting waste, and installation overlap. For construction applications: always verify calculated areas match your physical measurements before ordering materials. Standard practice is to measure the same dimension twice from different reference points and use the average. Express results in the units required by your supplier or building permit — conversions between cubic yards, cubic feet, and cubic meters must be precise for material orders.