Triangle Calculator

About Triangle Calculator

Solve any triangle from partial information. Enter three sides (SSS), two sides and the included angle (SAS), or two angles and the included side (ASA) to find all remaining sides, angles, area, perimeter, and heights. The calculator draws a scaled SVG diagram with labels and shows every formula step by step.

How Each Mode Works

Mode	You Know	Method Used	Example Input
SSS	All three sides	Law of Cosines for angles, Heron's formula for area	a=5, b=7, c=9
SAS	Two sides and included angle	Law of Cosines for third side, then SSS	a=5, b=7, C=60°
ASA	Two angles and included side	Third angle = 180° - A - B, Law of Sines for sides	A=40°, B=70°, c=8

Triangle Formulas Reference

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

Used when you know SSS (to find angles) or SAS (to find the third side). It generalizes the Pythagorean theorem - when C = 90°, the cos(C) term disappears and you get a² + b² = c².

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

Used when you know ASA or AAS. Each side is proportional to the sine of its opposite angle.

Heron's formula (area from three sides):

s = (a + b + c) / 2

A = √(s(s-a)(s-b)(s-c))

SAS area formula: A = (1/2) x a x b x sin(C)

Heights: h_a = 2A / a, h_b = 2A / b, h_c = 2A / c

Worked Examples

SSS: a = 5, b = 7, c = 9

• Find angle C: cos(C) = (5² + 7² - 9²) / (2 x 5 x 7) = (25 + 49 - 81) / 70 = -7/70 = -0.1
• C = arccos(-0.1) = 95.74°
• Heron's formula: s = (5 + 7 + 9) / 2 = 10.5
• Area = √(10.5 x 5.5 x 3.5 x 1.5) = √303.1875 = 17.41
• Perimeter = 21

SAS: a = 6, b = 8, C = 50°

• Find c: c² = 36 + 64 - 2(6)(8)cos(50°) = 100 - 96(0.6428) = 100 - 61.71 = 38.29
• c = √38.29 = 6.19
• Area = (1/2)(6)(8)sin(50°) = 24(0.766) = 18.39

Triangle Classification

By Sides	Definition	Properties
Equilateral	All three sides equal	All angles 60°, maximum area for given perimeter
Isosceles	Two sides equal	Two equal angles opposite the equal sides
Scalene	No equal sides	All angles different

By Angles	Definition	Properties
Acute	All angles less than 90°	All altitudes fall inside the triangle
Right	One angle equals 90°	Pythagorean theorem applies
Obtuse	One angle greater than 90°	One altitude falls outside the triangle

The Triangle Inequality

Three lengths can form a triangle only if the sum of any two sides is greater than the third. Formally: a + b > c, a + c > b, and b + c > a. If any of these fail, the triangle is impossible. The calculator validates your inputs and warns you if they do not form a valid triangle.

Valid: 3, 4, 5 (3+4=7 > 5, 3+5=8 > 4, 4+5=9 > 3)

Invalid: 1, 2, 5 (1+2=3 < 5 - these cannot form a triangle)

Practical Applications

• Construction: Calculating roof angles, rafter lengths, and triangular structural supports
• Surveying: Triangulation uses known distances and angles to determine unknown positions
• Navigation: Finding distances and bearings between three points
• Design: Calculating material needed for triangular shapes (sails, banners, architectural features)

For right triangles specifically, the Pythagorean theorem calculator provides a focused tool. For deeper trigonometry work, the law of cosines calculator and law of sines calculator show detailed workings for each formula.

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