Geometric Shapes in Solid Geometry

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Geometric Shapes Formulas

Capsule

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

Also called a stadium of revolution.

Circular Cone

Circular Cone shape
Circular Cone Calculator
  • 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

Circular Cylinder shape
Circular Cylinder Calculator
  • 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

Conical Frustum shape
Conical Frustum Calculator
  • 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

Cube Solid Diagram with a side length, f face diagonal and d solid diagonal
Cube Calculator
  • Volume = a3
  • Surface Area = 6a2
  • Face Diagonal (f) = a√2
  • Diagonal (d) = a√3

Hemisphere

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

Pyramid : Square

Pyramid Diagram with h = height, l = length and w = width and s = slant height
Pyramid Calculator
  • 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

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

Sphere

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

Spherical Cap

Sphere Cap with radius R, height h
Spherical Cap Formulas
  • 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

Spherical segment
Spherical Segment Formulas
  • 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

Tube or Hollow Cylinder
Tube Calculator (Hollow Cylinder)
  • 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 = πr12 for the area enclosed by C1
    • A2 = πr22 for the area enclosed by C2
    • A = A1 - A2 = π(r12 - r22) for the area of the solid cross section of the tube, the end, an annulus.
  • Volume, V, (using volume for a cylinder):
    • V1 = πr12h for the volume enclosed by C1
    • V2 = πr22h for the volume enclosed by C2
    • V = V1 - V2 = π(r12 - r22)h for the volume of the solid, the tube.
  • Thickness of the tube wall, t:
    • t = r1 - r2
 

Cite this content, page or calculator as:

Furey, Edward "Geometric Shapes and Formulas in Solid Geometry" From http://www.CalculatorSoup.com - Online Calculator Resource.