Free · Mathematical relations and geometry

Scalene Triangle Area from Measurements

Use a perpendicular height, three sides or selected angles to calculate a nondegenerate triangle area.

  • Formula & worked example
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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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Formula and method

given = 0, base = 8, height = 3 gives area_bh = 12.

A=bh/2; Heron A=√[s(s−a)(s−b)(s−c)]; A=ab sin γ/2

Worked example

Enter these known values and leave the other values blank.

Given
0.0
Base
8 cm
Height
3 cm
Area
12 cm²

Assumptions and limitations

  • Height is perpendicular to the chosen base.
  • Three positive sides must satisfy strict triangle inequalities. Heron’s formula covers arbitrary nondegenerate triangles.
  • SAS uses the included angle. ASA uses a side between two angles whose positive sum is below 180 degrees.
  • Positive inverse roots are retained. Area and two sides can admit supplementary angle solutions; the inverse chooses the principal acute angle.
  • The equations also work for equal-side triangles; the calculator does not enforce three unequal sides.

Common questions

How can I solve for a missing quantity?

Leave the unknown control empty and enter the quantities needed by the selected relation. Read the assumptions for any branch or domain restrictions.

How should I interpret the worked example?

It is an exact mathematical model with independently checked numbers. Display rounding does not change the values used for subsequent calculations.

References

The captured ASA inverse angle variants incorrectly returned constant PI and inferred no dependencies. The area identity cot α+cot β=a²/(2A) gives the positive interior-angle inverse atan2(1,cot α), removing fabricated 180-degree results after clearing. The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.

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