Free · Triangles
Oblique Triangle Sides, Angles and Area: Four Methods
Solve non-right triangles from three sides, two sides and an included angle, or two angles and a side.
- 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
An equilateral triangle with5cm sides has three60° angles and area25√3/4≈10.8253cm².
a/sin(α)=b/sin(β)=c/sin(γ); c²=a²+b²−2ab cos(γ); area=ab sin(γ)/2.
Worked example
Enter these known values and leave the other values blank.
- Given
- Three sides (SSS)
- Side a
- 5 cm
- Side b
- 5 cm
- Side c
- 5 cm
- Angle α
- 60 deg
- Angle ß
- 60 deg
- Angle γ
- 60 deg
- Area
- 10.825 cm²
Assumptions and limitations
- Choose one of the four captured input methods. All19 captured directions are forward calculations; read-only result controls do not become editable inverses.
- Every side is positive and smaller than the sum of the other two. Interior angles are between0° and180°, with total180°; supplied pairs must leave a positive third angle.
- This oblique-triangle model excludes right angles. At64-digit working precision it requires separation greater than1e-60 radian from90°, and accepts angle-sum residuals at most1e-60 radian. These are numerical resolution checks, not measurement uncertainty or an exact geometric classification.
- In SAS mode the smaller computed angle belongs opposite the shorter supplied side; the displayed larger-angle branch completes the triangle. The source switches those labels when the supplied sides change order.
- Use consistent length units; area scales with the square of length. All captured unit choices and four method controls remain available. Nearly degenerate or extreme-scale triangles may be numerically unresolved and should not be treated as verified geometry.
- Very small nonzero values may appear in scientific notation when fixed decimal display cannot represent them. Their value is preserved rather than shown as zero.
Common questions
What makes a triangle oblique?
It has no right angle. Acute and obtuse triangles both qualify; a right triangle should use a right-triangle calculator.
Why can’t I enter any three positive sides?
Each side must be shorter than the other two combined. Equality collapses the triangle to a line, with no positive interior area.
References
- OpenStax: law of sines and non-right triangles
- OpenStax: law of cosines
- Calculation definition and unit reference
Use an unexpanded four-factor Heron area and atan2 of four times area against the cosine-law dot product for the SSS angles. This avoids the captured expanded Heron cancellation and acos endpoint sensitivity while preserving the four original forward targets and their side inputs. Other fifteen equation variants remain captured. Retain four captured non-right-triangle modes and all19 forward equation directions. Express input boundaries by sign, validate every triangle inequality and both supplied-angle sums, and replace unsupported equal/unequal predicates with explicit64-digit numerical policies: angle-sum residual at most1e-60 radian and every angle separated from a right angle by more than1e-60 radian. These are computational resolution checks, not measurement tolerances or exact geometric certification. When ordinary fixed-decimal rounding cannot show a finite nonzero value, preserve that value in scientific notation. This fallback is bound to the complete reviewed source and compiled schemas; ordinary rounding, equations and controls stay unchanged.