Right Triangle Calculator: Sides, Angles and Trig Ratios

Give any two of the opposite side, adjacent side, hypotenuse and angle, and get every other part of the right triangle.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

In a right triangle the side opposite the right angle is the hypotenuse. For the angle you are working from, the side facing it is the opposite side and the side next to it (other than the hypotenuse) is the adjacent side. Sine, cosine and tangent are the ratios of these sides: opposite over hypotenuse, adjacent over hypotenuse, and opposite over adjacent. The memory aid is SOH CAH TOA.

Pick which two parts you know. The calculator works out the other sides with Pythagoras and the ratios, the other acute angle (90 degrees minus the first), the six trigonometric ratios, the area and perimeter, and the slope as a percentage and as a ratio, which is useful for ramps, roofs, falls and drainage.

Formulas used

  • \sin\theta = \frac{P}{H},\;\; \cos\theta = \frac{B}{H},\;\; \tan\theta = \frac{P}{B}
  • H^2 = P^2 + B^2
  • \text{slope \%} = 100\,\frac{P}{B}

Worked example

With the default numbers in the calculator above:

  1. Angle θ: 36.87°
  2. The other acute angle: 53.13°
  3. Opposite side P: 3
  4. Adjacent side B: 4
  5. Hypotenuse H: 5
  6. Pythagoras check: 25 = 25
  7. sin θ: 0.6
  8. cos θ: 0.8
  9. tan θ: 0.75
  10. cot θ: 1.3333
  11. sec θ: 1.25
  12. cosec θ: 1.6667

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

Which side is opposite and which is adjacent?

It depends on the angle you are using. The opposite side faces the angle. The adjacent side touches the angle and is not the hypotenuse. Swap to the other acute angle and the two swap.

How do I convert a slope to degrees?

Take the rise over the run and find the angle whose tangent it is. A slope of 1 in 12 (8.33 percent) is about 4.76 degrees.

Does it matter what unit I use for the sides?

No. Use the same unit for every side. The answers come out in that unit, and the angles are always in degrees.

Why can the angle not be 0 or 90 degrees?

The triangle collapses to a line. The angle must be between 0 and 90 degrees, not including either.

Related calculators

Further reading: Standard engineering mathematics texts (right-triangle trigonometry and Pythagoras' theorem).

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