Areas of Geometric Shapes

Square

A = a²     (1a)

a = A^1/2    (1b)

d = a 2^1/2    (1c)

Rectangle

A = a b          (2a)

d = (a² + b²)^1/2     (2b)

Parallelogram

A = a h 

  = a b sin α       (3a)

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

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

Equilateral Triangle

A = a²/3 3^1/2    (4a)

h = a/2 3^1/2     (4b)

Triangle

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

A = 1/2 (a + b) h  

  = m h       (6a)

m = (a + b) / 2      (6b)

Hexagon

A = 3/2 a² 3^1/2      (7a)

d= 2 a 

  =  2 / 3^1/2 s 

  = 1.155 s      (7b)

s = 3^1/2 / 2 d  

   = 0.866 d      (7c)

Circle

A = π/4 d²

  = π r² 

  = 0.785.. d²        (8a)

U = 2 π r 

  =  π d      (8b)

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)