How Do You Find the Volume and Surface Area of a Pyramid, Cone, or Sphere?

Use the correct formula for the solid, substitute the perpendicular dimensions, and evaluate with your GDC. For surface area, include every exposed face and distinguish a cone’s curved surface area from its total surface area.

Formulas and Dimensions

This is SL 3.1-3D geometry in Mathematics: Applications and Interpretation. Let $B$ be the base area, $P$ the base perimeter, $h$ the perpendicular height, $l$ the slant height, and $r$ the radius.

Choosing the Formula

First, identify whether the question asks for volume, curved or lateral area, or total surface area. Draw attention to the dimensions: the height used in volume is measured perpendicular to the base, while a face height or slant height lies along the surface.

For a composite solid, calculate each component separately, remove hidden contact areas from the surface-area total, and then add the remaining exposed areas. Never include an internal circular base where a cone joins another solid. A quick dimensional check helps: area formulas involve squared lengths, whereas volume formulas involve cubed lengths.

For a cone, the slant height is not the perpendicular height. If it is not given, use Pythagoras:

$$l=\sqrt{r^2+h^2}.$$

For a pyramid, $\frac{1}{2}Pl$ applies when all triangular side faces share the same slant height. Otherwise, calculate each triangular face separately using $\frac{1}{2}\times\text{base}\times\text{face height}$.

Worked Example

A cone has radius $3\text{ cm}$ and perpendicular height $4\text{ cm}$. First find its slant height:

$$l=\sqrt{3^2+4^2}=5\text{ cm}.$$

Its volume is

$$V=\frac{1}{3}\pi(3)^2(4)=12\pi\approx37.7\text{ cm}^3.$$

Its total surface area is

$$A=\pi(3)^2+\pi(3)(5)=24\pi\approx75.4\text{ cm}^2.$$

A common misconception is using $l$ in the volume formula. Volume always uses the perpendicular height $h$, while the cone’s curved surface area uses $l$.

Exam Technique

Write the formula and substitution before the GDC value, keep full precision until the final answer, and normally give an exact answer or one correct to three significant figures. Surface area requires square units, such as $\text{cm}^2$, while volume requires cubic units, such as $\text{cm}^3$.