Free · Triangles
Hypotenuse, Right Triangle Sides and Area
Find a right triangle’s hypotenuse, legs, acute angles, area or perimeter from compatible known measurements.
- 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
A right triangle with legs 3 cm and 4 cm has a 5 cm hypotenuse, 6 cm² area and 12 cm perimeter.
c²=a²+b²; A=ab/2; P=a+b+c; α+β=90°.
Worked example
Enter these known values and leave the other values blank.
- Leg a, opposite α
- 3 cm
- Leg b, opposite β
- 4 cm
- Angle α, opposite a
- 36.87 deg
- Angle β, opposite b
- 53.13 deg
- Area
- 6 cm²
- Perimeter
- 12 cm
- Hypotenuse c
- 5 cm
Assumptions and limitations
- This model describes a flat, nondegenerate right triangle with positive sides and acute α and β. The hypotenuse is longer than either leg.
- Enter compatible independent measurements; angle pairs alone leave the size undetermined. The captured controls retain their editable and unlock behavior.
- When only hypotenuse and area, or perimeter and area, determine the two legs, the captured inverse selects the larger leg as a. Exchanging the legs produces the same area and hypotenuse; the calculator does not enumerate a second orientation.
- At fixed hypotenuse c the maximum area is c²/4. At fixed perimeter P it is P²(3−2√2)/4. Relative boundary residuals at most 1e-60 are treated as equality at working precision, not as measurement tolerance.
- Extremely narrow triangles may be unresolved at 64-digit working precision. An indicated domain or consistency error must be resolved before using the result.
- Very small finite results use scientific notation when ordinary fixed-decimal display would lose the nonzero value.
- The retained acre conversion uses the source approximation 4046.856 m² per acre; the exact international acre is 4046.8564224 m². Soccer field is a fixed 7140 m² comparison, not every regulation field. Use square metres for the primary SI model.
- Hidden area and perimeter bounds follow their corresponding displayed unit controls. No separate hidden unit choice is presented.
Common questions
Which measurements determine the hypotenuse?
Two legs give c=√(a²+b²). A leg and its opposite acute angle give c=leg/sin(angle); compatible area and size information can also determine it.
Why does an area-based inverse choose the longer leg?
Area and hypotenuse, or area and perimeter, do not label the legs uniquely. This retained branch reports the larger leg as a; swapping a and b gives the other orientation.
References
Factor the two area-based larger-leg inverse discriminants. Clamp only a negative discriminant inside the separately enforced relative 1e-60 isosceles boundary; no positive discriminant is rounded away. Preserve all twelve captured relationships and all thirty-four directions. Factor both ordinary Pythagorean inverse differences before squaring to retain representable near-equal input differences. Evaluate equivalent angle consistency identities with atan2 and the complementary-angle sum; direct tan near a right angle and subtraction of nearly complementary angles otherwise amplify harmless64digit angle rounding. Preserve both source-linked presentation unit dependencies. Express positive lengths, acute angles and strict side/perimeter ordering using exact signs. Reject incompatible area/hypotenuse and area/perimeter pairs with a declared relative numerical boundary of 1e-60; rounded equality at the isosceles limit is accepted within that computational band, not as measurement uncertainty.