Right Triangle Calculator
Solve any right triangle from two known values using the Pythagorean theorem and trigonometric ratios. Shows all sides, angles, area, perimeter, inradius, and circumradius.
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How It Works
Given legs a and b: hypotenuse c = √(a²+b²). Angles: A = arctan(a/b), B = 90°-A.
c=√(a²+b²), A=arctan(a/b)a=3, b=4 → c=5, A=36.87°, B=53.13°Given leg a and hyp c: other leg b = √(c²-a²). Angle A = arcsin(a/c).
b=√(c²-a²), A=arcsin(a/c)a=5, c=13 → b=12, A=22.62°sin(A)=opposite/hyp, cos(A)=adjacent/hyp, tan(A)=opposite/adjacent. Use these to find sides from angles or angles from sides.
sin/cos/tan and their inversessin(30°)=0.5 → if hyp=10, opposite=545-45-90: if leg=1, hyp=√2≈1.414. Both legs equal. 30-60-90: if short leg=1, long leg=√3≈1.732, hyp=2.
45-45-90: 1:1:√2; 30-60-90: 1:√3:2leg=5 in 45-45-90 → hyp=5√2≈7.07Area = (1/2) × a × b (product of legs, divided by 2). Perimeter = a + b + c. Circumradius = c/2 for any right triangle.
Area=(a×b)/2, R=c/2a=3,b=4,c=5: Area=6, R=2.5Right triangles appear in: ramp slopes, ladder placement, staircase design, navigation angles, and surveying. The Pythagorean theorem is the foundation of all these.
Slope = rise/run = tan(angle)Ramp: rise=3, run=4 → length=5, angle=36.87°Quick Reference
Common examples — verify instantly above.
Two legs
a=3, b=4
c=5, A=36.87°
Two legs
a=5, b=12
c=13, A=22.62°
Leg+hyp
a=5, c=13
b=12
Leg+hyp
a=8, c=17
b=15
45-45-90
leg=5
hyp=7.071
30-60-90
short leg=6
hyp=12
Area
a=3, b=4
Area=6
Circumradius
a=3,b=4,c=5
R=2.5
Tips & Shortcuts
SOHCAHTOA: Sin=Opposite/Hypotenuse, Cos=Adjacent/Hypotenuse, Tan=Opposite/Adjacent.
For a 45-45-90 triangle: both legs are equal and hypotenuse = leg × √2 ≈ leg × 1.414.
For a 30-60-90 triangle: sides are in ratio 1:√3:2. If the hypotenuse is 2, legs are 1 and √3.
The circumradius of a right triangle always equals hypotenuse/2 — the hypotenuse is the diameter of the circumscribed circle.
The inradius of a right triangle: r = (a + b - c)/2. For a 3-4-5 triangle: r = (3+4-5)/2 = 1.
Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (20,21,29).
Common Mistakes
Applying Pythagorean theorem to non-right triangles
a²+b²=c² only works for right triangles. For other triangles, use the Law of Cosines.
Confusing adjacent and opposite sides for trig ratios
Adjacent is next to the angle; opposite is across from it. The hypotenuse is always across from the 90° angle.
Forgetting that the hypotenuse is always the longest side
If your calculated hypotenuse is shorter than a leg, check your input values.
Using degrees in radians mode or vice versa
Make sure your calculator is in degree mode when angles are in degrees. Radian results will be very different.
Rounding angles before computing sides
Round only at the final step. Intermediate angle rounding compounds into significant side length errors.
Forgetting that the two non-right angles sum to 90°
In a right triangle, A+B=90°. If you find A, then B=90°-A. No need to calculate B separately.
Frequently Asked Questions
A right triangle has one angle exactly equal to 90°. The side opposite the right angle is the hypotenuse — always the longest side.
sin(A) = opposite/hypotenuse, cos(A) = adjacent/hypotenuse, tan(A) = opposite/adjacent. These ratios relate angles to side lengths.
The hypotenuse is the side opposite the 90° angle. It is always the longest side. c = √(a²+b²).
Special right triangles. 45-45-90: legs ratio 1:1:√2. 30-60-90: sides ratio 1:√3:2. Memorizing these speeds up calculations.
Use inverse trig: A = arctan(opposite/adjacent), A = arcsin(opposite/hypotenuse), or A = arccos(adjacent/hypotenuse).
For a right triangle, the circumradius R = hypotenuse/2. The hypotenuse is the diameter of the circumscribed circle.
Use inverse trig functions. Opposite and hypotenuse known: angle = sin⁻¹(opposite/hypotenuse). Opposite and adjacent known: angle = tan⁻¹(opposite/adjacent). Adjacent and hypotenuse known: angle = cos⁻¹(adjacent/hypotenuse). Example: opposite=3, hypotenuse=5. Angle = sin⁻¹(0.6) = 36.87°. The other acute angle = 90°−36.87° = 53.13°. Both angles must sum to 90°.
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