BIZOMAT MAKEOS · Geometry
Triangle Solver
What are the missing sides, angles and area of this triangle?
Solve any triangle from three sides, two sides and the included angle, or two angles and the included side, using the laws of sines and cosines.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Known values | choice | Choose the combination you know. |
| Side a | mm | Side opposite angle A. |
| Side b | mm | Side opposite angle B. |
| Side c | mm | Side opposite angle C. |
| Angle A (opposite a) | ° | Interior angle opposite side a (used with two angles and the included side). |
| Angle B (opposite b) | ° | Interior angle opposite side b (used with two angles and the included side). |
| Angle C (opposite c) | ° | Interior angle opposite side c (used with two sides and the included angle). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Known values
- What it is
- Which combination of sides and angles you know. The tool uses only the inputs that belong to the chosen combination and calculates the rest of the triangle.
- Units and format
- Choose one of 3 options from the list: Three sides (a, b, c); Two sides and included angle (a, b, C); Two angles and included side (A, B, c).
- Example
- The form opens with “Three sides (a, b, c)” selected as a worked example. Choose the option that matches your case.
- Get it right
- For two sides, the angle used is the one between them (C between a and b); for two angles, the side used is the one between them (c).
What this means: Side a
- What it is
- The length of side a, the side opposite angle A. Used when three sides or two sides and the included angle are known.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Label the triangle on your sketch first so each side sits opposite the angle with the same letter.
What this means: Side b
- What it is
- The length of side b, the side opposite angle B. Used when three sides or two sides and the included angle are known.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- With three sides, each side must be shorter than the sum of the other two or no triangle exists.
What this means: Side c
- What it is
- The length of side c, the side opposite angle C. Used when three sides are known, or as the side between angles A and B.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Not used for two sides and the included angle; there c is calculated.
What this means: Angle A (opposite a)
- What it is
- The interior angle at the corner opposite side a. Used only when two angles and the side between them are known.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 179.999°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Angles A and B must add to less than 180°, because the three interior angles add to exactly 180°.
What this means: Angle B (opposite b)
- What it is
- The interior angle at the corner opposite side b. Used only when two angles and the side between them are known.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 179.999°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the inside angle of the triangle, not the outside angle of a sketch line.
What this means: Angle C (opposite c)
- What it is
- The interior angle between sides a and b, opposite side c. Used only when two sides and this included angle are known.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 179.999°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must be the angle between the two known sides; an angle at another corner gives a different triangle.
Method and formulas
- Three sides: A = arccos((b² + c² − a²) ÷ 2bc), and similarly for B; C = 180° − A − B.
- Two sides and angle C: c = √(a² + b² − 2ab cos C), then the remaining angles by the law of cosines.
- Two angles and side c: C = 180° − A − B; a = c sin A ÷ sin C; b = c sin B ÷ sin C.
- Area = ½ab sin C; circumradius R = a ÷ (2 sin A); inradius r = area ÷ semi-perimeter.
- Example: sides 30, 40 and 50 mm give angles of 36.87°, 53.13° and 90° and an area of 600 mm².
Limitations
- Two sides with a non-included angle (the ambiguous case) is not offered; convert it to one of the listed cases.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find hole positions and distances on brackets and frames
- Check angles of gussets and braces
- Solve layout problems without CAD
Who it is for
- Design engineers
- Toolmakers
- Estimators
What you get
- All three sides
- All three angles
- Area
- Perimeter
- Circumscribed and inscribed circle radii
Interpreting the result
- Use the inradius and circumradius to check fit of round features in triangular spaces.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 95–102. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Solving triangles for hidden points
In Drafting Geometry by the Numbers
What it covers: Find construction points that no dimension gives directly: chamfers by length and angle, the sine and cosine rules, line intersections and perpendicular distances, with checks.
Method explained here · BIZOMAT original book
Every pattern starts with a solid
In Sheet Metal Patterns by the Numbers
What it covers: Read a fitting as cylinders, prisms, cones and planes, choose parallel line, radial line or triangulation, and set the conventions the rest of the book uses.
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