Triangle Calculator Guide: Area, Perimeter, Angles & All Triangle Types

Every triangle has three sides (a, b, c) and three angles (A, B, C) that always sum to 180°.

Triangle types by angles: • Acute triangle: all angles < 90° • Right triangle: one angle = exactly 90° • Obtuse triangle: one angle > 90°

Triangle types by sides: • Equilateral: all three sides equal (all angles = 60°) • Isosceles: two sides equal (two base angles equal) • Scalene: all three sides different (all angles different)

The 180° rule: A + B + C = 180° always. • If two angles are known: third angle = 180° − A − B • Example: angles 45° and 72° → third angle = 180° − 45° − 72° = 63°

Triangle inequality theorem: any side must be less than the sum of the other two sides: • a < b + c | b < a + c | c < a + b • Example: sides 3, 4, 8 → 3 + 4 = 7 < 8 → NOT a valid triangle

Similar triangles: same shape (equal angles), different size. Ratios of corresponding sides are equal.

• All triangle angles always sum to 180°
• Third angle = 180° − (sum of known angles)
• Right triangle: one 90° angle | Equilateral: all angles 60°
• Triangle inequality: each side < sum of the other two

Right triangles have a 90° angle (C = 90°) and special properties:

Pythagorean theorem: c² = a² + b² • c = hypotenuse (side opposite the 90° angle) • a, b = legs • c = √(a² + b²)

Examples: • a = 3, b = 4: c = √(9 + 16) = √25 = 5 (3-4-5 right triangle) • a = 5, b = 12: c = √(25 + 144) = √169 = 13 • a = 8, b = 15: c = √(64 + 225) = √289 = 17

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29

Finding legs from hypotenuse: a = √(c² − b²)

Trigonometry in right triangles: • sin(A) = opposite / hypotenuse = a/c • cos(A) = adjacent / hypotenuse = b/c • tan(A) = opposite / adjacent = a/b

Finding angles: • A = arcsin(a/c) = arccos(b/c) = arctan(a/b)

• Pythagorean theorem: c² = a² + b² (c = hypotenuse)
• Common triples: 3-4-5, 5-12-13, 8-15-17
• sin = opposite/hypotenuse | cos = adjacent/hypotenuse | tan = opposite/adjacent
• Find angle: A = arctan(opposite/adjacent)

Multiple formulas for triangle area, depending on what's known:

1. Base and height (most common): A = ½ × base × height • Height must be perpendicular to the base • Example: base = 10, height = 6 → A = ½ × 10 × 6 = 30

2. Two sides and included angle (SAS): A = ½ × a × b × sin(C) • C is the angle between sides a and b • Example: a = 5, b = 8, C = 60°: A = ½ × 5 × 8 × sin(60°) = 20 × 0.866 = 17.32

3. Heron's formula (all three sides known, SSS): s = (a + b + c) / 2 (semi-perimeter) A = √[s(s−a)(s−b)(s−c)] • Example: sides 7, 8, 9: s = 12; A = √[12 × 5 × 4 × 3] = √720 = 26.83

4. Equilateral triangle: A = (√3 / 4) × side² • Side 10: A = (√3/4) × 100 = 43.3

5. Right triangle: A = ½ × leg₁ × leg₂ • Legs 6 and 8: A = ½ × 6 × 8 = 24

• Base × height ÷ 2: most common formula (height must be perpendicular to base)
• SAS: A = ½ab × sin(C) — use when two sides and included angle are known
• Heron's formula: A = √[s(s−a)(s−b)(s−c)] — use when all three sides are known
• Equilateral: A = (√3/4) × side²

For triangles that aren't right triangles, use these general laws:

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Use when given: • AAS (two angles and non-included side) • ASA (two angles and included side) • SSA (two sides and non-included angle — ambiguous case)

Example: A = 40°, B = 75°, a = 10 • C = 180° − 40° − 75° = 65° • b/sin(75°) = 10/sin(40°) • b = 10 × sin(75°)/sin(40°) = 10 × 0.966/0.643 = 15.02

Law of Cosines: c² = a² + b² − 2ab × cos(C)

Use when given: • SAS (two sides and included angle) • SSS (all three sides — to find angles)

Finding angle from SSS: cos(C) = (a² + b² − c²) / (2ab) • C = arccos[(a² + b² − c²) / (2ab)]

Example: a = 7, b = 10, C = 53°: • c² = 49 + 100 − 2(7)(10)cos(53°) = 149 − 140 × 0.602 = 149 − 84.28 = 64.72 • c = √64.72 ≈ 8.04

• Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) — use for AAS, ASA, SSA
• Law of Cosines: c² = a² + b² − 2ab·cos(C) — use for SAS, SSS
• Find angle from 3 sides: cos(C) = (a² + b² − c²) ÷ (2ab)
• Law of Cosines reduces to Pythagorean theorem when C = 90° (cos 90° = 0)

Perimeter: P = a + b + c (sum of all three sides)

Equilateral triangle: P = 3 × side Isosceles triangle: P = 2 × equal side + base Right triangle: P = leg₁ + leg₂ + hypotenuse

Special right triangles: • 30-60-90: sides in ratio 1 : √3 : 2 − Short leg = x, long leg = x√3, hypotenuse = 2x − Example: hypotenuse 10 → legs 5 and 5√3 ≈ 8.66 • 45-45-90 (isosceles right): sides in ratio 1 : 1 : √2 − Legs = x, hypotenuse = x√2 − Example: legs 6 → hypotenuse = 6√2 ≈ 8.49

Centroid, incenter, circumcenter: • Centroid: intersection of medians (center of mass), at 2/3 of each median from vertex • Circumcenter: center of circumscribed circle (equal distance from all vertices) • Incenter: center of inscribed circle (equal distance from all sides) • In an equilateral triangle, all three coincide at the same point

Inradius and circumradius: • Inradius r = Area / s (s = semi-perimeter) • Circumradius R = (a × b × c) / (4 × Area)

• Perimeter: P = a + b + c
• 30-60-90 triangle: sides ratio 1 : √3 : 2
• 45-45-90 triangle: sides ratio 1 : 1 : √2 (hypotenuse = leg × √2)
• Inradius = Area ÷ semi-perimeter | Circumradius = (a×b×c) ÷ (4×Area)

A decision guide for triangle problems:

Given → Formula to use: • Base + height → Area = ½ × b × h • Right triangle, two sides → Pythagorean theorem • Right triangle, side + angle → SOH-CAH-TOA • Three sides (SSS) → Heron's formula for area; law of cosines for angles • Two sides + included angle (SAS) → Law of cosines for third side; ½ab·sin(C) for area • Two angles + any side (AAS or ASA) → Law of sines for remaining sides • Two sides + non-included angle (SSA) → Law of sines (check ambiguous case) • Equilateral (one side known) → All sides equal; all angles 60°; A = (√3/4)s²

Ambiguous case (SSA): • May have 0, 1, or 2 valid triangles • If a ≥ b: one solution exists • If a < b: check if a < b×sin(A) (no solution), a = b×sin(A) (one solution), or a > b×sin(A) (two solutions)

Practical tip: Draw a rough sketch first. Label known values. Identify what combination (SSS, SAS, ASA, AAS, SSA) you have, then select the matching formula.

• SSS: use Heron's formula (area) and law of cosines (angles)
• SAS: use law of cosines for missing side; ½ab·sin(C) for area
• AAS/ASA: use law of sines for missing sides
• Right triangle: use Pythagorean theorem + SOH-CAH-TOA