Surface Area and Volume of 3D Solids
Surface Area and Volume of 3D Solids
Surface area is the total area of all outer surfaces. Volume is the total space enclosed. For pyramids, cones and spheres the formulae involve a third of the base area or a cube of the radius — these must be memorised (they are given on the AQA formula sheet, but working with them fluently requires familiarity).
Formulae
| Solid | Volume | Surface Area |
|---|---|---|
| Sphere | \( \frac{4}{3}\pi r^3 \) | \( 4\pi r^2 \) |
| Cone (right circular) | \( \frac{1}{3}\pi r^2 h \) | \( \pi r l + \pi r^2 \) where \( l=\sqrt{r^2+h^2} \) |
| Pyramid (any base) | \( \frac{1}{3} \times \text{base area} \times h \) | base area + sum of lateral face areas |
| Cylinder | \( \pi r^2 h \) | \( 2\pi r h + 2\pi r^2 \) |
Slant Height of a Cone
The slant height \( l \) connects the apex to the circumference of the base. It is found using Pythagoras:
\[ l = \sqrt{r^2 + h^2} \]The curved surface area uses \( l \), not \( h \). Do not confuse them.
Composite Solids
For solids built from standard parts (e.g. cylinder with hemispherical cap, cone on a cylinder), find each volume separately and add. For surface area, identify which faces are exposed — internal joins are not counted.
Worked Examples
Find the volume and surface area of a sphere with radius 6 cm.
\[ V = \frac{4}{3}\pi\times216 = 288\pi\approx905 \text{ cm}^3 \] \[ SA = 4\pi\times36 = 144\pi\approx452 \text{ cm}^2 \]A cone has base radius 5 cm and perpendicular height 12 cm. Find the volume and total surface area.
\[ l = \sqrt{25+144} = \sqrt{169} = 13 \text{ cm} \] \[ V = \frac{1}{3}\pi\times25\times12 = 100\pi\approx314 \text{ cm}^3 \] \[ SA = \pi\times5\times13 + \pi\times25 = 65\pi + 25\pi = 90\pi\approx283 \text{ cm}^2 \]A solid consists of a cylinder (radius 4 cm, height 10 cm) with a hemisphere (radius 4 cm) on top. Find the total volume.
\[ V_{\text{cyl}} = \pi\times16\times10 = 160\pi \] \[ V_{\text{hemi}} = \frac{1}{2}\times\frac{4}{3}\pi\times64 = \frac{128\pi}{3} \] \[ V_{\text{total}} = 160\pi + \frac{128\pi}{3} = \frac{480\pi+128\pi}{3} = \frac{608\pi}{3}\approx636 \text{ cm}^3 \]Key Takeaways
- Sphere: \( V=\frac{4}{3}\pi r^3 \), \( SA=4\pi r^2 \).
- Cone: \( V=\frac{1}{3}\pi r^2 h \), \( SA=\pi r l+\pi r^2 \), \( l=\sqrt{r^2+h^2} \).
- Pyramid: \( V=\frac{1}{3}\times\text{base area}\times h \).
- Composite solids: volumes add; surface areas require identifying exposed faces only.
- Slant height \( l \neq h \) — always use Pythagoras to find \( l \) from \( r \) and \( h \).