Triangle Formulas

Every triangle formula in one place — area, perimeter, angles, theorems

Reviewed by [email protected], Geometry Calculator Developer & Online Math Educator Last updated May 14, 2026

Triangles are the most-formula-rich shape in geometry — area alone has half a dozen forms depending on what you know. This page lists every triangle formula you'll meet in school, organized by what they compute (area, perimeter, angles, similarity, congruence) with the conditions for each. Use the related calculators when you need a numeric answer.

The Formulas

Name Formula Notes
Area — Base × Height A = ½ × b × h b = base; h = perpendicular height to that base. Simplest form.
Area — Heron's Formula A = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2 When all three sides are known. No height needed. See full Heron's formula reference.
Area — SAS (two sides + angle) A = ½ × a × b × sin(C) a, b = two sides; C = the included angle between them.
Area — Coordinate Form A = ½ × |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| When the three vertices are given as coordinates.
Area — Inradius A = r × s r = inscribed circle radius; s = semi-perimeter. Inversion: r = A / s.
Area — Circumradius A = (a × b × c) / (4R) R = circumscribed circle radius. Inversion: R = abc / (4A).
Perimeter P = a + b + c Sum of all three sides. Equilateral: P = 3a.
Angle Sum ∠A + ∠B + ∠C = 180° Interior angles of any triangle sum to 180°.
Exterior Angle Theorem ext = sum of 2 opposite interior angles An exterior angle equals the sum of the two non-adjacent interior angles.
Pythagorean Theorem a² + b² = c² For right triangles only. c is the hypotenuse (opposite the right angle). See full Pythagorean theorem reference.
30-60-90 Triangle sides ratio 1 : √3 : 2 Short leg : long leg : hypotenuse, opposite 30°, 60°, 90° respectively. Derivation + examples on the special right triangles page.
45-45-90 Triangle sides ratio 1 : 1 : √2 Isosceles right triangle. Legs equal, hypotenuse = leg × √2. Derivation + examples on the special right triangles page.
Law of Sines a/sin(A) = b/sin(B) = c/sin(C) = 2R For ANY triangle. Best when you have ASA, AAS, or SSA configurations.
Law of Cosines c² = a² + b² − 2ab × cos(C) For ANY triangle. Best for SSS (solve for angles) or SAS (solve for third side). Generalizes Pythagoras (when C = 90°, cos C = 0).
Similarity — AA two pairs of equal angles → similar If two pairs of corresponding angles are equal, all three are (angle sum), and the triangles are similar.
Similarity Ratio k = corresponding sides ratio For similar triangles: sides scale by k, area scales by k², volume (if scaled to solids) by k³.
Congruence Postulates SSS, SAS, ASA, AAS, HL Triangles are congruent (identical shape and size) if any one of these 5 conditions holds. HL is for right triangles only.
Equilateral — All Formulas A = (√3/4)·a², h = (√3/2)·a, P = 3a When all sides equal a. Height bisects the base and is also a median, angle bisector, and altitude. See full equilateral triangle reference.

Worked Examples

Example 1: Triangle with sides 5, 6, 7 — find area and angles

  1. s = (5+6+7)/2 = 9 ; A = √[9·4·3·2] = √216 ≈ 14.70 (Heron)
  2. cos(C) = (a²+b²−c²) / (2ab) = (25+36−49)/60 = 12/60 = 0.2
  3. ∠C = arccos(0.2) ≈ 78.46°
  4. Use Law of Sines for the others: sin(A)/5 = sin(78.46°)/7

Example 2: Right triangle with legs 9 and 12 — find hypotenuse, area

  1. c = √(81 + 144) = √225 = 15
  2. A = ½ × 9 × 12 = 54 unit²
  3. P = 9 + 12 + 15 = 36 units

Example 3: Similar triangles: AB/DE = 4/6 = 2/3. AC = 9. Find DF.

  1. Similarity ratio k = 2/3
  2. AC and DF are corresponding: AC/DF = 2/3
  3. 9/DF = 2/3 → DF = 27/2 = 13.5

Example 4: Equilateral triangle side 10 — area and height

  1. A = (√3/4) × 10² = 25√3 ≈ 43.30 unit²
  2. h = (√3/2) × 10 = 5√3 ≈ 8.66
  3. P = 30

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