Geometry / Unit 11: Area, Surface Area and Volume
11.4 Surface Area of Pyramids and Cones
Notes
The slant height $\ell$ is measured along the face, from the apex down to the base edge. It is longer than the height $h$.
- Regular pyramid: $\displaystyle SA = \frac{1}{2}P\ell + B$.
- Cone: $SA = \pi r \ell + \pi r^2$. Lateral area is $\pi r \ell$.
- Find $\ell$ with the Pythagorean Theorem: for a cone, $\displaystyle \ell = \sqrt{r^2 + h^2}$.
- Example: a cone with $r = 3$ and $h = 4$ has $\ell = 5$, so $SA = 15\pi + 9\pi = 24\pi$.
Practice
A square pyramid has a base edge of 10 and a slant height of 13. Find its surface area.
Four triangles: $\displaystyle 4 \cdot \frac{1}{2}(10)(13) = 260$. Base: 100. Total 360.
A cone has radius 5 and slant height 13. Find its surface area.
$\pi(5)(13) + \pi(5)^2 = 65\pi + 25\pi = 90\pi$.
A cone has radius 6 and height 8. Find its slant height.
$\displaystyle \sqrt{36 + 64} = 10$.
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