Free · Triangle geometry
Right Triangle Sides, Area and Consistency Checks
Find a right triangle leg, hypotenuse or area, or compare entered sides and angles with right-triangle relationships.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
Legs of 3 and 4 give hypotenuse 5 and area 6 square units.
c²=a²+b²; area=ab/2; sin α=a/c; cos α=b/c; tan α=a/b; α+β=90°.
Worked example
Enter these known values and leave the other values blank.
- Leg (a)
- 3 cm
- Leg (b)
- 4 cm
- Hypotenuse (c)
- 5 cm
- Area
- 6 cm²
Assumptions and limitations
- The two legs meet at a right angle, and side c is the hypotenuse. For a specified right triangle, c² = a²+b² and area = ab/2. All side lengths and area must be positive.
- Checking three sides treats c as the designated hypotenuse candidate. Checking two acute angles uses their complementary sum; it is not a test of arbitrary obtuse angles.
- The side/angle checking mode retains the captured absolute ratio discrepancy cutoff of 0.005. Passing this approximate check does not prove an exact right triangle. The comparison uses the computed discrepancy at supported numerical precision; values extremely close to 0.005 can change classification with rounding.
- Very thin triangles and extreme magnitudes can exceed finite numerical resolution. Boundary warnings may reflect representation limits rather than impossible ideal geometry. Clear an unknown or use locks to keep supplied values fixed.
Common questions
Which side is the hypotenuse?
It is the side opposite the right angle and longer than either leg. In the checking diagram it is labeled c.
Does the angle-and-side check prove an exact right triangle?
The side/angle checking mode retains the captured absolute ratio discrepancy cutoff of 0.005. Passing this approximate check does not prove an exact right triangle.
References
Factor the second leg inverse and the Pythagorean checking residual to preserve small differences between large nearby side values. Retain all visible controls and inverse branches. Use strict positive lengths and acute-angle boundaries, validate all retained two-side checking inputs, and restrict approximate ratio notes to their own mode. The Pythagorean note uses a strict residual; complementary-angle matching allows a documented 1e-60-radian numerical resolution.