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Geometric Dimensions

Common formulas for calculating areas, volumes, and perimeters of geometric shapes.

Triangles

Triangles are one of the most fundamental shapes in geometry with various properties based on their angles and sides.

Right Triangle (Pythagorean Theorem)

Pythagorean Theorem

a² + b² = c²

Where:

  • a = Base length
  • b = Height length
  • c = Hypotenuse length

Area of a Right Triangle

A = (1/2)ab

Where:

  • A = Area
  • a = Base length
  • b = Height length
abca² + b² = c²

45 degree sling angle

45 degree sling angle

c = a√2

Where:

  • a = Equal sides length (legs)
  • c = Hypotenuse length

Area of 45° Right Triangle

A = (1/2)a²

Where:

  • A = Area
  • a = Equal sides length (legs)
45°45°abba = b√2

Equilateral Triangle (60°)

Area of Equilateral Triangle

A = (√3/4)a²

Where:

  • A = Area
  • a = Side length

Height of Equilateral Triangle

h = (√3/2)a

Where:

  • h = Height
  • a = Side length

All Angles

∠A = ∠B = ∠C = 60°

Where:

  • ∠A, ∠B, ∠C = All angles equal 60 degrees
60°60°60°aaaArea = (√3/4)a²

30 degree sling angle

30 degree sling angle

Angles: 120° + 30° + 30° = 180°

Where:

  • 120° = Obtuse angle
  • 30° = Each base angle

Law of Cosines (for side a)

a² = 2b² - 2b²cos(120°)

Where:

  • a = Base length
  • b = Equal sides length
  • cos(120°) = -0.5 (negative value)

Simplified Relation

a² = 2b² - 2b²(-0.5) = 2b² + b² = 3b²

Where:

  • a = Base length
  • b = Equal sides length

Relation between a and b

a = b√3

Where:

  • a = Base length
  • b = Equal sides length
120°30°30°abba² = 2b² - 2b²cos(120°)

Basic Shapes

Common formulas for basic 2D and 3D geometric shapes.

Square

Area of a Square

A = s²

Where:

  • A = Area
  • s = Side length

Perimeter of a Square

P = 4s

Where:

  • P = Perimeter
  • s = Side length
ssArea = s²

Cube

Volume of a Cube

V = s³

Where:

  • V = Volume
  • s = Side length

Surface Area of a Cube

SA = 6s²

Where:

  • SA = Surface Area
  • s = Side length
s³ss

Circle

Area of a Circle

A = πr²

Where:

  • A = Area
  • r = Radius
  • π = Pi (≈ 3.14159)

Circumference of a Circle

C = 2πr

Where:

  • C = Circumference
  • r = Radius
  • π = Pi (≈ 3.14159)
rπr²

Cylinder

Volume of a Cylinder

V = πr²h

Where:

  • V = Volume
  • r = Radius of base
  • h = Height
  • π = Pi (≈ 3.14159)

Lateral Surface Area

LSA = 2πrh

Where:

  • LSA = Lateral Surface Area
  • r = Radius
  • h = Height
  • π = Pi (≈ 3.14159)

Total Surface Area

TSA = 2πr(r+h)

Where:

  • TSA = Total Surface Area
  • r = Radius
  • h = Height
  • π = Pi (≈ 3.14159)
rhπr²h

Sphere

Volume of a Sphere

V = (4/3)πr³

Where:

  • V = Volume
  • r = Radius
  • π = Pi (≈ 3.14159)

Surface Area of a Sphere

SA = 4πr²

Where:

  • SA = Surface Area
  • r = Radius
  • π = Pi (≈ 3.14159)
r(4/3)πr³
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