Calculator Inputs
Enter any two sides, or one side with one acute angle. You may also enter one angle only to get a ratio triangle.
Example Data Table
| Known Values | Input Example | Expected Use | Output Focus |
|---|---|---|---|
| Two legs | a = 3, b = 4 | Classic side solution | Hypotenuse, angles, ratios |
| Leg and hypotenuse | a = 5, c = 13 | Missing leg by theorem | Side b, area, perimeter |
| Side and angle | c = 10, A = 35° | Trigonometric projection | Both legs and ratios |
| Angle only | A = 60° | Ratio triangle | Unit hypotenuse comparison |
Formula Used
Pythagorean rule: c² = a² + b²
Trigonometric ratios: sin(A) = a / c, cos(A) = b / c, tan(A) = a / b
Acute angles: A + B = 90°
Area: Area = (a × b) / 2
Perimeter: P = a + b + c
Altitude to hypotenuse: h = (a × b) / c
Inradius: r = (a + b - c) / 2
Circumradius: R = c / 2
How to Use This Calculator
- Enter any two known side lengths, or enter one side with one acute angle.
- Use side a as the side opposite Angle A.
- Use side b as the side adjacent to Angle A.
- Use side c only for the hypotenuse.
- Select degrees or radians for angle input.
- Choose your preferred decimal precision.
- Press the calculate button and review the result above the form.
- Use the CSV or PDF button to save the calculation.
Right Triangle Trigonometry Guide
Why Right Triangle Trigonometry Matters
A right triangle has one fixed angle of ninety degrees. This simple shape supports many practical calculations. It appears in surveying, architecture, navigation, physics, machining, roofing, and classroom geometry. When one side and one angle are known, trigonometric ratios can describe the full triangle.
Understanding the Three Sides
The hypotenuse is always the longest side. It sits opposite the right angle. The opposite and adjacent sides depend on the chosen acute angle. In this calculator, side a is opposite Angle A. Side b is adjacent to Angle A. This naming keeps the sine, cosine, and tangent ratios clear.
Using Sine, Cosine, and Tangent
Sine compares the opposite side with the hypotenuse. Cosine compares the adjacent side with the hypotenuse. Tangent compares the opposite side with the adjacent side. These ratios help solve missing sides when an acute angle is known. They also help find angles from known side lengths.
Solving from Side Lengths
When two sides are known, the Pythagorean rule finds the third side. After all sides are known, inverse trigonometric functions find the acute angles. This method is useful when measurements come from a drawing, a field survey, or a construction plan. It also checks whether three entered sides truly form a right triangle.
Area and Extra Measures
Area uses half the product of the two legs. Perimeter adds all three sides. The altitude to the hypotenuse shows the height from the right angle. The inradius gives the radius of the inscribed circle. The circumradius equals half the hypotenuse. These values support deeper geometry work.
Best Input Practice
Use positive values only. Keep units consistent for every side. Do not mix meters with feet in one calculation. If using radians, make sure both entered angles use radians. Review the diagram after calculation. The chart helps confirm side placement and triangle shape before exporting.
FAQs
1. What values are required?
Enter two sides, or one side and one acute angle. You can also enter one acute angle to create a unit ratio triangle.
2. Which side is the hypotenuse?
The hypotenuse is side c. It is opposite the right angle and must be the longest side in a real right triangle.
3. What is side a?
Side a is treated as the side opposite Angle A. This lets the calculator use sin(A) = a / c clearly.
4. What is side b?
Side b is treated as adjacent to Angle A. It forms the cosine ratio with the hypotenuse and tangent ratio with side a.
5. Can I use radians?
Yes. Select radians in the angle unit field. The calculator converts entered radians internally before solving and displaying degree-based results.
6. Why did I get an error?
An error appears when values are impossible. Common causes include a hypotenuse shorter than a leg or two acute angles not totaling ninety degrees.
7. What does the chart show?
The chart draws the right triangle from the solved side values. It helps you confirm orientation, side lengths, and the hypotenuse position visually.
8. Are CSV and PDF exports included?
Yes. After calculation, export buttons appear in the result panel. Use them to save the solved values for reports or study notes.