Solid Geometric Shapes Reference

Complete reference guide for common 3D solid geometric shapes with their volume and surface area formulas.

Basic Shapes

Cube

Volume: $V = s³$

Surface Area: $SA = 6s²$

Space Diagonal: $d = s√3$

Rectangular Prism (Cuboid)

Volume: $V = lwh$

Surface Area: $SA = 2(lw + wh + lh)$

Space Diagonal: $d = √(l² + w² + h²)$

Sphere

Volume: $V = \frac{4}{3}πr³$

Surface Area: $SA = 4πr²$

Hemisphere

Volume: $V = \frac{2}{3}πr³$

Surface Area: $SA = 3πr²$ (including base)

Curved Surface: $CSA = 2πr²$

Cylinders and Cones

Circular Cylinder

Volume: $V = πr²h$

Surface Area: $SA = 2πr(r + h)$

Lateral Area: $LA = 2πrh$

Cone

Volume: $V = \frac{1}{3}πr²h$

Surface Area: $SA = πr(r + l)$

Slant Height: $l = √(r² + h²)$

Conical Frustum

Volume: $V = \frac{πh}{3}(R² + Rr + r²)$

Surface Area: $SA = π(R² + r² + (R+r)l)$

Slant Height: $l = √(h² + (R-r)²)$

Capsule

Volume: $V = πr²(\frac{4}{3}r + h)$

Surface Area: $SA = 2πr(2r + h)$

Pyramids and Prisms

Square Pyramid

Volume: $V = \frac{1}{3}s²h$

Surface Area: $SA = s² + 2sl$

Slant Height: $l = √(h² + (\frac{s}{2})²)$

Triangular Prism

Volume: $V = \frac{1}{2}abh$

Surface Area: $SA = ab + (a+b+c)h$

Pentagonal Prism

Volume: $V = \frac{5}{4}s²h√(5+2√5)$

Surface Area: $SA = 2A_{base} + P_{base} × h$

Other Shapes

Tube (Hollow Cylinder)

Volume: $V = πh(R² - r²)$

Surface Area: $SA = 2π(R+r)h + 2π(R²-r²)$

Ellipsoid

Volume: $V = \frac{4}{3}πabc$

Surface Area: Complex formula (no simple expression)

Torus

Volume: $V = 2π²Rr²$

Surface Area: $SA = 4π²Rr$

Octahedron

Volume: $V = \frac{√2}{3}a³$

Surface Area: $SA = 2√3a²$

Variable Definitions

Common Variables:

r = radius
R = larger radius
h = height
l = length or slant height
w = width
s = side length

Additional Variables:

a, b, c = semi-axes lengths
π = pi ≈ 3.14159
P = perimeter
A = area
V = volume
SA = surface area

Quick Reference Tips

Remember:

Volume is always measured in cubic units (unit³)
Surface area is always measured in square units (unit²)
Linear measurements (radius, height) are in base units
For composite shapes, break them down into basic components
Always check your units for consistency

Common Conversions:

1 cubic yard = 27 cubic feet
1 cubic meter = 1000 liters
1 cubic foot = 7.48 gallons (US)