Volumes & Surface Areas of Common Solids

Introduction to Volume & Surface Area

Volume is the measure of three-dimensional space occupied by a substance or enclosed by a surface.

The standard SI (International System of Units) unit for volume is the cubic meter, denoted as $m^3$.

Other common units include:

  • Cubic centimeter ($cm^3$): $1 cm^3 = 10^{-6} m^3$
  • Liter (L): $1 L = 1 dm^3 = 1000 cm^3 = 0.001 m^3$
  • Milliliter (mL): $1 mL = 1 cm^3$

Surface area is the measure of the total area that the surface of the object occupies.

Solid Calculators

Enter the required dimensions for each solid shape to calculate its volume and total surface area. Each step of the calculation will be shown below.

Cuboid (Rectangular Prism)

Volume: $V = lwh$

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

Results will appear here with step-by-step calculations.

Cylinder

Volume: $V = \pi r^2 h$

Total Surface Area: $A = 2\pi r h + 2\pi r^2 = 2\pi r (h+r)$

Results will appear here with step-by-step calculations.

Triangular Prism

Base Triangle Area ($A_b$ using Heron's if sides a,b,c given, or $\frac{1}{2} \times \text{base} \times \text{height}$)

Volume: $V = A_b \times L$ (where L is prism length/height)

Surface Area: $A = 2A_b + (P_b \times L)$ (where $P_b$ is perimeter of base triangle)

Enter side lengths of the triangular base (a, b, c) and prism length (L).

Results will appear here with step-by-step calculations.

Square Pyramid

Volume: $V = \frac{1}{3} \times (\text{base side})^2 \times h$

Slant Height ($s_l$): $\sqrt{h^2 + (\frac{\text{base side}}{2})^2}$

Surface Area: $A = (\text{base side})^2 + 2 \times \text{base side} \times s_l$

Results will appear here with step-by-step calculations.

Cone

Slant Height ($s_l$): $\sqrt{r^2 + h^2}$

Volume: $V = \frac{1}{3}\pi r^2 h$

Total Surface Area: $A = \pi r (r + s_l)$

Results will appear here with step-by-step calculations.

Sphere

Volume: $V = \frac{4}{3}\pi r^3$

Surface Area: $A = 4\pi r^2$

Results will appear here with step-by-step calculations.

Frustum of Square Pyramid

Volume: $V = \frac{1}{3}h(A_1 + A_2 + \sqrt{A_1 A_2})$ where $A_1 = B^2, A_2 = b^2$

Slant Height ($s_l$): $\sqrt{h_f^2 + (\frac{B-b}{2})^2}$ (where $h_f$ is frustum height)

Surface Area: $A = A_1 + A_2 + \frac{1}{2}(P_1 + P_2)s_l$ (where $P_1=4B, P_2=4b$)

Results will appear here with step-by-step calculations.

Frustum of a Cone

Volume: $V = \frac{1}{3}\pi h (R^2 + Rr + r^2)$

Slant Height ($s_l$): $\sqrt{h^2 + (R-r)^2}$

Total Surface Area: $A = \pi (R^2 + r^2 + s_l(R+r))$

Results will appear here with step-by-step calculations.

Zone of a Sphere (Spherical Cap)

h r_cap R

Volume: $V = \frac{1}{3}\pi h^2 (3R - h)$ (where R is sphere radius, h is cap height)

Curved Surface Area: $A_{curved} = 2\pi R h$

Results will appear here with step-by-step calculations.

Frustum of a Sphere (Two Bases)

r1 r2 h

Volume: $V = \frac{1}{6}\pi h (3r_1^2 + 3r_2^2 + h^2)$ (where $r_1, r_2$ are radii of bases, $h$ is height of frustum)

Curved Surface Area: $A_{curved} = 2\pi R h$ (where R is sphere radius, $h$ is frustum height)

Note: Sphere radius R needed for curved area. If not given, it can be complex to find from $r_1, r_2, h$. This calculator assumes R is known for SA.

Results will appear here with step-by-step calculations.

Prismoidal Rule

The prismoidal rule (or Simpson's first rule for volumes) is a numerical method for approximating the volume of various solids. It is particularly useful for solids with parallel end faces and a varying cross-section.

The formula is: $V = \frac{h}{6}(A_1 + 4A_m + A_2)$

Where:
  • $V$ is the volume of the solid.
  • $h$ is the perpendicular distance between the two parallel end faces ($A_1$ and $A_2$).
  • $A_1$ is the area of one end face.
  • $A_2$ is the area of the other end face.
  • $A_m$ is the area of the cross-section midway between $A_1$ and $A_2$.

This rule gives exact volumes for prisms, cylinders, pyramids, cones, spheres, and frusta of pyramids/cones if the areas are correctly determined.

Volume by Prismoidal Rule

A1 A2 h Am

Formula: $V = \frac{h}{6}(A_1 + 4A_m + A_2)$

Results will appear here with step-by-step calculations.