Free · Triangle geometry

Obtuse Triangle Area Methods

Find triangle area with all four captured input methods, retaining branch-specific inverses and explaining obtuse-angle and height conventions.

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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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Formula and method

A base of3 and perpendicular height4 gives area6. The corresponding area unit is the square of the length unit.

A=bh/2; A=ab sin(γ)/2; A=√(s(s−a)(s−b)(s−c)); A=a² sinβ sinγ/(2 sin(β+γ)).

Worked example

Enter these known values and leave the other values blank.

Given
base and height
Base
3 cm
Height
4 cm
Area
6 cm²

Assumptions and limitations

  • The four area modes use base/height, three sides, two sides with their included angle, or two angles with the included side. All lengths must use compatible units, and the area unit must match the squared length unit.
  • An obtuse triangle has an angle above90°. The area identities also apply to acute and right triangles; base and height alone do not determine which kind you have.
  • Every triangle length and area is positive. Each triangle angle is between0 and180°, and the two ASA angles have sum below180°. Three sides must satisfy all triangle inequalities.
  • Inverse SAS returns the acute principal angle. Its supplement has the same sine and area and may be the intended obtuse angle; enter a known included angle directly when choosing between the two shapes.
  • A Heron inverse selects the captured larger unknown-side root. The smaller-side root may also fit the same area. The calculator retains that branch rather than asserting a unique reconstruction.
  • The ASA inverse has been corrected from the source’s constant180° placeholder. It solves the angle within the physical triangle domain using the area, included side and other angle.
  • The sketches are original fixed reference geometry. An altitude can meet the extended base outside an obtuse triangle; it still measures perpendicular height.
  • Three-side area uses a stable, parenthesized Heron rearrangement. Nearly degenerate triangles can remain sensitive to64 digit input conversion or arithmetic. If the positive triangle gap is not numerically distinguishable, the area is withheld; this does not establish that the exact real triangle is impossible.
  • Lengths and areas may be extremely small or large without a physical magnitude cap. Scientific display preserves computed nonzero values when ordinary decimal formatting cannot represent them. Inverse values use the retained numerical controls and the displayed algebraic branch; they do not recover input digits already lost during unit conversion.

Common questions

Can I recover an obtuse angle from the area?

Sine gives the same value for an angle and its supplement. The retained SAS inverse reports the acute principal angle; the supplement is the second possible included angle.

Why is an ASA angle not180°?

Two positive triangle angles must sum to less than180°. The corrected inverse solves the angle from the area and side instead of returning the source placeholder.

References

Replace the captured constant-pi ASA angle placeholders with the unique physical angle from cot(beta)=a²/(2A)−cot(gamma), using atan2 to retain the correct quadrant. Preserve all captured mathematical modes and inverse branches, applying the separately described stable algebra to the three-side computation. SAS inverse retains its acute principal branch and Heron inverse the larger-side branch; these are disclosed rather than claimed unique. Use sorted, parenthesized Kahan Heron forward area and factored larger-root inverse to avoid expanded-quartic cancellation, preserving the captured larger-side inverse branch. Guard only direct trigonometric evaluations by known physical angles, including forward consistency identities. No working precision increase or magnitude cap. Use exact pi for strict triangle-angle bounds instead of the rounded3.1415 cutoff; require missing positive area/side/angle domains and the geometric bound A<=ab/2 for supplied side pairs. These are mathematical domains, not tolerance relaxations. Sign-based positivity and sorted minimum gap preserve tiny positive lengths while rejecting unresolved degenerate bounds. Compare supplied area directly to half the side product without a lossy normalized ratio.

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