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Triangle Equations Calculator

Triangle equation formula

Solution

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Triangle Equations

Select a triangle type and enter the known measurements. The calculator handles right triangles, equilateral triangles, scalene triangles, law of cosines, and law of sines.

K = ½ab, c = √(a² + b²)

How It Works

Select a triangle type and enter the known measurements. For right triangles the calculator uses the Pythagorean theorem. For equilateral triangles it applies s-based formulas. The law of cosines and law of sines handle oblique triangles.

Example Problem

A right triangle has legs 3 and 4. Find the hypotenuse, area, and perimeter.

  1. Hypotenuse = √(3² + 4²) = √25 = 5
  2. Area = ½ × 3 × 4 = 6
  3. Perimeter = 3 + 4 + 5 = 12

Key Concepts

Triangle calculations depend on the known measurements. Right triangles use the Pythagorean theorem (c² = a² + b²) and basic trigonometry. Equilateral triangles have all sides equal, so a single side determines everything. For oblique triangles, the Law of Cosines (c² = a² + b² − 2ab·cos C) handles SAS and SSS cases, while the Law of Sines (a/sin A = b/sin B) handles AAS and ASA cases. Area can always be computed from ½·base·height or from Heron's formula.

Applications

  • Surveying and land measurement: calculating plot areas and distances from angle and side measurements taken in the field
  • Construction: determining rafter lengths, roof pitch angles, and gable heights for framing
  • Navigation: solving distance and bearing problems using triangle geometry and the law of cosines
  • Computer graphics: decomposing polygons into triangles for rendering and collision detection

Common Mistakes

  • Using the Law of Sines for the ambiguous case (SSA) without checking for two possible solutions — when the given angle is acute and opposite a shorter side, two valid triangles may exist
  • Forgetting to convert degrees to radians when using trig functions in programming or scientific calculators set to radian mode
  • Applying the Pythagorean theorem to non-right triangles — use the Law of Cosines instead, which reduces to Pythagorean when the angle is 90°
  • Confusing altitude (height) with side length — the height must be perpendicular to the chosen base for the area formula ½·b·h

Frequently Asked Questions

How to find the hypotenuse of a right triangle?

Use the Pythagorean theorem c = √(a² + b²). For legs of 5 and 12, the hypotenuse is √(25 + 144) = √169 = 13.

When do you use the law of cosines vs. the law of sines?

Use the law of cosines when you know two sides and the included angle (SAS), or all three sides (SSS). Use the law of sines when you know a side and its opposite angle plus one more measurement (AAS or ASA).

What is the area formula for an equilateral triangle?

A = (√3/4)s². For a side length of 6, the area is (√3/4) × 36 = 9√3 ≈ 15.59 square units.

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