Triangle Calculator
Find area, angles, and all sides of any triangle — SSS, SAS, or base & height.
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Triangle Formulas and Properties
A triangle is a polygon with three sides, three vertices, and three interior angles. The sum of all interior angles always equals 180°. The longest side is always opposite the largest angle.
Area Formulas
Base and height: A = ½ × b × h. SAS: A = ½ × a × b × sin(C). Heron's formula (SSS): A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.
Law of Cosines
Relates sides and angles: c² = a² + b² − 2ab×cos(C). Useful for SSS (finding angles) and SAS (finding the third side). For a right triangle (C=90°): cos(90°)=0, giving c²=a²+b² — the Pythagorean theorem.
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C). Useful for ASA and AAS problems. Each side divided by the sine of its opposite angle gives the same ratio (the diameter of the circumscribed circle).
Triangle Types
By sides: equilateral (all sides equal, all angles 60°), isosceles (two sides equal), scalene (all different). By angles: acute (all <90°), right (one 90°), obtuse (one >90°).
Worked Example: SSS Triangle Using Heron's Formula
Find the area of a triangle with sides a = 7, b = 8, c = 9.
Step 1 — Calculate the semi-perimeter: s = (7 + 8 + 9) / 2 = 24 / 2 = 12.
Step 2 — Apply Heron's formula: A = √(s × (s−a) × (s−b) × (s−c)) = √(12 × 5 × 4 × 3) = √720 ≈ 26.83 square units.
Step 3 — Verify triangle inequality: 7+8=15 > 9 ✓, 7+9=16 > 8 ✓, 8+9=17 > 7 ✓. The triangle is valid.
Triangle Type Summary
| Type | Side condition | Angle condition |
|---|---|---|
| Equilateral | a = b = c | All angles = 60° |
| Isosceles | Two sides equal | Two base angles equal |
| Scalene | All sides different | All angles different |
| Right | c² = a² + b² | One angle = 90° |
| Obtuse | c² > a² + b² | One angle > 90° |
Frequently Asked Questions
A = ½ × base × height. The height must be perpendicular to the base. For base=10, height=6: A = 30 sq units.
For sides a,b,c: s=(a+b+c)/2, A=√(s(s-a)(s-b)(s-c)). Use when height is unknown but all three sides are known.
180°. Always. This is a fundamental property of triangles in Euclidean geometry.
c² = a² + b² − 2ab×cos(C). Used to find the third side given two sides and the included angle, or to find angles given all three sides.
The triangle inequality: the sum of any two sides must be greater than the third side. a+b>c, a+c>b, b+c>a.
Use Heron's formula to find the area A, then rearrange: h = 2A / base. For a 3-4-5 right triangle, area = 6, so the height relative to the longest side (5) is h = 2×6/5 = 2.4 units.
For an equilateral triangle with side s, area = (√3/4)s². Setting (√3/4)s² = 9√3 gives s² = 36, so s = 6. Perimeter = 3 × 6 = 18 units.