Right Triangle Trigonometry Calculator

Enter one acute angle and one known side of a right triangle to solve for both legs, the hypotenuse, and the triangle's area using sin, cos, and tan.

Quick Facts

SOH-CAH-TOA
sin A = opp/hyp, cos A = adj/hyp, tan A = opp/adj
The three basic trig ratios for a right triangle's acute angles.
Pythagorean theorem
a² + b² = c²
Relates the two legs (a, b) to the hypotenuse (c) — use it to check your answer.
Angle sum
A + B = 90°
The two acute angles in a right triangle are always complementary.

Your Results

Calculated
Side Opposite A (a)
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a = c · sin A
Side Adjacent to A (b)
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b = c · cos A
Hypotenuse (c)
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c² = a² + b²
Triangle Area
-
Area = ½ · a · b

Ready

Enter angle A and one known side, then press Calculate.

Formula and Method for Right Triangle Trigonometry

A right triangle has one 90° angle and two acute angles, A and B, that add up to 90° (A + B = 90°). The side opposite the right angle is the hypotenuse (c) — the longest side — and the other two sides are called legs. Relative to angle A, the leg across from it is the "opposite" side (a) and the leg next to it is the "adjacent" side (b). This calculator uses the three basic trigonometric ratios — sine, cosine, and tangent — together with the Pythagorean theorem to solve for every side and angle from just one known angle and one known side.

How the calculation works

The three ratios, remembered with the mnemonic SOH-CAH-TOA, are sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, and tan A = opposite/adjacent. Given angle A and one side, the calculator rearranges whichever ratio applies: if you know the opposite side a, the hypotenuse is c = a / sin A and the adjacent side is b = a / tan A; if you know the adjacent side b, the hypotenuse is c = b / cos A and the opposite side is a = b × tan A; if you know the hypotenuse c, then a = c × sin A and b = c × cos A. Once all three sides are known, the triangle's area follows from Area = ½ × a × b (the two legs act as base and height because they meet at the 90° angle), and the second acute angle is simply B = 90° − A.

Common mistakes

  • Mislabeling opposite vs. adjacent: "opposite" and "adjacent" are always relative to the angle you're using — the same leg is "opposite" to one acute angle and "adjacent" to the other.
  • Degrees vs. radians: make sure your angle is entered in degrees (this calculator expects degrees, not radians) — a formula that works for 30° gives a very different answer if 30 is mistakenly treated as radians.
  • Confusing the hypotenuse with a leg: the hypotenuse is always the longest side and is always opposite the 90° angle, never adjacent to it.

Real-world applications

  • Construction and roofing use right-triangle trig to find rafter lengths, roof pitch, and stair rise/run from an angle and a known length
  • Surveying and navigation use angle-of-elevation and angle-of-depression problems to find distances and heights that can't be measured directly
  • Physics uses these ratios to resolve forces or velocities into perpendicular components
  • Ladder-safety and ramp-design calculations use the angle and one side to confirm a safe reach or slope

Frequently Asked Questions

What is SOH-CAH-TOA?
SOH-CAH-TOA is a memory aid for the three basic right-triangle trig ratios: sin A = opposite/hypotenuse (SOH), cos A = adjacent/hypotenuse (CAH), and tan A = opposite/adjacent (TOA), where A is one of the two acute angles.
How do I find the missing sides of a right triangle from one angle and one side?
If you know the side opposite angle A, the hypotenuse is c = a / sin A and the adjacent side is b = a / tan A. If you know the adjacent side, the hypotenuse is c = b / cos A and the opposite side is a = b × tan A. If you know the hypotenuse, the opposite side is a = c × sin A and the adjacent side is b = c × cos A.
How are the two acute angles in a right triangle related?
They are complementary: A + B = 90°. Once you know one acute angle, the other is simply 90° minus that angle.
How can I check my right-triangle answer?
Use the Pythagorean theorem: a² + b² should equal c² (within rounding). If the two sides don't square-sum to the hypotenuse squared, recheck which side you labeled as opposite, adjacent, or hypotenuse.

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