Right Triangle Trigonometry Calculator
Solve a right triangle from any two independent measurements, then export the current sides, angles, area, and calculation notes to a real Excel workbook.
Known measurements
Edit any two independent fields. The two most recently edited measurements become the known pair.
Leg opposite angle α. Enter a positive decimal.
Leg adjacent to angle α. Enter a positive decimal.
Hypotenuse, opposite the 90° angle. Must be longest.
Acute angle opposite side a. Use a value above 0 and below 90.
Acute angle opposite side b. α + β always equals 90°.
Half the product of the two legs. Enter a positive decimal.
Accepted format: positive plain decimals such as 5, 3.25, or .75. Commas, unit symbols, and scientific notation are rejected to avoid ambiguous interpretation.
Solved triangle
All outputs use the same canonical model as the table and Excel export.
Hypotenuse c
5.0000 units
Side a
3.0783 units
Side b
3.9401 units
Angle α
38.00°
Angle β
52.00°
Area
6.0643 units²
Perimeter
12.0184 units
Calculation detail
| Quantity | Symbol | Value | Relationship |
|---|---|---|---|
| Opposite leg | a | 3.0783 units | a = c × sin(α) |
| Adjacent leg | b | 3.9401 units | b = c × cos(α) |
| Hypotenuse | c | 5.0000 units | c² = a² + b² |
| Acute angle | α | 38.00° | α = atan2(a, b) |
| Complementary angle | β | 52.00° | β = 90° – α |
| Area | A | 6.0643 units² | A = a × b ÷ 2 |
| Perimeter | P | 12.0184 units | P = a + b + c |
Values are shown to four decimal places for lengths and area and two decimal places for angles. The Excel workbook stores the unrounded canonical numbers.
How to use the right triangle trigonometry calculator
What this calculator does
This calculator solves a right triangle when you know any two independent measurements. It returns the two legs, the hypotenuse, both acute angles, the area, and the perimeter. The model assumes one angle is exactly 90 degrees, side a is opposite angle α, side b is adjacent to angle α, and side c is the hypotenuse. It does not solve general oblique triangles or account for measurement uncertainty, rounding error in field instruments, or three-dimensional geometry. For the underlying ratios and inverse functions, review the OpenStax right-triangle trigonometry lesson.
When to use it
Use the calculator to check homework, lay out a diagonal brace, estimate an inaccessible height from a measured distance and angle, or verify dimensions in a drawing. It is also useful when a project gives area plus one side, or when you need to convert a side ratio into an acute angle.
How to calculate
- The calculator opens with a complete demonstration: Side length c = 5 and Angle α = 38°. The other fields are already solved, and Download Excel is immediately available.
- Edit any field. The two most recently edited independent measurements receive a Known badge and drive the next solution. For example, enter Side length a and Side length b to solve by the Pythagorean theorem, or enter Area and Angle α to solve the triangle's scale and shape together.
- Read the primary Hypotenuse c result, then compare Side a, Side b, Angle α, Angle β, Area, and Perimeter. The Calculation detail table shows each symbol, displayed value, and the relationship used.
- Select Download Excel to export the current validated model. Reset clears the demonstration and all solved values; Download Excel then remains disabled until two complete, valid, independent measurements are entered again.
Input guide
Side length a, Side length b, and Side length c are required only when they belong to the active known pair. Enter positive plain decimals in one consistent length unit, such as 3, 4.5, or .75. Side c is the hypotenuse and must exceed either leg when paired with that leg. Increasing a or b generally increases c, area, and perimeter. A common mistake is mixing centimeters and inches or entering a hypotenuse shorter than a leg.
Angle α and Angle β are acute angles in degrees. Each must be greater than 0 and less than 90. Angle α is opposite side a; Angle β is opposite side b; together they equal 90°. Entering only α and β cannot determine the triangle's size, so that pair is rejected. A common mistake is entering radians or pairing two complementary angles without a side or area.
Area is measured in square units and must be positive. It equals one-half of Side a multiplied by Side b. Area can be paired with a leg, an acute angle, or the hypotenuse. For Area plus Side c, two mirror-image labelings are possible; this calculator uses the deterministic convention a ≤ b, so α ≤ 45°. Area must not exceed c² ÷ 4 for a right triangle with that hypotenuse.
Output guide
Known pair identifies the two measurements currently used. Status reports Solved, Incomplete, or Needs correction. Angle check is the exact identity α + β = 90°. Hypotenuse c is the longest side and the primary result. Side a and Side b are the legs; Angle α and Angle β are complementary estimates derived from the chosen pair. Area is a square-unit result, while Perimeter is the total boundary length. A zero or negative result is never valid; extremely large finite inputs may be rejected if the derived calculation would overflow.
The Calculation detail table has four columns: Quantity names the result, Symbol gives its standard shorthand, Value shows the current formatted result, and Relationship states the identity used to interpret that row. These are exact mathematical identities applied to the entered measurements, but the displayed decimals are rounded estimates unless the result is exact.
Worked example
With c = 5 units and α = 38°, the opposite leg is a = 5 × sin(38°) = 3.0783 units, while the adjacent leg is b = 5 × cos(38°) = 3.9401 units. The second acute angle is β = 90° – 38° = 52.00°. Area is 3.0783 × 3.9401 ÷ 2 = 6.0643 square units, and perimeter is 3.0783 + 3.9401 + 5 = 12.0184 units. These values match the first-open controls, results, table, and workbook checkpoints.
Learn more
The MathWorld right-triangle reference summarizes the Pythagorean and area identities. When attaching real units, remember that area uses squared units; the NIST guide to SI area explains that distinction.
How the model chooses a formula
Two sides use the Pythagorean theorem and inverse tangent. A side plus an acute angle uses sine, cosine, or tangent according to whether the side is opposite, adjacent, or the hypotenuse. Area plus one leg first recovers the other leg from A = ab/2. Area plus an angle uses the ratio a/b = tan(α), and Area plus the hypotenuse solves the product and sum-of-squares constraints with the a ≤ b convention.
Keep full precision during intermediate steps, then round only for display. That is why the on-page values can differ by a few ten-thousandths from a calculation performed with already-rounded legs. The downloadable workbook stores the current canonical numbers and applies cell number formats rather than replacing those numbers with display text.