Quiz: Right Triangle Side and Angle Calculator

Enter one acute angle and one known side of a right triangle to solve for the two remaining sides and the other acute angle using sine, cosine, and tangent.

Quick Facts

Angle sum
A + B = 90°
The right angle is always 90°, so the two acute angles are complementary.
SOH-CAH-TOA
sin A = a/c, cos A = b/c, tan A = a/b
Opposite, adjacent, and hypotenuse relative to angle A.
Pythagorean theorem
a² + b² = c²
Use it to double-check the three computed side lengths.

Your Results

Calculated
Side a (opposite A)
-
a = c × sin A
Side b (adjacent to A)
-
b = c × cos A
Side c (hypotenuse)
-
c² = a² + b²
Angle B
-
B = 90° − A

Ready

Enter angle A, one side, and which side it is, then press Calculate.

Formula and Method for Solving a Right Triangle by Side and Angle

A right triangle has one 90° angle. Once you know one of the other two acute angles (call it A) and any one of the three side lengths, the triangle is fully determined — the remaining two sides and the remaining angle follow directly from the trigonometric ratios sine, cosine, and tangent (remembered by the mnemonic SOH-CAH-TOA) and from the Pythagorean theorem.

How the calculation works

Label the side opposite angle A as a, the side adjacent to angle A (that is not the hypotenuse) as b, and the hypotenuse (opposite the right angle, always the longest side) as c. Depending on which side you enter, the calculator solves for the other two using: sin A = a/c, cos A = b/c, and tan A = a/b. If side a is known, c = a / sin A and b = a / tan A. If side b is known, a = b × tan A and c = b / cos A. If the hypotenuse c is known, a = c × sin A and b = c × cos A. The remaining acute angle is always B = 90° − A, since the three interior angles of any triangle sum to 180° and one of them is already 90°. As a check, the three resulting sides always satisfy the Pythagorean theorem: a² + b² = c².

Common mistakes

  • Mixing up opposite and adjacent: "opposite" and "adjacent" are always relative to the angle you are using. Swapping them flips which trig ratio applies and gives a wrong side.
  • Degrees vs. radians: this calculator expects angle A in degrees. Entering a radian value (like 0.52 instead of 30) will produce a nonsensical result.
  • Forgetting the hypotenuse is longest: the hypotenuse is always the largest side of a right triangle. If a computed leg comes out longer than the hypotenuse, double-check which side was entered.

Real-world applications

  • Surveying and construction use angle-and-side triangle solving to find distances or heights that can't be measured directly, such as a building's height from a known distance and elevation angle.
  • Navigation and aviation use right-triangle trigonometry to compute bearings, altitude, and ground distance from angle-of-climb or angle-of-descent readings.
  • Carpentry and roofing use it to cut rafters and braces at the correct angle given a run (horizontal leg) and a pitch angle.
  • Physics and engineering resolve forces or vectors into perpendicular components using the same sine/cosine relationships.

Frequently Asked Questions

How do I solve a right triangle with one angle and one side?
Use the trigonometric ratios sine, cosine, and tangent (SOH-CAH-TOA). If you know acute angle A and the side opposite it (a), the hypotenuse is c = a / sin(A) and the adjacent side is b = a / tan(A). If you know the adjacent side (b), then a = b × tan(A) and c = b / cos(A). If you know the hypotenuse (c), then a = c × sin(A) and b = c × cos(A). The remaining acute angle is B = 90° − A.
What is the relationship between the two acute angles in a right triangle?
The three angles of any triangle sum to 180°. Since one angle in a right triangle is already 90°, the other two acute angles must add up to 90° — they are complementary. So angle B = 90° − angle A.
How can I check the answer with the Pythagorean theorem?
Once you have all three sides, verify that a² + b² = c², where c is the hypotenuse (the side opposite the 90° angle, and always the longest side). If the two sides don't match the squared hypotenuse within rounding error, recheck which side or angle was entered.

Related calculators