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Triangle Area from Two Sides and Included Angle

Generally no. An acute angle and its obtuse supplement have the same sine and triangle area. The backward angle control returns the principal acute or right angle; area cannot exceed ab/2.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Example in initial displayed units: Side a 4, Side b 3, Angle γ 90 gives Area 6.

A = ab sin γ/2

Worked example

Enter these known values and leave the other values blank.

Side a
4 cm
Side b
3 cm
Angle γ
90 deg
Area
6 cm²

Assumptions and limitations

  • The entered angle lies between the two positive side lengths. It is strictly between 0° and 180°. Area-to-angle solving returns the principal acute/right arcsine branch; the supplementary obtuse angle can have the same area.
  • Use compatible units and enough independent values. Clear unknowns before solving backward. Original reference drawings are schematic and do not scale to entered values.

Common questions

Does area determine a unique included angle?

Generally no. An acute angle and its obtuse supplement have the same sine and triangle area. The backward angle control returns the principal acute or right angle; area cannot exceed ab/2.

References

Require an area no greater than half the product of the two known sides for a real included-angle inverse. The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.

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