Surface Area and Volume of Solids

Understanding how to calculate the surface area and volume of 3D shapes is essential in geometry. Volume measures the amount of space inside a solid, while Surface Area measures the total area of the solid's outer surfaces.

Prisms and Cylinders

Prisms and cylinders have two parallel, congruent bases.

• Prism:
  • Volume: $V = Bh$
  • Surface Area: $SA = 2B + Ph$ (Where $B$ is the base area, $P$ is the base perimeter, and $h$ is the height)
• Cylinder:
  • Volume: $V = \pi r^2 h$
  • Surface Area: $SA = 2\pi r^2 + 2\pi rh$

Pyramids and Cones

Pyramids and cones have one base and taper to a point (apex).

• Pyramid:
  • Volume: $V = \frac{1}{3}Bh$
  • Surface Area: $SA = B + \frac{1}{2}Pl$ (Where $l$ is the slant height)
• Cone:
  • Volume: $V = \frac{1}{3}\pi r^2 h$
  • Surface Area: $SA = \pi r^2 + \pi rl$ (Slant height $l = \sqrt{r^2 + h^2}$)

Spheres

A sphere is perfectly round, with every point on its surface equidistant from the center.

• Volume: $V = \frac{4}{3}\pi r^3$
• Surface Area: $SA = 4\pi r^2$

Composite Solids

Composite solids are made by combining two or more basic shapes.

• To find the volume, simply add (or subtract) the volumes of the individual solids.
• To find the surface area, add the surface areas of the exposed faces. Be careful not to include the hidden areas where the shapes overlap!

Example Problems

Example 1: Find the volume and surface area of a cone with radius $5$ and height $12$.

1. First, find the slant height ($l$) using the Pythagorean theorem: $l = \sqrt{r^2 + h^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
2. Calculate Volume: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (25)(12) = 100\pi$
3. Calculate Surface Area: $SA = \pi r^2 + \pi rl = \pi(25) + \pi(5)(13) = 25\pi + 65\pi = 90\pi$

Example 2: A sphere has a surface area of $100\pi$. Find its volume.

1. Find the radius using the surface area formula: $4\pi r^2 = 100\pi$ $r^2 = 25 \implies r = 5$
2. Calculate Volume: $V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (5)^3 = \frac{500\pi}{3}$