Triangle Calculator

Calculate triangle area, perimeter, and angles

Triangle Calculation Methods

SSS: Calculate all angles from three sides
SAS: Calculate from two sides and the angle between them
ASA: Calculate from two angles and the side between them
SSA: Ambiguous case - may have multiple solutions
Right Triangle: Uses Pythagorean theorem
Area: Simple area calculation using base and height

How to Use the Triangle Calculator

Triangle Calculation Methods

1. SSS (Side-Side-Side)

  1. Enter all three side lengths
  2. The calculator will find all angles using the Law of Cosines
  3. View the complete triangle solution including area and perimeter

2. SAS (Side-Angle-Side)

  1. Enter two side lengths
  2. Enter the angle between those two sides (in degrees)
  3. The calculator will find the third side and remaining angles

3. ASA (Angle-Side-Angle)

  1. Enter two angles (in degrees)
  2. Enter the side length between those angles
  3. The calculator will find the remaining sides and angle

4. SSA (Side-Side-Angle) - Ambiguous Case

  1. Enter two side lengths
  2. Enter the angle opposite to the first side
  3. This may produce 0, 1, or 2 valid solutions
  4. The calculator will show all possible solutions

5. Right Triangle

  1. Enter two known sides
  2. Check the box if one side is the hypotenuse
  3. The calculator uses the Pythagorean theorem to find the third side

6. Area Calculation

  1. Enter the base length
  2. Enter the height (perpendicular distance from base to opposite vertex)
  3. Get the triangle area using the simple formula: Area = ½ × base × height

Understanding Triangle

Triangles are fundamental geometric shapes with three sides, three angles, and many practical applications. Understanding triangle calculations is essential for construction, engineering, navigation, and many other fields.

Triangle Properties

  • Angle Sum: The sum of all angles in any triangle is always 180°
  • Triangle Inequality: The sum of any two sides must be greater than the third side
  • Longest Side: The longest side is always opposite the largest angle
  • Shortest Side: The shortest side is always opposite the smallest angle

Types of Triangles

By Side Length:

  • Equilateral: All three sides are equal (all angles = 60°)
  • Isosceles: Two sides are equal (two angles are equal)
  • Scalene: All sides are different lengths (all angles are different)

By Angle Measure:

  • Acute: All angles are less than 90°
  • Right: One angle is exactly 90°
  • Obtuse: One angle is greater than 90°

The Ambiguous Case (SSA)

When given two sides and a non-included angle (SSA), the triangle may not be uniquely determined. This "ambiguous case" can result in zero, one, or two valid triangles depending on the relationship between the given measurements.

Real-World Applications

  • Construction: Roof trusses, structural supports, land surveying
  • Navigation: Triangulation for GPS and mapping
  • Engineering: Force analysis, mechanical design
  • Art and Design: Composition, perspective drawing
  • Sports: Field layouts, trajectory calculations

Formula & Calculation Method

Triangle Calculation Formulas

Law of Cosines

c² = a² + b² - 2ab cos(C)

Used to find a side when two sides and included angle are known

Law of Sines

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

Used to find angles or sides when some combinations are known

Pythagorean Theorem (Right Triangles)

c² = a² + b²

Where c is the hypotenuse and a, b are the legs

Area Formulas

Area = ½ × base × heightArea = ½ × a × b × sin(C)Area = √[s(s-a)(s-b)(s-c)] (Heron's Formula)

Where s = (a+b+c)/2 is the semiperimeter

Perimeter

Perimeter = a + b + c

Sum of all three side lengths

Frequently Asked Questions

Tips & Best Practices

  • Always check that your triangle satisfies the triangle inequality: each side must be less than the sum of the other two
  • For right triangles, use the dedicated right triangle mode for more accurate calculations
  • Remember that SSA (ambiguous case) might give you two different valid triangles
  • When measuring angles, ensure they're in degrees, not radians
  • The largest angle is always opposite the longest side
  • For construction projects, double-check your calculations with multiple methods
  • Use the area calculation for quick estimates when you only need the triangle's area
  • Equilateral triangles have all angles equal to 60° and can be used to check your calculations