Lesson 5 — Pyramids
Big idea: A pyramid has one polygonal base and triangular lateral faces meeting at an apex, and its volume is one-third of the base area times the altitude.
Words to know
| Word | Meaning |
|---|---|
| Pyramid | The solid formed by all segments connecting a point P outside a plane to the points of a polygonal region R in that plane. |
| Base | The polygonal face of a pyramid that lies in the plane of R. |
| Lateral faces | The triangular faces of a pyramid other than the base. |
| Lateral edges | The edges where the lateral faces meet; they all meet at the apex. |
| Apex | The common vertex of the lateral faces. |
| Altitude of a pyramid | The perpendicular segment from the apex to the plane of the base; also its length. |
Concepts to know
- A pyramid’s base can be any polygon; the lateral faces are triangles that meet at the apex.
- The altitude may lie inside the pyramid, outside it, or even be one of its lateral edges.
- A prism with the same base and altitude as a pyramid can be cut into three pyramids of equal volume.
- Therefore each pyramid has one-third the volume of that prism.
- Theorem 80: The volume of any pyramid is one-third of the product of the area of its base and its altitude: V = ⅓Bh.
- A square pyramid has a square base; the Great Pyramid is an example.
- The height of a pyramid is measured by the length of its altitude.
Formulas / rules
- Pyramid: V = ⅓Bh
- Square base: B = s²
- Rectangular base: B = lw
- Triangular base: B = ½bh
- B is the area of the base; h is the perpendicular altitude.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find the volume of a pyramid | Find the base area B, multiply by the altitude h, then divide by 3. |
| Find the base area | Use the formula for that polygon: square s², rectangle lw, triangle ½bh. |
| Find a missing altitude | Substitute V and B into V = ⅓Bh and solve for h. |
| Work with a square pyramid | Use B = s², then V = ⅓s²h. |
| Compare a pyramid and a prism | Same base and altitude: the pyramid is ⅓ the prism’s volume. |
| Find a lateral area | Add the areas of the triangular lateral faces; do not include the base. |
Watch out
- Do not forget the factor ⅓.
- The altitude is the perpendicular height, not the slant height of a lateral face.
- The base is one face of the pyramid, but it does not have to be the “bottom” face.
- The altitude can fall outside the pyramid; use the perpendicular distance to the base plane.
- Lateral area and total area are different: total area includes the base.