Triangle Calculator — Find Sides, Angles, Area & Perimeter

The triangle calculator is one of the most searched math tools online — because triangles appear everywhere: geometry homework, construction, surveying, navigation, and engineering. Give the CalcHub Triangle Calculator any three known values (sides or angles) and it finds everything else.

Triangle Area Formulas

Known Values	Area Formula
Base and height	A = ½ × base × height
Two sides and included angle	A = ½ × a × b × sin(C)
Three sides (Heron's formula)	A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2
Three vertices (coordinates)	A = ½\	x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)\

Quick Area Examples

Base	Height	Area
6	4	12
10	5	25
8	3	12
15	8	60
20	12	120

Solving Triangles — What You Need

What You Know	Method	What You Can Find
SSS (3 sides)	Law of Cosines	All angles, area
SAS (2 sides + included angle)	Law of Cosines	Third side, other angles, area
ASA (2 angles + included side)	Law of Sines	Other sides, third angle, area
AAS (2 angles + non-included side)	Law of Sines	Other sides, third angle, area
SSA (2 sides + non-included angle)	Law of Sines	Ambiguous case (0, 1, or 2 solutions)

Key Formulas

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)

Use when you know: an angle and its opposite side, plus one more value.

Law of Cosines

c² = a² + b² − 2ab × cos(C)

Use when you know: three sides (SSS) or two sides and included angle (SAS).

Pythagorean Theorem (Right Triangles Only)

a² + b² = c² (where c is the hypotenuse)

a	b	c (hypotenuse)
3	4	5
5	12	13
8	15	17
7	24	25
6	8	10

Triangle Types

Type	Property	Angle Sum
Equilateral	All sides equal, all angles 60°	180°
Isosceles	Two sides equal, two angles equal	180°
Scalene	All sides different, all angles different	180°
Right	One angle = 90°	180°
Acute	All angles < 90°	180°
Obtuse	One angle > 90°	180°

Every triangle's angles sum to exactly 180°. If you know two angles, the third = 180° − sum of the other two.

Worked Example

Given: Triangle with sides a = 7, b = 10, c = 12 Step 1: Find angles using Law of Cosines:

• cos(C) = (7² + 10² − 12²) / (2 × 7 × 10) = (49 + 100 − 144) / 140 = 5/140
• C = cos⁻¹(0.0357) ≈ 87.9°

Step 2: Find angle A:

• cos(A) = (10² + 12² − 7²) / (2 × 10 × 12) = 195/240
• A = cos⁻¹(0.8125) ≈ 35.7°

Step 3: Angle B = 180° − 87.9° − 35.7° = 56.4° Step 4: Area using Heron's formula:

• s = (7 + 10 + 12)/2 = 14.5
• A = √(14.5 × 7.5 × 4.5 × 2.5) = √(1,222.97) ≈ 34.97 sq units

Practical Applications

Construction: Calculating roof angles, staircase stringers, and diagonal bracing all require triangle math. Surveying: Measuring land area by breaking irregular shapes into triangles (triangulation). Navigation: GPS positioning uses trilateration — calculating your position from distances to three known points. Art and Design: The "rule of thirds" in photography and golden triangle composition are triangle-based design principles.

How to Use the Calculator

1. Open the CalcHub Triangle Calculator
2. Enter any 3 known values (sides and/or angles)
3. See: all missing sides, all angles, area, perimeter, and triangle type

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Can a triangle have two right angles?

No. The angles must sum to 180°. If two angles were 90°, the third would be 0°, which isn't a triangle. A triangle can have at most one right angle or one obtuse angle.

What's the SSA ambiguous case?

When you know two sides and a non-included angle, there can be zero, one, or two valid triangles. This happens because the given angle might correspond to different configurations. The calculator handles this automatically and shows all valid solutions.

How do I find the height of a triangle if I only know the sides?

Use the area formula: Area = ½ × base × height. Find the area using Heron's formula (from 3 sides), then solve: height = 2 × Area / base.

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Related Calculators

• Pythagorean Theorem Calculator — right triangles
• Area Calculator — all geometric shapes
• Slope Calculator — rise over run
• Square Footage Calculator — practical area