Geometric Shapes - Areas

Square

 

The area of a square can be calculated as

A = a²                            (1a)

The side of a square can be calculated as

a = A^1/2                            (1b)

The diagonal of a square can be calculated as

d = a 2^1/2                        (1c)

Rectangle

The area of a rectangle can be calculated as

A = a b          (2a)

The diagonal of a rectangle can be calculated as

d = (a² + b²)^1/2     (2b)

Parallelogram

The area of a parallelogram can be calculated as

A = a h

  = a b sin α                        (3a)

The diameters of a parallelogram can be calculated as

d₁ = ((a + h cot α)² + h²)^1/2                      (3b)

d₂ = ((a - h cot α)² + h²)^1/2                     (3b)

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are equal.

The area of an equilateral triangle can be calculated as

A = a²/3 3^1/2                                 (4a)

The area of an equilateral triangle can be calculated as

h = a/2 3^1/2                              (4b)

Triangle

The area of a triangle can be calculated as

A = a h / 2  

  = r s                                 (5a)

r = a h / 2s                          (5b)

R = b c / 2 h                        (5c)

s = (a + b + c) / 2                     (5d)

x = s - a                           (5e)

y = s - b                           (5f)

z = s - c                          (5g)

Trapezoid

The area of a trapezoid can be calculated as

A = 1/2 (a + b) h  

  = m h                           (6a)

m = (a + b) / 2                      (6b)

Hexagon

The area of a hexagon can be calculated as

A = 3/2 a² 3^1/2                             (7a)

d = 2 a 

  =  2 / 3^1/2 s 

  = 1.1547005 s                              (7b)

s = 3^1/2 / 2 d  

   = 0.866025 d                              (7c)

Circle

 

The area of a circle can be calculated as

A = π/4 d²

  = π r² 

  = 0.785.. d²                       (8a)

C = 2 π r 

  =  π d                           (8b)

where

C = circumference

• How many circles fits within a rectangle?
• How many smaller circles fits witin a larger circle?

Sector and Segment of a Circle

Sector of Circle

Area of a sector of circle can be expressed as

A = 1/2 θ_r r²                            (9)

= 1/360 θ_d π r²

where

θ_r = angle in radians

θ_d = angle in degrees

Segment of Circle

Area of a segment of circle can be expressed as

A = 1/2 (θ_r - sin θ_r) r²

= 1/2 (π θ_d/180 - sin θ_d) r²                             (10)

Right Circular Cylinder

Lateral surface area of a right circular circle can be expressed as

A = 2 π r h                                      (11)

where

h = height of cylinder (m, ft)

r = radius of base (m, ft)

Right Circular Cone

Lateral surface area of a right circular cone can be expressed as

A = π r l

= π r (r² + h²)^1/2                                   (12)

where

h = height of cone (m, ft)

r = radius of base (m, ft)

l = slant length (m, ft)

Sphere

Lateral surface area of a sphere can be expressed as

A = 4 π r²                                    (13)