Right Triangle Side and Angle Calculator
Solve right triangle side angle problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
opp = hyp * sin(A), adj = hyp * cos(A), opp = adj * tan(A)
Given one acute angle A and one side, use trigonometric ratios to find the other sides. Sine relates the opposite side to the hypotenuse, cosine relates the adjacent side to the hypotenuse, and tangent relates the opposite side to the adjacent side.
Worked Examples
Example 1: Finding a Building Height
Problem:Standing 50 meters from a building, you measure an angle of elevation of 32 degrees. Find the building height.
Solution:The adjacent side is 50m, the angle is 32 degrees.\nOpposite (height) = adjacent * tan(angle) = 50 * tan(32) = 50 * 0.6249 = 31.24 meters\nHypotenuse (line of sight) = adjacent / cos(angle) = 50 / cos(32) = 50 / 0.8480 = 58.95 meters\nOther angle = 90 - 32 = 58 degrees
Result:Building height = 31.24 meters. Line of sight distance = 58.95 meters.
Example 2: Ladder Angle Calculation
Problem:A 20-foot ladder must reach a window 15 feet high. What angle should it make with the ground?
Solution:Hypotenuse = 20 ft (ladder), Opposite = 15 ft (height)\nAngle = arcsin(15/20) = arcsin(0.75) = 48.59 degrees\nBase distance = 20 * cos(48.59) = 20 * 0.6614 = 13.23 feet\nThis is a safe angle (OSHA recommends 75.5 degrees for extension ladders)\nArea of triangle = (1/2)(13.23)(15) = 99.21 sq ft
Result:Angle with ground = 48.59 degrees. Base = 13.23 ft from wall. Not steep enough for OSHA compliance.
Frequently Asked Questions
How do you find the sides of a right triangle from one side and one angle?
To find all sides of a right triangle when you know one side and one acute angle, use the trigonometric ratios. If you know the angle A and the opposite side, the adjacent side = opposite / tan(A) and the hypotenuse = opposite / sin(A). If you know the adjacent side, the opposite = adjacent times tan(A) and hypotenuse = adjacent / cos(A). If you know the hypotenuse, the opposite = hypotenuse times sin(A) and adjacent = hypotenuse times cos(A). The key principle is that trigonometric functions relate angles to side ratios in a fixed way, so knowing any angle-side pair gives you all other sides. The other acute angle is simply 90 minus the known angle.
What do opposite and adjacent mean in a right triangle?
The terms opposite and adjacent describe the relationship between a specific angle and the sides of the right triangle. The opposite side is directly across from the angle being considered, while the adjacent side is next to the angle (but is not the hypotenuse). These designations change depending on which angle you are referencing. For example, in a 3-4-5 triangle, if you are looking at the angle opposite the side of length 3, then 3 is the opposite side and 4 is the adjacent side. If you switch to the other acute angle, then 4 becomes the opposite and 3 becomes the adjacent. The hypotenuse is always the side opposite the 90-degree angle and never changes based on which acute angle is being considered.
How accurate is solving a triangle with angle and side measurements?
The accuracy of solving a right triangle depends on the precision of your input measurements and the behavior of the trigonometric functions involved. Small angles amplify errors in certain calculations: when the angle is very small, the opposite side is small relative to the hypotenuse, so small measurement errors in the angle cause large percentage errors in the calculated opposite side. Similarly, for angles close to 90 degrees, the adjacent side is very small and its calculation becomes sensitive to angle errors. The most stable calculations occur with angles between 30 and 60 degrees. In practical applications, measurement precision typically limits accuracy more than computational precision. Modern calculators provide about 15 significant digits, but a protractor typically measures angles to plus or minus 0.5 degrees.
How do you find an angle if you know two sides?
If you know any two sides of a right triangle, use inverse trigonometric functions to find the angles. Given the opposite and hypotenuse: angle = arcsin(opposite/hypotenuse). Given the adjacent and hypotenuse: angle = arccos(adjacent/hypotenuse). Given the opposite and adjacent: angle = arctan(opposite/adjacent). The arctan function is most commonly used because it does not require knowing the hypotenuse. For example, with legs of 5 and 8: angle = arctan(5/8) = 32.01 degrees. These inverse functions are written as sin^(-1), cos^(-1), tan^(-1) or asin, acos, atan on calculators. In programming, the atan2 function is preferred because it correctly handles all quadrants and avoids division by zero issues.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy