Free · Mathematical relations and geometry

Triangle Area by Height, Sides or Angles

Calculate triangle area using perpendicular height, three sides, an included angle or two angles around a known side.

  • Formula & worked example
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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.

Private by default

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

Formula and method

Base 8 with perpendicular height 3 gives area 12. A 3–4–5 triangle has area 6 by Heron’s formula.

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
Base (b)
8 cm
Height (h)
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.
  • For ASA and SAS, the upper angle bound is 3.1415 radians, slightly below π.

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

Exact π supplements the source rounded 3.1415-radian limits without widening the inherited angle domain. The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.

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