Geometric Shapes in Solid Geometry

Capsule

• Volume = πr²((4/3)r + a)
• Surface Area = 2πr(2r + a)
• Circumference = 2πr

Also called a stadium of revolution.

Circular Cone

• Volume = (1/3)πr²h
• Slant Height = √(r² + h²)
• Lateral Surface Area = πrs = πr√(r² + h²)
• Base Surface Area = πr²
• Total Surface Area
  = L + B = πrs + πr² = πr(s + r) = πr(r + √(r² + h²))

Circular Cylinder

• Volume = πr²h
• Lateral Surface Area = 2πrh
• Top Surface Area = πr²
• Bottom Surface Area = πr²
• Total Surface Area
  = L + T + B = 2πrh + 2(πr²) = 2πr(h+r)

Conical Frustum

• Volume = (1/3)πh (r₁² + r₂² + (r₁ * r₂))
• Slant Height = √((r₁ - r₂)² + h²)
• Lateral Surface Area
  = π(r₁ + r₂)s = π(r₁ + r₂)√((r₁ - r₂)² + h²)
• Top Surface Area = πr₁²
• Base Surface Area = πr₂²
• Total Surface Area
  = π(r₁² + r₂² + (r₁ * r₂) * s)
  = π[ r₁² + r₂² + (r₁ * r₂) * √((r₁ - r₂)² + h²) ]

Cube

• Volume = a³
• Surface Area = 6a²
• Face Diagonal (f) = a√2
• Diagonal (d) = a√3

Hemisphere

• Volume = (2/3)πr³
• Circumference = 2πr
• Curved Surface Area = 2πr²
• Base Surface Area = πr²
• Total Surface Area= (2πr²) + (πr²) = 3πr²

Pyramid : Square

• Volume = (1/3)a²h
• Slant Height (s) = √(h² + (1/4)a²)
• Lateral Surface Area = a√(a² + 4h²)
• Base Surface Area = a²
• Total Surface Area
  = L + B = a² + a√(a² + 4h²))
  = a(a + √(a² + 4h²))

Rectangular Prism

• Volume = lwh
• Surface Area = 2(lw + lh + wh)
• Diagonal (d) = √(l² + w² + h²)

Sphere

• Volume = (4/3)πr³
• Circumference = 2πr
• Surface Area = 4πr²

Spherical Cap

• Volume
  = (1/6)πh(3a² + h²)
  = (1/3)πh²(3R - h)
• Radius Base Circle = √h(2R - h)
• Circumference Base Circle = 2π√h(2R - h)
• Surface Area = 2πRh = π(a² + h²)

Spherical Segment

• Volume = (1/6)πh(3a² + 3b² + h²)
• Top Surface Area = πb²
• Bottom Surface Area = πa²
• Lateral Surface Area = 2πRh
• Where R = sphere radius and
• R = √{ [ [(a-b)² + h²] [(a+b)² + h²] ] / 4h² }

Tube

• Circumference, C:
  • C₁ = 2πr₁
  • C₂ = 2πr₂
• Lateral Surface Area, L, for a cylinder:
  • L₁ = 2πr₁h, the external surface area
  • L₂ = 2πr₂h, the internal surface area
• Area, A, for the end cross section of the tube:
  • A₁ = πr^₁2 for the area enclosed by C₁
  • A₂ = πr^₂2 for the area enclosed by C₂
• A = A₁ - A₂ = π(r^₁2 - r^₂2) for the area of the solid cross section of the tube, the end, an annulus.
• Volume, V, (using volume for a cylinder):
  • V₁ = πr^₁2h for the volume enclosed by C₁
  • V₂ = πr^₂2h for the volume enclosed by C₂
• V = V₁ - V₂ = π(r^₁2 - r^₂2)h for the volume of the solid, the tube.
• Thickness of the tube wall, t:
  • t = r₁ - r₂