Any triangle
A triangle does not need a right angle. If you know three of its measurements — and at least one of them is a side — the sine and cosine rules can find the rest.
The explanation is shown. Quick: SSS, SAS and ASA. Advanced: also the ambiguous case SSA — and heights, perimeter and the triangle's circles.
Your numbers
Side a is opposite angle A, b opposite B and c opposite C.
More options: unit
Answer
Angle A
38.21°
Angle B
60°
Angle C
81.79°
Area
17.32cm²
Circumference
20cm
Drawn to scale.
How to work it out
- 1cos A = (b² + c² − a²) / (2 · b · c)The cosine rule: if you know all three sides, it gives each angle. It also works for obtuse angles.
- 2cos A = (49 + 64 − 25) / 112 = 0.7857 ⇒ A = cos⁻¹(0.7857) = 38.21°
- 3cos B = (25 + 64 − 49) / 80 = 0.5 ⇒ B = 60°
- 4C = 180° − 38.21° − 60° = 81.79°The angles in a triangle always add up to 180°.
- 5T = ½ · a · b · sin C = ½ · 5 · 7 · sin 81.79° = 17.32 cm²The area without knowing the height: two sides and the sine of the angle between them.
The formula
a / sin A = b / sin B = c / sin C
The sine rule — when you know a side and the angle opposite it.
a² = b² + c² − 2 · b · c · cos A
The cosine rule — when you know three sides, or two sides and the angle between them.
T = ½ · a · b · sin C
The area from two sides and the angle between them.
In depth: more numbers and “what if”
More quantities
| Size | Value |
|---|---|
| Circumference | 20 cm |
| Height to a | 6.93 cm |
| Height to b | 4.95 cm |
| Height to c | 4.33 cm |
| Circumradius (a / (2 · sin A)) | 4.04 cm |
| Inradius (2T / perimeter) | 1.73 cm |
More: Geometry
- PythagorasFind the missing side, set out a right angle, check whether a room is square, and find the space diagonal.
- Area and perimeterRectangle, triangle, trapezium, circle and composite shapes — area and perimeter.
- Volume and surface areaBox, cylinder, cone, sphere, pyramid, prism, pipe and frustum — also in litres.
- The circleRadius, circumference and area from one measurement — plus sector, annulus, chord and rise.
- Right triangleSine, cosine and tangent: all sides and angles from two known values.
- AnglesThe third angle in a triangle and the angle sum of a polygon.
- Points in the coordinate systemDistance, midpoint and the line between two points — drawn in.
- Similar figuresScale factor and missing sides — area is multiplied by k², volume by k³.
- TransformationsReflect, rotate and translate a figure on a coordinate grid — before and after.