Right Triangle Side and Angle Calculator
Enter any two values of a right triangle and find all remaining sides, angles, area, perimeter, and all six trigonometric ratios instantly.
📐 What is a Right Triangle Side and Angle Calculator?
A right triangle is a triangle with one angle equal to exactly 90 degrees. The side opposite the right angle is called the hypotenuse, always the longest side. The other two sides are called legs, and their two angles (called acute angles) each measure between 0 and 90 degrees and always sum to exactly 90 degrees together.
This calculator solves a right triangle completely from any two known values. In practice, you always know two things and need to find the rest. A contractor measuring a roof pitch knows the run and rise (two legs). A student given a trig problem might know one angle and the hypotenuse. A surveyor might know one angle and the distance across a river (one leg). All four scenarios are handled here with four dedicated input modes.
Beyond finding the three sides and two acute angles, the calculator also computes area (one half times leg a times leg b), perimeter (sum of all three sides), the altitude from the right angle to the hypotenuse (a measure used in geometric proofs and similarity theorems), and all six trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) for both acute angles. These ratios are the foundation of trigonometry and appear in physics, engineering, navigation, and architecture.
The core relationships are the Pythagorean theorem (c squared = a squared + b squared) and the SOH-CAH-TOA ratios: Sin = Opposite over Hypotenuse, Cos = Adjacent over Hypotenuse, Tan = Opposite over Adjacent. By applying inverse trig functions (arcsin, arccos, arctan), any unknown angle can be recovered from two known sides. Together these six equations can find any unknown in the triangle as long as two independent knowns are provided.
📐 Formula
c = √(a² + b²) | A = arctan(a ÷ b)
a, b = legs (shorter sides). c = hypotenuse (side opposite 90°)
A = angle opposite leg a. B = angle opposite leg b = 90° − A
sin A = a/c cos A = b/c tan A = a/b
csc A = c/a sec A = c/b cot A = b/a
Area = ½ × a × b Perimeter = a + b + c
Altitude to hypotenuse = (a × b) ÷ c
Example (3-4-5): a=3, b=4 → c=5, A=36.87°, B=53.13°, Area=6, Perimeter=12, Altitude=2.4
📖 How to Use This Calculator
Steps
1
Choose your input mode — Select which two values you know: Two Legs (a and b), Leg plus Hypotenuse, Angle plus Hypotenuse, or Angle plus One Leg. The relevant input fields appear immediately.
2
Enter your known values — Type your two known values. Lengths can be in any consistent unit. Angles must be in degrees between 0 and 90 (exclusive). The calculator accepts decimals for precision.
3
Click Calculate — Press Calculate to instantly see all three sides, both acute angles, area, perimeter, altitude to hypotenuse, and all nine listed trig ratios.
4
Read all results — The top result boxes show the primary unknowns. The trig ratio grid below shows sin, cos, tan, csc, sec, and cot for angle A, plus sin, cos, and tan for angle B (which is 90° minus A).
💡 Example Calculations
Example 1 — Classic 3-4-5 Right Triangle (Two Legs)
The most famous Pythagorean triple
1
Legs: a = 3, b = 4. Hypotenuse: c = √(3² + 4²) = √(9 + 16) = √25 = 5.
2
Angle A = arctan(3/4) = arctan(0.75) = 36.87°. Angle B = 90 − 36.87 = 53.13°.
3
Area = ½ × 3 × 4 = 6 sq units. Perimeter = 3 + 4 + 5 = 12 units. Altitude to hyp = (3 × 4)/5 = 2.4 units.
4
Trig ratios: sin A = 3/5 = 0.6, cos A = 4/5 = 0.8, tan A = 3/4 = 0.75, csc A = 5/3 = 1.6667, sec A = 5/4 = 1.25, cot A = 4/3 = 1.3333.
c = 5, A = 36.87°, B = 53.13°, Area = 6 sq units
Try this example →Example 2 — 30-60-90 Triangle (Angle + Hypotenuse)
Special triangle with exact trig values
1
Angle A = 30°, Hypotenuse c = 10. Opposite leg a = 10 × sin(30°) = 10 × 0.5 = 5.
2
Adjacent leg b = 10 × cos(30°) = 10 × 0.8660 = 8.660. Angle B = 90 − 30 = 60°.
3
Area = ½ × 5 × 8.660 = 21.65 sq units. Perimeter = 5 + 8.660 + 10 = 23.660 units.
a = 5, b = 8.660, A = 30°, B = 60°
Try this example →Example 3 — Roof Pitch Problem (Angle + Opposite Leg)
Roof rafter length from pitch angle and rise
1
A roof rises at 22° pitch. The vertical rise (opposite leg a) = 8 feet. Find the rafter length (hypotenuse) and horizontal run (adjacent leg).
2
Hypotenuse c = a / sin(A) = 8 / sin(22°) = 8 / 0.3746 = 21.36 feet.
3
Adjacent leg b = a / tan(A) = 8 / tan(22°) = 8 / 0.4040 = 19.80 feet. Angle B = 90 − 22 = 68°.
Rafter = 21.36 ft, Run = 19.80 ft, Angle B = 68°
Try this example →❓ Frequently Asked Questions
How do you find all sides of a right triangle?
You need exactly two known values. With two legs (a, b): hypotenuse c = sqrt(a squared + b squared). With one leg and the hypotenuse: missing leg = sqrt(c squared minus known leg squared). With an angle and any side: use sin, cos, or tan to find the others. This calculator handles all four cases and shows all three sides, both angles, area, and all trig ratios.
How do you find angles of a right triangle from two sides?
Use inverse trig functions. Given legs a and b: angle A = arctan(a/b). Given one leg a and hypotenuse c: angle A = arcsin(a/c). Given adjacent leg b and hypotenuse c: angle A = arccos(b/c). The right angle is always exactly 90 degrees, and the remaining acute angle B = 90 degrees minus A. Most calculators and phones have arctan, arcsin, and arccos as the inv or second function of tan, sin, and cos.
What is the Pythagorean theorem?
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c squared = a squared + b squared. For the classic 3-4-5 triangle: 5 squared = 3 squared + 4 squared, or 25 = 9 + 16. This theorem only works for right triangles. For any two sides, c = sqrt(a squared + b squared) gives the hypotenuse, and a = sqrt(c squared minus b squared) gives a missing leg.
What does SOH-CAH-TOA mean?
SOH-CAH-TOA is a mnemonic for the three primary trig ratios in a right triangle relative to angle A. SOH: Sine = Opposite leg divided by Hypotenuse. CAH: Cosine = Adjacent leg divided by Hypotenuse. TOA: Tangent = Opposite leg divided by Adjacent leg. To find a side when an angle is known: opposite = hyp times sin(A), adjacent = hyp times cos(A), opposite = adjacent times tan(A). To find an angle from two sides: use inverse sin, cos, or tan.
What are all six trigonometric ratios of a right triangle?
For angle A in a right triangle (opposite = a, adjacent = b, hypotenuse = c): sin A = a/c, cos A = b/c, tan A = a/b. The reciprocal functions: csc A = c/a (1/sin A), sec A = c/b (1/cos A), cot A = b/a (1/tan A). These six ratios are used throughout trigonometry, calculus, and physics. This calculator computes all six for both acute angles after solving the triangle.
What are the special right triangles?
The 45-45-90 triangle has equal legs and a hypotenuse equal to the leg times sqrt(2) = 1.4142. If a leg is 1, hypotenuse = 1.4142. The 30-60-90 triangle has sides in ratio 1 : sqrt(3) : 2. If the shortest leg is 1, the other leg is 1.732 and the hypotenuse is 2. These triangles produce exact trig values: sin 30 = 0.5, sin 45 = sqrt(2)/2 = 0.7071, cos 60 = 0.5, tan 45 = 1.
How is the altitude to the hypotenuse calculated?
The altitude h drawn from the right angle perpendicular to the hypotenuse equals (leg a times leg b) divided by the hypotenuse c: h = (a times b) / c. For the 3-4-5 triangle: h = (3 times 4) / 5 = 2.4. This altitude is also the geometric mean: h squared equals the product of the two segments it creates on the hypotenuse. It is used in proofs of the Pythagorean theorem and in geometric mean altitude theorems.
What is the area formula for a right triangle?
Area = 0.5 times leg a times leg b. The two legs of a right triangle are perpendicular, so one serves as the base and the other as the height of the standard triangle area formula (Area = 0.5 times base times height). For the 5-12-13 right triangle: Area = 0.5 times 5 times 12 = 30 square units. For non-right triangles, the formula changes because height and base are not the same as the sides.
Can you solve a right triangle with only one known value?
No. With only one side and the knowledge that one angle is 90 degrees, the triangle is not uniquely determined. There are infinitely many right triangles with a given hypotenuse (all sizes of similar triangles). You need exactly two pieces of independent information to fully determine all sides and angles. The one exception: if you know two angles (one is 90 degrees, another is given), you know the shape but not the scale.
What is the difference between the opposite and adjacent legs?
Opposite and adjacent are defined relative to a specific angle. For angle A: the opposite leg is the side directly across the triangle from A (not touching angle A), and the adjacent leg is the side that forms angle A along with the hypotenuse. The hypotenuse is always opposite the right angle. If you switch focus to angle B, opposite and adjacent swap: what was opposite A is now adjacent to B, and vice versa.
How do right triangles appear in everyday life?
Right triangles appear everywhere. Builders use the 3-4-5 rule to check square corners. Ramps, stairs, and roof pitches form right triangles. Ladders leaning against a wall create right triangles. GPS and navigation systems use trigonometric triangulation. Surveyors measure inaccessible distances using right triangle relationships. Carpenters use the diagonal of a rectangle (the hypotenuse of a right triangle) to check for squareness. Electrical conduit runs at angles through walls form right triangles with the wall and floor.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a squared + b squared = c squared, meaning they form the sides of a right triangle with no irrational numbers. Common examples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples also work: 6-8-10 and 9-12-15 are both scaled versions of 3-4-5. Pythagorean triples are used in construction and carpentry because measuring in whole-number lengths avoids rounding errors when checking for square corners.