BIZOMAT MAKEOS · Geometry
Triangle from Two Sides and an Opposite Angle
Which triangles have these two sides and the angle opposite one of them?
Solve the ambiguous case of a triangle (two sides and an angle opposite one of them), giving none, one or two possible triangles with all sides, angles and areas.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Side a (opposite the known angle) | mm | Side opposite the known angle. |
| Side b | mm | The other known side. |
| Angle A (opposite side a) | ° | Known angle, opposite side a. |
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: Side a (opposite the known angle)
- What it is
- The known side that lies opposite the known angle A, across the triangle from the corner where A is measured.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. 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
- If side a is shorter than b, two different triangles can fit; check the result against the drawing.
What this means: Side b
- What it is
- The other known side, opposite the unknown angle B. It runs from the corner of angle A to the corner where it meets side a, at angle C.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. 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
- Make sure side b touches the corner of angle A; if it does not, this is not the two-sides-and-opposite-angle case.
What this means: Angle A (opposite side a)
- What it is
- The known angle, at the corner opposite side a, between side b and the unknown side c.
- 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
- Give the angle in degrees; an angle not opposite side a needs a different triangle method.
Method and formulas
- sin B = b sin A ÷ a (law of sines).
- If sin B > 1 there is no triangle; if A is acute and a < b, B and 180° − B may both work.
- C = 180° − A − B; c = a sin C ÷ sin A; area = ½ ab sin C.
- Example: a = 40, b = 50, A = 40° gives 2 triangle(s).
Limitations
- Plane triangles only.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Solve layout problems where only two sides and one angle are known
- Check whether a linkage can assemble
- Find both positions of a mechanism
Who it is for
- Design engineers
- Toolmakers
- Estimators
What you get
- Number of solutions
- Angles B and C and side c for each
- Area of each
Interpreting the result
- Two solutions mean two possible shapes (for example two positions of a link); pick the one that matches the design.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 95–102. Topic reference; no source tables or recommended settings are reproduced.
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Open the Triangle from Two Sides and an Opposite Angle calculator