---
title: "Pyramids and Cones"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 62
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones
---

# Chapter 62 — Pyramids and Cones

After the [prisms](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) and [cylinders](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) of [Chapter 53](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#ch-g7-areas) — [solids](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) with constant cross-section — come the [solids](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) with a point: [pyramids](#def-g8-solids-def) and [cones](#def-g8-solids-def). Their [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) formula carries a famous factor $\frac13$, which this chapter makes plausible and uses; the study of [solids](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) continues with spheres and sections in [Chapter 70](https://one-course.com/books/math/1/en/chapter/70-solids-and-volumes#ch-g9-solids).

## 62.1 Describing pyramids and cones

**Definition 62.1 (Pyramid, cone).**

- A *pyramid* has a [polygon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) as its *base* and a point, the *apex* , joined to every [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of the base by triangular [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) . 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](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) (square, [equilateral triangle](https://one-course.com/books/math/1/en/chapter/41-triangles-and-quadrilaterals#def-g6-shapes-triangles), …) 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.](https://one-course.com/images/onecourse/chapters/math-1/g8-solids/fig-68ffa524b0ec.svg)

*A square-based [pyramid](#def-g8-solids-def) and a [cone](#def-g8-solids-def): one polygonal or circular base, one apex, and the height measured [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to the base.*

**Example 62.2 (Counting faces, edges, vertices).**

A [pyramid](#def-g8-solids-def) with a square base has $5$ [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ($1$ square $+ 4$ triangles), $8$ edges and $5$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def). With an $n$-sided base: $n + 1$ [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), $2n$ edges, $n + 1$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def). Check Euler’s little pattern: [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $+$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $=$ edges $+ 2$.

## 62.2 Volume

**Theorem 62.3 (Volume of a pyramid or cone).**

For a [pyramid](#def-g8-solids-def) or a [cone](#def-g8-solids-def) with base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) $B$ and height $h$:

$$
V = \frac{1}{3}\, B \times h
$$

— one third of the [prism](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) or [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) with the same base and height. For a [cone](#def-g8-solids-def) of base [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $r$: $V = \frac13 \pi r^2 h$.

**Proof.** *Admitted at this level.* ∎

**Remark 62.4 (Why one third?).**

Fill a hollow [pyramid](#def-g8-solids-def) with water and pour it into the [prism](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) of same base and height: it takes exactly three fills. Even better: a [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) can be cut into three identical [pyramids](#def-g8-solids-def), each having a [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) as base and one [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) as apex (try to picture it!). A general proof uses integration, a tool of the High School volume.

**Example 62.5.**

A [pyramid](#def-g8-solids-def) has a square base of side $5$ cm and height $9$ cm:

1. base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) : $B = 5^2 = 25$ cm $^2$ ;
2. [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) : $V = \frac13 \times 25 \times 9 = 25 \times 3 = 75$ cm $^3$ .

An ice-cream [cone](#def-g8-solids-def) of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ 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](#def-g8-solids-def), do not confuse the height $h$ (apex to center of the base, [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp)) with the *slant height* $s$ (apex to the rim). They are related by Pythagoras ([Theorem 58.1](https://one-course.com/books/math/1/en/chapter/58-the-pythagorean-theorem#thm-g8-pythagoras-direct)) in the [right triangle](https://one-course.com/books/math/1/en/chapter/41-triangles-and-quadrilaterals#def-g6-shapes-triangles) apex–center–rim:

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

A [cone](#def-g8-solids-def) with $r = 3$ and $s = 5$ therefore has height $h = \sqrt{25 - 9} = 4$, and [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) $\frac13 \pi \times 9 \times 4 = 12\pi$.

**Method 62.7 (Volume problems).**

1. Identify the [solid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ( [prism](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) / [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) : $V = Bh$ ; [pyramid](#def-g8-solids-def) / [cone](#def-g8-solids-def) : $V = \frac13 Bh$ );
2. compute the base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) $B$ first, as a separate step;
3. check that the height is [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to the base (use Pythagoras if the slant height is given);
4. multiply, keep units consistent, convert at the end if needed ( $1$ L $= 1000$ cm $^3$ ).

## 62.3 Nets

**Example 62.8 (Net of a pyramid).**

Unfolding a regular square-based [pyramid](#def-g8-solids-def) flattens it into its *[net](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-net)*: the square base with four identical [isosceles triangles](https://one-course.com/books/math/1/en/chapter/41-triangles-and-quadrilaterals#def-g6-shapes-triangles) attached to its sides. The triangles’ equal sides are the *lateral edges* of the [pyramid](#def-g8-solids-def) — their length is neither $h$ nor the slant height of a [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), so label carefully before cutting cardboard.

![The net of a square-based pyramid: fold the four triangles up until their apexes meet.](https://one-course.com/images/onecourse/chapters/math-1/g8-solids/fig-da4b2a2f34de.svg)

*The [net](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-net) of a square-based [pyramid](#def-g8-solids-def): fold the four triangles up until their apexes meet.*

## 62.4 Exercises

**Exercise 62.1 ★.**

How many [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), edges and [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) has a [pyramid](#def-g8-solids-def) with a triangular base (a *tetrahedron*)? With a hexagonal base?

**Solution of Exercise 62.1.**

Tetrahedron: $4$ [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), $6$ edges, $4$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ($4 + 4 = 6 + 2$). Hexagonal base: $7$ [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), $12$ edges, $7$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ($7 + 7 = 12 + 2$).

**Exercise 62.2 ★.**

Compute the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of a [pyramid](#def-g8-solids-def) with base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) $36$ cm$^2$ and height $10$ cm; of a [pyramid](#def-g8-solids-def) with rectangular base $6 \times 4$ cm and height $7.5$ cm.

**Solution of Exercise 62.2.**

$V = \frac13 \times 36 \times 10 = 120$ cm$^3$.

Rectangular base: $B = 24$ cm$^2$, so $V = \frac13 \times 24 \times 7.5 = 60$ cm$^3$.

**Exercise 62.3 ★.**

Compute the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of a [cone](#def-g8-solids-def) of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $5$ cm and height $9$ cm (exact with $\pi$, then rounded to the cm$^3$).

**Solution of Exercise 62.3.**

$V = \frac13 \pi \times 25 \times 9 = 75\pi \approx 236$ cm$^3$.

**Exercise 62.4 ★.**

A [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) and a [cone](#def-g8-solids-def) have the same [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm and the same height $12$ cm. Compute both [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume). What is their ratio?

**Solution of Exercise 62.4.**

[Cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism): $\pi \times 16 \times 12 = 192\pi \approx 603$ cm$^3$. [Cone](#def-g8-solids-def): $\frac{192\pi}{3} = 64\pi \approx 201$ cm$^3$. Ratio: the [cone](#def-g8-solids-def) is one third of the [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism).

**Exercise 62.5 ★.**

A [pyramid](#def-g8-solids-def) has [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) $56$ cm$^3$ and height $8$ cm. What is its base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area)? (Write the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) formula and solve.)

**Solution of Exercise 62.5.**

$56 = \frac13 \times B \times 8$, so $B = \frac{56 \times 3}{8} = 21$ cm$^2$.

**Exercise 62.6 ★.**

Sketch the [net](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-net) of a [cone](#def-g8-solids-def) (a disk for the base, and a sector of a larger disk for the lateral surface). Which measurement of the [cone](#def-g8-solids-def) is the [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) of that sector: the height $h$ or the slant height $s$?

**Solution of Exercise 62.6.**

The lateral surface unrolls into a disk sector whose [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) is the *slant height* $s$ (the distance from the apex to the rim — that [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) lies on the surface), not the height $h$.

**Exercise 62.7 ★★.**

The Louvre [pyramid](#def-g8-solids-def) has a square base of side about $35$ m and a height of about $22$ m. Estimate its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume), to the nearest hundred cubic meters.

**Solution of Exercise 62.7.**

$V = \frac13 \times 35^2 \times 22 = \frac13 \times 1225 \times 22
= \frac{26\,950}{3} \approx 8\,983$ m$^3$ — about $9\,000$ cubic meters.

**Exercise 62.8 ★★.**

A [cone](#def-g8-solids-def) has slant height $13$ cm and base [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $5$ cm. Compute its height, then its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) (exact with $\pi$).

**Solution of Exercise 62.8.**

Height: $h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ cm. [Volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume): $V = \frac13 \pi \times 25 \times 12 = 100\pi$ cm$^3$.

**Exercise 62.9 ★★.**

A conical glass of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm and height $9$ cm is filled to the brim, then poured into a cylindrical glass of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm. What height does the liquid reach in the [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism)?

**Solution of Exercise 62.9.**

Same [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes), so the [cone](#def-g8-solids-def)’s [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) is one third of the [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism)’s [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) *for the same height*: the liquid reaches $\frac{9}{3} = 3$ cm in the [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism).

**Exercise 62.10 ★★.**

An hourglass is made of two identical [cones](#def-g8-solids-def) ([radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $2$ cm, height $4.5$ cm) joined at their apexes. All the sand fills exactly one [cone](#def-g8-solids-def). Compute the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of sand (exact, then in cm$^3$ rounded to the tenth), and the [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) of the hourglass’s total inner [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) occupied by the sand.

**Solution of Exercise 62.10.**

Sand $=$ one [cone](#def-g8-solids-def): $V = \frac13 \pi \times 4 \times 4.5 = 6\pi
\approx 18.8$ cm$^3$. The hourglass holds two such [cones](#def-g8-solids-def), so the sand fills exactly [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the total inner [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume).

**Exercise 62.11 ★★★.**

A square-based [pyramid](#def-g8-solids-def) with base side $6$ cm has its apex directly above a corner of the base, at height $8$ cm.

1. Does the formula $V = \frac13 Bh$ still apply? (It does — the apex need not be centered. Compute $V$ .)
2. Compute the length of the longest lateral edge, from the apex to the opposite corner of the base (two Pythagoras steps: the base diagonal first).

**Solution of Exercise 62.11.**

*1.* Yes: the formula $V = \frac13 Bh$ holds for any position of the apex, as long as $h$ is the [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) distance to the base plane. $V = \frac13 \times 36 \times 8 = 96$ cm$^3$.

*2.* Base diagonal: $\sqrt{6^2 + 6^2} = 6\sqrt2$ cm. The longest lateral edge is the hypotenuse of a [right triangle](https://one-course.com/books/math/1/en/chapter/41-triangles-and-quadrilaterals#def-g6-shapes-triangles) with legs $6\sqrt2$ (in the base plane) and $8$ (vertical):

$$
\sqrt{(6\sqrt2)^2 + 8^2} = \sqrt{72 + 64} = \sqrt{136} \approx 11.7
\text{ cm}.
$$

## 62.5 Problem: A cube cut into three pyramids

**Problem 62.1.**

Weekend problem — where the factor $\frac13$ comes from, and what happens to a pyramid cut at half height

The remark after [Theorem 62.3](#thm-g8-solids-volume) claims that a [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) can be cut into three identical [pyramids](#def-g8-solids-def) — “try to picture it!”. This problem pictures it precisely, gets the famous factor $\frac13$ out of it honestly, then cuts the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) a second way into *six* [pyramids](#def-g8-solids-def), and finishes by slicing a [pyramid](#def-g8-solids-def) at [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) its height — with a result few people guess right on the first try.

Throughout, $ABCDEFGH$ is a [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of side $a$: base $ABCD$, top [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $EFGH$, with $E$ above $A$, $F$ above $B$, $G$ above $C$ and $H$ above $D$.

**Part I — Three [pyramids](#def-g8-solids-def) sharing an apex.**

1. The [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $G$ belongs to three [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) . List the three [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) that do *not* contain $G$ , and describe the three [pyramids](#def-g8-solids-def) obtained by taking each of them as a base, with apex $G$ every time. How many [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) , edges and [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) has each [pyramid](#def-g8-solids-def) ( [Example 62.2](#ex-g8-solids-count) )?
2. Explain why turning the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) by a third of a turn around its long diagonal $(AG)$ leaves the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) unchanged but shuffles the three bases of question 1 in a cycle — and why the three [pyramids](#def-g8-solids-def) are therefore identical copies of one another.
3. Admitting that the three [pyramids](#def-g8-solids-def) fill the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) without overlapping (a cardboard model is quite convincing), deduce the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of each one.
4. Check this against the formula of [Theorem 62.3](#thm-g8-solids-volume) : the [pyramid](#def-g8-solids-def) with base $ABCD$ and apex $G$ has its apex directly above a *corner* of its base, exactly as in [Exercise 62.11](#exo-g8-solids-11) . Identify its height, apply the formula, and compare.
5. For a [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of side $6$ cm, give the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of each of the three [pyramids](#def-g8-solids-def) — and explain what this decomposition has to do with the water-pouring experiment of the chapter (three fills of the [pyramid](#def-g8-solids-def) for one [prism](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) ).

**Part II — Six [pyramids](#def-g8-solids-def) sharing the center.** Now let $O$ be the center of the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), and join $O$ to the four [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of each of the six [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def).

6. Describe the six [pyramids](#def-g8-solids-def) obtained, and explain why they are identical copies of one another. What is the height of each one?
7. Deduce, by sharing the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ’s [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) , the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of each of the six [pyramids](#def-g8-solids-def) .
8. Recompute that [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) with the formula $V = \frac13 B h$ , and check the two answers agree.
9. The two decompositions confirm the factor $\frac13$ for some very special [pyramids](#def-g8-solids-def) . Explain why they do not yet *prove* [Theorem 62.3](#thm-g8-solids-volume) for every [pyramid](#def-g8-solids-def) and [cone](#def-g8-solids-def) — and say how the chapter deals with that gap: which statement is admitted, which experiment supports it, and in which [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of the series is it honestly proved.
10. A [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of side $6$ cm is cut into its six center [pyramids](#def-g8-solids-def) . Compute the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of each, both ways.

**Part III — Cutting a [pyramid](#def-g8-solids-def) at [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) height.** $SABCD$ is a regular square-based [pyramid](#def-g8-solids-def): base side $c$, height $h$, apex $S$ above the center of the base. Cut it by the plane through the *midpoints* of the four lateral edges $[SA]$, $[SB]$, $[SC]$, $[SD]$.

11. Apply the midpoint theorem ( [Theorem 59.1](https://one-course.com/books/math/1/en/chapter/59-midpoints-and-parallels#thm-g8-midpoints-direct) ) in the triangle $SAB$ : what are the direction and length of the [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) joining the midpoints of $[SA]$ and $[SB]$ ? Deduce that the cut is a square of side $\frac{c}{2}$ , [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to the base, at height $\frac{h}{2}$ .
12. The piece above the cut is itself a square-based [pyramid](#def-g8-solids-def) . Give its base side and height, and show that its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) is exactly *one eighth* of the original [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) $V$ .
13. Deduce the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of the lower piece (the *frustum* , the shape of an unfinished [pyramid](#def-g8-solids-def) ). In what ratio does the half-height cut share the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) ?
14. The Louvre [pyramid](#def-g8-solids-def) ( [Exercise 62.7](#exo-g8-solids-7) : base side $35$ m, height $22$ m) is cleaned in two campaigns: the glass above [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) height, then the glass below. What [fraction](https://one-course.com/books/math/1/en/chapter/39-fractions-first-steps#def-g6-fractions-def) of the *[volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume)* sits below [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) height? Estimate it in cubic meters, to the nearest hundred.
15. The general moral: if *all* the dimensions of a [pyramid](#def-g8-solids-def) are multiplied by $2$ , what happens to its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) ? By what factor does the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) grow when every dimension is multiplied by $k$ — and how does this explain, in one line, the “one eighth” of question 12?

**Solution of Problem 62.1.**

**1.** $G$ belongs to the top [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $EFGH$ and to the two side [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $BCGF$ and $DCGH$. The three [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) *not* containing $G$ are the bottom $ABCD$, the front $ABFE$ and the left [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $ADHE$ (they all share the [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $A$, the corner opposite $G$). The three [pyramids](#def-g8-solids-def) are $ABCDG$, $ABFEG$ and $ADHEG$: each has a square [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) as base and the far [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $G$ as apex. Each is a square-based [pyramid](#def-g8-solids-def): $5$ [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), $8$ edges, $5$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ([Example 62.2](#ex-g8-solids-count)).

**2.** The long diagonal $[AG]$ joins the only two [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) belonging to none of the three bases. A third-of-a-turn rotation about the line $(AG)$ sends the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) to itself (the three edges leaving $A$ — towards $B$, $D$, $E$ — are shuffled in a cycle, and so are the three edges arriving at $G$). It carries the base $ABCD$ to $ADHE$, then $ADHE$ to $ABFE$, and back: apexes fixed at $G$, bases cycling. Each [pyramid](#def-g8-solids-def) is thus carried onto the next one: the three are identical copies.

**3.** Three identical pieces filling the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) share its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) equally:

$$
V_{\text{pyramid}} = \frac{a^3}{3} .
$$

**4.** The [pyramid](#def-g8-solids-def) $ABCDG$ has base $ABCD$, of [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) $a^2$, and its apex $G$ sits vertically above the corner $C$, at height $CG = a$ (a vertical edge of the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def)). As in [Exercise 62.11](#exo-g8-solids-11), the formula applies with the [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) height $h = a$:

$$
V = \frac13 \times a^2 \times a = \frac{a^3}{3} ,
$$

in perfect agreement with question 3 — the decomposition and the formula confirm each other.

**5.** For $a = 6$: $V = \frac{6^3}{3} = \frac{216}{3} = 72$ cm$^3$ each. The water-pouring experiment is this decomposition made liquid: the [prism](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) over the base $ABCD$ with height $a$ is the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) itself, and it holds exactly three pyramid-fills — here the three fills [even](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) assemble geometrically into the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def).

**6.** Joining the center $O$ to the four corners of each [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) produces six [pyramids](#def-g8-solids-def), one per [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def): square base, apex $O$. The center is equally placed with respect to the six [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) (any rotation or symmetry of the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) fixes $O$ and permutes the [faces](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def)), so the six [pyramids](#def-g8-solids-def) are identical. The height of each is the distance from $O$ to a [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def): [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the side, $\frac{a}{2}$.

**7.** Six identical pieces share the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def): $V = \frac{a^3}{6}$ each.

**8.** Formula: base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) $a^2$, height $\frac a2$:

$$
V = \frac13 \times a^2 \times \frac{a}{2} = \frac{a^3}{6} .
$$

The two answers agree.

**9.** Both decompositions concern [pyramids](#def-g8-solids-def) of very special proportions carved from a [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def); a slender [pyramid](#def-g8-solids-def), a lopsided one, or a [cone](#def-g8-solids-def) cannot be assembled this way (no three [cones](#def-g8-solids-def) fill a [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism)). This is why [Theorem 62.3](#thm-g8-solids-volume) is *admitted* at this level: the decompositions and the water-pouring experiment make the $\frac13$ entirely believable, and the honest general proof — integration — is given in the High School volume.

**10.** $V = \frac{6^3}{6} = 36$ cm$^3$; and by the formula, $V = \frac13 \times 36 \times 3 = 36$ cm$^3$. They agree.

**11.** In the triangle $SAB$, the midpoints of the sides $[SA]$ and $[SB]$ are joined by a [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to $(AB)$ and of length $\frac{AB}{2} = \frac{c}{2}$ ([Theorem 59.1](https://one-course.com/books/math/1/en/chapter/59-midpoints-and-parallels#thm-g8-midpoints-direct)). The same holds on each lateral [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), so the four midpoints form a square of side $\frac c2$, with sides [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to the base. On the vertical line through the apex, the cut passes at the midpoint of the height (midpoint theorem again, in a triangle through $S$, the base center and a base [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def)): the cutting plane sits at height $\frac h2$.

**12.** The top piece is a regular square-based [pyramid](#def-g8-solids-def) of base side $\frac c2$ and height $\frac h2$ (its base is the cut, its apex is $S$). Its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) is

$$
\frac13 \times \left(\frac c2\right)^2 \times \frac h2
= \frac13 \times \frac{c^2}{4} \times \frac h2
= \frac{1}{8} \times \frac{c^2 h}{3}
= \frac{V}{8} :
$$

one eighth of the original — not one [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half).

**13.** The frustum keeps the rest: $V - \frac V8 = \frac{7V}{8}$. The half-height cut shares the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) in the ratio $1 : 7$ — the humble-looking bottom slab holds seven times the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of the pointed top.

**14.** Below [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) height lies $\frac78$ of the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume). With $V = \frac13 \times 35^2 \times 22 = \frac{26\,950}{3} \approx
8\,983$ m$^3$ ([Exercise 62.7](#exo-g8-solids-7)):

$$
\frac78 \times \frac{26\,950}{3} \approx 7\,860 \approx
7\,900 \text{ m}^3
$$

to the nearest hundred — the “lower [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)” campaign cleans almost eight times the glass [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of the upper one.

**15.** Doubling every dimension multiplies the base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) by $2^2 = 4$ and the height by $2$, hence the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) by $4 \times 2 = 8 = 2^3$. Scaling by $k$ multiplies [areas](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) by $k^2$ and one more length by $k$: [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) grow by the factor $k^3$. The top piece of question 12 is a scaled copy of the whole [pyramid](#def-g8-solids-def) with $k = \frac12$, so its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) is $\left(\frac12\right)^3 = \frac18$ of the whole — the one-line explanation.