What is Pyramid, cone? — Mathematics

Definition 62.1 Primary & Middle School Mathematics · Chapter 62 — Pyramids and Cones

A pyramid has a polygon as its base and a point, the apex, joined to every vertex of the base by triangular faces. Its height is the distance from the apex to the plane of the base.
A cone is the same construction over a disk: a base disk, an apex, and a curved lateral surface.

A pyramid is regular when its base is a regular polygon (square, equilateral triangle, …) and its apex sits vertically above the center of the base.

A square-based pyramid and a cone: one polygonal or circular base, one apex, and the height measured perpendicular to the base.

Examples

Example 62.2 (Counting faces, edges, vertices)

A pyramid with a square base has $5$ faces ( $1$ square $+ 4$ triangles), $8$ edges and $5$ vertices. With an $n$ -sided base: $n + 1$ faces, $2n$ edges, $n + 1$ vertices. Check Euler’s little pattern: faces $+$ vertices $=$ edges $+ 2$ .

Example 62.5

A pyramid has a square base of side $5$ cm and height $9$ cm:

base area: $B = 5^2 = 25$ cm $^2$ ;
volume: $V = \frac13 \times 25 \times 9 = 25 \times 3 = 75$ cm $^3$ .

An ice-cream cone of radius $3$ cm and height $10$ cm:

V = \frac13 \times \pi \times 3^2 \times 10 = 30\pi \approx 94 \text{ cm}^3 .

Example 62.6 (Careful with the height)

For a cone, do not confuse the height $h$ (apex to center of the base, perpendicular) with the slant height $s$ (apex to the rim). They are related by Pythagoras (Theorem 58.1) in the right triangle apex–center–rim:

s^2 = h^2 + r^2 .

A cone with $r = 3$ and $s = 5$ therefore has height $h = \sqrt{25 - 9} = 4$ , and volume $\frac13 \pi \times 9 \times 4 = 12\pi$ .

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Examples

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