Triangle solver
Math & numbersHow do I solve a triangle?
Give three parts of a triangle and get the other three, its area and a drawing to scale, including the case with two answers.
What you enter stays on your device while you use this tool. Opening a shared link sends the settings in its address to HowToolbox to prepare the page.
Missing A, B and C
36.87° · 53.13° · 90°
Right scalene · area 6 m² · perimeter 12 m
The whole triangle
- Side a
- 3 m
- Side b
- 4 m
- Side c
- 5 m
- Angle A (opposite a)
- 36.87°
- Angle B (opposite b)
- 53.13°
- Angle C (opposite c)
- 90°
- Area
- 6 m²
- Perimeter
- 12 m
- Kind
- Right scalene
- Inscribed circle radius
- 1 m
- Circumscribed circle radius
- 2.5 m
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Give three parts of a triangle and get the other three, its area and a drawing to scale, including the case with two answers.
How to use
- Choose what you know: three sides, or a mix of sides and angles.
- Enter the three parts. Side a is opposite angle A, b opposite B and c opposite C.
- Read the missing parts, the area and the kind of triangle; the drawing shows it to scale.
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How it works
Three parts fix a triangle, as long as one of them is a side. The angles always add up to 180°, so two angles give the third. Two laws connect the sides and angles: the law of sines, a / sin A = b / sin B = c / sin C, and the law of cosines, c² = a² + b² − 2ab × cos C, which is Pythagoras' theorem with a correction for angles other than 90°.
Example: sides 3, 4 and 5. The law of cosines gives cos C = (3² + 4² − 5²) / (2 × 3 × 4) = 0, so C = 90°: a right triangle. Then sin A = 3 / 5 gives A ≈ 36.87° and B = 180° − 90° − 36.87° ≈ 53.13°. The area is ½ × a × b × sin C = 6.
Two sides and an angle that is not between them (SSA) is the ambiguous case: the side opposite the angle can swing to meet the third side in two places, in one, or not at all. When two triangles fit, both are shown. Source: OpenStax Precalculus 2e, sections 8.1 and 8.2 (CC BY 4.0).
Worked example
What angle lies opposite the longest side of a 3–4–5 triangle?
- cos C = (3² + 4² − 5²) ÷ (2 × 3 × 4) = 0.
- C = arccos(0) = 90°.
The angle is 90°, so the triangle is right-angled.
Frequently asked questions
Why can't I solve a triangle from three angles?
Angles fix only the shape, not the size: a small and a large triangle can have the same angles. You need at least one side.
Why are there sometimes two answers?
With two sides and an angle opposite one of them, the shorter side can reach the base at two points. Both triangles are correct; pick the one that matches your drawing.
How do I check whether three lengths make a triangle?
The two shorter sides together must be longer than the longest one. Sides of 1, 2 and 5 never meet.
Is this the same as Pythagoras?
For a right triangle, yes: the law of cosines with C = 90° becomes c² = a² + b². The solver works for every other triangle too.
Do all three sides need the same unit?
Yes. Convert the measurements to one unit before comparing sides. The unit selector changes how the stored lengths are displayed.
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