Quick answer: A triangle with sides 3, 4 and 5 has angles 36.87°, 53.13° and 90°; Heron's formula gives s = 6 and area √(6 × 3 × 2 × 1) = 6.
How it's calculated
Side a is opposite angle A, b opposite B and c opposite C. The law of cosines, c² = a² + b² − 2ab·cos C, finds a side from two sides and the angle between them, or an angle from three sides.
The law of sines, a ÷ sin A = b ÷ sin B = c ÷ sin C, finds the remaining sides once two angles are known; the third angle is always 180° minus the other two.
SSA is ambiguous: sin B = b·sin A ÷ a can give an acute and an obtuse angle B, so there may be two triangles, one, or none. The table shows every triangle that fits.
The area uses Heron's formula with the semi-perimeter s = (a + b + c) ÷ 2: area = √(s(s − a)(s − b)(s − c)). The table also gives the inradius r = area ÷ s and the circumradius R = abc ÷ (4 × area).
Worked examples
- SSS 3, 4, 5: A = 36.87°, B = 53.13°, C = 90°, area 6
- SAS a = 5, b = 7, C = 60°: c = √(25 + 49 − 35) = √39 = 6.245
- AAS A = 30°, B = 45°, a = 10: C = 105°, b = 14.1421, c = 19.3185
Frequently asked questions
Why does SSA sometimes give two triangles?
With two sides and an angle that is not between them, the side opposite the angle can swing to two positions. The calculator shows both when they exist.
What is the triangle inequality?
Each side must be shorter than the sum of the other two. Sides 2, 3 and 6 cannot form a triangle because 2 + 3 < 6.
Are the angles in degrees?
Yes, all angles are entered and shown in degrees.
Is there a faster tool for right triangles?
Yes, the right triangle calculator solves from any two values, and the Pythagorean theorem calculator finds a missing side.
What's the area of a triangle with sides 3, 4 and 5?
6. Heron's formula gets there, and so does ½ × 3 × 4, since it's a right triangle.
What's the area of an equilateral triangle?
√3 ÷ 4 × side². With 10 cm sides that's about 43.3 cm².