Triangle Calculator

Calculate triangle properties including area, perimeter, sides, and angles using various methods. Supports SSS, SAS, ASA, and base-height calculations for all your geometry needs.

Triangle Calculator

Calculate triangle properties from sides, angles, or base and height

Enter valid values to calculate triangle properties. Make sure the triangle inequality theorem is satisfied: the sum of any two sides must be greater than the third side.

How to Use the Triangle Calculator

  1. Select Calculation Method: Choose from Base & Height, SSS (three sides), SAS (two sides and included angle), or ASA (two angles and included side)
  2. Enter Known Values: Input the measurements you have - the calculator adapts fields based on your selected method
  3. Review Results: Get comprehensive results including area, perimeter, all sides, all angles, and triangle type classification
  4. Verify Triangle Validity: The calculator checks triangle inequality theorem to ensure entered values create a valid triangle

Triangle Calculation Methods

Base and Height Method

Simplest calculation requiring only base length and perpendicular height. Area equals half the base multiplied by height. This method works for any triangle when you know these two measurements. Commonly used in basic geometry and real-world applications like land surveying.

Formula:
Area = (base × height) / 2
Example: Triangle with base 10 and height 6 has area of 30 square units.

SSS Method (Three Sides)

When you know all three side lengths, the calculator uses law of cosines to find angles and Heron's formula for area. This method completely defines the triangle. Essential for construction, engineering, and surveying where you can measure distances but not angles directly.

Requirements:
• All three sides must be positive
• Must satisfy triangle inequality: a + b > c
• Calculates all angles using law of cosines

SAS Method (Two Sides, One Angle)

Given two sides and the angle between them, the calculator determines the third side using law of cosines, then finds remaining angles. The included angle must be between the two known sides. This method appears frequently in navigation and physics problems involving vectors.

Key Points:
• Angle must be between the two sides
• Angle must be greater than 0° and less than 180°
• Uses law of cosines for third side

ASA Method (Two Angles, One Side)

With two angles and the side between them, the third angle is found by subtracting from 180°. Law of sines then calculates remaining sides. This method works well when angles are easier to measure than sides, such as in astronomy and optical measurements.

Validation:
• Sum of angles must be less than 180°
• Each angle must be positive
• Uses law of sines for remaining sides

Triangle Types and Properties

Equilateral Triangle

All three sides equal length and all angles measure 60 degrees. Most symmetric triangle type. Common in nature, architecture, and engineering due to structural stability.

Properties:
• All sides equal: a = b = c
• All angles 60°
• Height = (side × √3) / 2
• Area = (side² × √3) / 4

Isosceles Triangle

Two sides equal length with two equal angles opposite those sides. Line from apex to base creates two mirror-image right triangles. Appears in roof trusses and bridge construction.

Properties:
• Two sides equal length
• Two angles equal measure
• Symmetrical along altitude
• Base angles are equal

Scalene Triangle

All three sides different lengths with all angles different measures. Most general triangle form without special properties or symmetries. Requires full calculation for area and angles.

Properties:
• All sides different: a ≠ b ≠ c
• All angles different
• No lines of symmetry
• Can be acute, right, or obtuse

Right Triangle

One angle exactly 90 degrees with sides following Pythagorean theorem. Most important triangle in practical applications. Foundation of trigonometry and appears throughout mathematics and engineering.

Properties:
• One 90° angle
• Pythagorean: a² + b² = c²
• Area = (leg₁ × leg₂) / 2
• Basis for trigonometry

Acute Triangle

All three angles less than 90 degrees. Every equilateral triangle is also acute. Triangle appears wider and more spread out compared to obtuse triangles. All altitudes fall inside the triangle.

Properties:
• All angles less than 90°
• a² + b² > c² for all sides
• Circumcenter inside triangle
• All altitudes inside triangle

Obtuse Triangle

One angle greater than 90 degrees with remaining two angles acute. Longest side opposite the obtuse angle. Two altitudes fall outside the triangle. Common in architecture for dramatic angular designs.

Properties:
• One angle greater than 90°
• a² + b² < c² (c is longest)
• Circumcenter outside triangle
• Two altitudes outside triangle

Important Triangle Formulas

Heron's Formula (Area)

s = (a + b + c) / 2
Area = √[s(s-a)(s-b)(s-c)]

Calculate area when you know all three sides. Variable s represents semi-perimeter (half the perimeter).

Law of Cosines

c² = a² + b² - 2ab·cos(C)

Relates three sides and one angle. Generalizes Pythagorean theorem to all triangles, not just right triangles.

Law of Sines

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

Ratio of side length to sine of opposite angle stays constant. Essential for solving ASA and AAS triangle problems.

Pythagorean Theorem

a² + b² = c²

Only applies to right triangles where c is hypotenuse. Foundation of distance calculations and coordinate geometry.

Triangle Inequality

a + b > c
b + c > a
a + c > b

Sum of any two sides must exceed third side. Validates whether three lengths can form a triangle.

Angle Sum Property

A + B + C = 180°

Interior angles always sum to 180 degrees. If you know two angles, third angle equals 180° minus their sum.

Real-World Applications

Construction and Architecture

Roof trusses use triangles for structural strength. Calculating rafter lengths and angles requires triangle geometry. Triangular bracing prevents building deformation under load.

Navigation and Surveying

GPS systems triangulate position using satellite distances. Land surveyors measure property boundaries with angle and distance readings. Mariners use triangulation for coastal navigation.

Physics and Engineering

Force vectors resolve into components using right triangles. Bridge designs analyze stress through triangular structures. Projectile motion calculations employ triangle trigonometry.

Computer Graphics

3D models consist of millions of triangular polygons. Game engines calculate lighting and shadows using triangle normals. Screen coordinates transform through triangular matrices.

Astronomy

Stellar distances determined through parallax triangulation. Planetary positions calculated using orbital triangles. Telescope aiming requires precise angular calculations.

Sports and Recreation

Pool players angle shots using triangle geometry. Golfers estimate distances combining elevation angles with horizontal range. Sailing routes optimize using wind angle triangulation.

Frequently Asked Questions

What information do I need to calculate triangle area?

You need either base and height, all three sides (using Heron's formula), or two sides and the included angle. Most simple method uses base multiplied by height divided by two.

Can any three lengths form a triangle?

No, the triangle inequality theorem must be satisfied. Sum of any two sides must be greater than the third side. For example, sides 3, 4, 10 cannot form a triangle because 3 + 4 is less than 10.

How do I find the third angle if I know two angles?

Subtract the sum of known angles from 180 degrees. Interior angles of any triangle always total exactly 180 degrees. If two angles are 60° and 70°, third angle equals 180 - 60 - 70 = 50°.

What's the difference between altitude and median?

Altitude is perpendicular line from vertex to opposite side, used for area calculations. Median connects vertex to midpoint of opposite side, used for centroid location. They only coincide in isosceles and equilateral triangles.

When should I use law of sines vs law of cosines?

Use law of cosines when you know SSS or SAS (includes finding third side). Use law of sines for ASA, AAS, or SSA scenarios. Law of cosines works for any combination but requires more calculation for some cases.

Why are triangles the strongest shape in construction?

Triangle is only polygon that cannot deform without changing side lengths. Rectangles collapse into parallelograms under stress, but triangles remain rigid. This property makes triangular trusses essential in bridges and buildings.

Related Calculators

Geometry Calculator

Calculate area, perimeter, and properties of various geometric shapes.

Try calculator

Scientific Calculator

Perform advanced calculations with trigonometric and logarithmic functions.

Try calculator

Algebra Calculator

Solve algebraic equations and simplify expressions with step-by-step solutions.

Try calculator