---
title: "Solids and Volumes"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 70
exercises: 9
source: https://one-course.com/books/math/1/en/chapter/70-solids-and-volumes
---

# Chapter 70 — Solids and Volumes

Space geometry begins with a small family of [solids](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) — [prisms](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism), [cylinders](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism), [pyramids](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def), [cones](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def), spheres — and two questions: how much do they hold ([volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume)), and what do you see when you slice them (sections)? The chapter closes with a fact of great practical importance: scaling a [solid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) by $k$ multiplies its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) by $k^3$.

## 70.1 The classical solids and their volumes

**Theorem 70.1 (Volume formulas).**

Write $B$ for the [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) of the base and $h$ for the height (the distance between the base and the opposite [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) or apex).

| [solid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) | [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) |
| --- | --- |
| [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) | $V = B \times h$ |
| [5pt] [pyramid](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def), [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) | $V = \dfrac13\, B \times h$ |
| [5pt] sphere of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $r$ | $V = \dfrac43\,\pi r^3$ |

For a [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) and a [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $r$, the base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) is $B = \pi r^2$; the sphere’s surface [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) is $4\pi r^2$.

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

![The solids of this chapter. For each, the volume involves a base area and a height — except the sphere, which only needs its radius.](https://one-course.com/images/onecourse/chapters/math-1/g9-solids/fig-22857620fd30.svg)

*The [solids](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of this chapter. For each, the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) involves a base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) and a height — except the sphere, which only needs its [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes).*

**Example 70.2.**

A cylindrical can has [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm and height $10$ cm:

$$
V = \pi r^2 h = \pi \times 16 \times 10 = 160\pi \approx 503 \text{ cm}^3
\approx 0.5 \text{ L}.
$$

A [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) with the same base and height holds one third of that: $\frac{160\pi}{3} \approx 168$ cm$^3$. A sphere of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm: $V = \frac43 \pi \times 64 = \frac{256\pi}{3} \approx 268$ cm$^3$.

**Example 70.3 (A pyramid step by step).**

A [pyramid](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) has a square base of side $6$ m and height $5$ m.

1. Base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) : $B = 6^2 = 36$ m $^2$ .
2. [Volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) : $V = \frac13 \times 36 \times 5 = 60$ m $^3$ .

Mind the units: if the side were given in cm and the height in m, one of them would have to be converted first.

## 70.2 Sections by planes

**Proposition 70.4 (Sections of the classical solids).**

Cutting a [solid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) by a plane produces a flat figure, its *cross-section*:

- a [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) cut *[parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to its base* gives a copy of the base, at every height;
- a [pyramid](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) or [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) cut [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to its base gives a *reduction* of the base: at distance $d$ from the apex, the [scale factor](https://one-course.com/books/math/1/en/chapter/68-thales-theorem#def-g9-thales-scaling) is $k = \frac{d}{h}$ ;
- a sphere of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $r$ cut by a plane at distance $d < r$ from the center gives a circle of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $\sqrt{r^2 - d^2}$ (by the Pythagorean theorem).

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

![Left: cutting a cone parallel to its base at distance d from the apex gives a disk scaled by k = dh. Right: a plane at distance d from the center of a sphere cuts it in a circle of radius √r2 - d2.](https://one-course.com/images/onecourse/chapters/math-1/g9-solids/fig-9433cc5ad2f1.svg)

*Left: cutting a [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to its base at distance $d$ from the apex gives a disk scaled by $k = \frac dh$. Right: a plane at distance $d$ from the center of a sphere cuts it in a circle of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $\sqrt{r^2 - d^2}$.*

**Example 70.5.**

A [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) has base [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $6$ and height $9$. The section by a plane [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to the base, at distance $3$ from the apex, is a disk scaled by $k = \frac39 = \frac13$: its [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) is $6 \times \frac13 = 2$.

A sphere of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $5$ is cut by a plane at distance $3$ from its center: the section is a circle of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $\sqrt{25 - 9} = 4$.

## 70.3 Scaling solids

**Theorem 70.6 (Effect of a scaling on lengths, areas, volumes).**

When a [solid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) is enlarged or reduced by the [scale factor](https://one-course.com/books/math/1/en/chapter/68-thales-theorem#def-g9-thales-scaling) $k > 0$:

- all lengths are multiplied by $k$ ;
- all [areas](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) are multiplied by $k^2$ ;
- all [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) are multiplied by $k^3$ .

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

**Example 70.7.**

A model car at [scale](https://one-course.com/books/math/1/en/chapter/49-proportionality#def-g7-prop-scale) $\frac{1}{18}$: lengths are those of the real car multiplied by $k = \frac{1}{18}$, painted surface multiplied by $k^2 = \frac{1}{324}$, and [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) by $k^3 = \frac{1}{5832}$.

Doubling the [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) of a sphere ($k = 2$) multiplies its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) by $2^3 = 8$: check on the formula, $\frac43\pi (2r)^3 = \frac43 \pi \times 8r^3$.

**Example 70.8 (Truncated cone).**

A [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) of height $9$ and base [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $6$ ([volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) $\frac13 \pi \times 36 \times 9 = 108\pi$) is cut at distance $3$ from the apex, and the small [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) above the cut is removed. The small [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) is the reduction by $k = \frac13$, so its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) is $108\pi \times \left(\frac13\right)^3 = 4\pi$. The remaining [solid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) (a *truncated [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def)*) has [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) $108\pi - 4\pi = 104\pi \approx 327$.

## 70.4 Exercises

**Exercise 70.1 ★.**

Compute the [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume): a [box](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) (rectangular [prism](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism)) of dimensions $4 \times 5
\times 12$; a [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ and height $7$ (exact value with $\pi$, then rounded to the unit).

**Solution of Exercise 70.1.**

[Box](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def): $V = 4 \times 5 \times 12 = 240$.

[Cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism): $V = \pi r^2 h = \pi \times 9 \times 7 = 63\pi \approx 198$.

**Exercise 70.2 ★.**

Compute the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of a [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $5$ and height $12$, and of a [pyramid](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) with rectangular base $8 \times 3$ and height $10$.

**Solution of Exercise 70.2.**

[Cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def): $V = \frac13 \pi r^2 h = \frac13 \pi \times 25 \times 12 =
100\pi \approx 314$.

[Pyramid](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def): base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) $B = 8 \times 3 = 24$, so $V = \frac13 \times 24 \times 10 = 80$.

**Exercise 70.3 ★.**

Compute the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) and the surface [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) of a sphere of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $6$ (exact values with $\pi$).

**Solution of Exercise 70.3.**

[Volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume): $V = \frac43 \pi \times 6^3 = \frac43 \pi \times 216 = 288\pi$.

Surface [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area): $4\pi \times 6^2 = 144\pi$.

**Exercise 70.4 ★.**

A sphere of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $10$ is cut by a plane at distance $8$ from its center. What is the [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) of the section circle?

**Solution of Exercise 70.4.**

[Radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) of the section: $\sqrt{r^2 - d^2} = \sqrt{100 - 64} = \sqrt{36}
= 6$.

**Exercise 70.5 ★★.**

A [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) has base [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $8$ and height $12$. A plane [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to the base cuts it at distance $9$ from the apex. Compute the [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) of the section, then the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of the small [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) above the cut.

**Solution of Exercise 70.5.**

[Scale factor](https://one-course.com/books/math/1/en/chapter/68-thales-theorem#def-g9-thales-scaling) from the whole [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) to the small one: $k = \frac{9}{12} = \frac34$. Section [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes): $8 \times \frac34 = 6$.

[Volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of the whole [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def): $\frac13\pi \times 64 \times 12 = 256\pi$. [Volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of the small [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def): $256\pi \times \left(\frac34\right)^3 =
256\pi \times \frac{27}{64} = 108\pi \approx 339$.

**Exercise 70.6 ★★.**

A cylindrical glass of inner [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ cm contains water to a height of $10$ cm. A ball of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $2$ cm is fully submerged in it. By how much does the water level rise? (The added [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) spreads over the glass’s base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area); give the exact rise, then [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) to the millimeter.)

**Solution of Exercise 70.6.**

[Volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of the ball: $\frac43\pi \times 2^3 = \frac{32\pi}{3}$ cm$^3$. This [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) spreads over the base [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) $\pi \times 3^2 = 9\pi$ cm$^2$, so the level rises by

$$
\frac{32\pi/3}{9\pi} = \frac{32}{27} \approx 1.2 \text{ cm}
$$

(about $12$ mm).

**Exercise 70.7 ★★.**

A recipe fills a spherical mold of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $6$ cm. You only have a spherical mold of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ cm. How many small molds can you fill? (Answer without computing either [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) explicitly.)

**Solution of Exercise 70.7.**

The small mold is the reduction of the large one by $k = \frac36 =
\frac12$, 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 large one: the recipe fills $8$ small molds.

**Exercise 70.8 ★★.**

The Great [Pyramid](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) of Giza has a square base of side about $230$ m and a height of about $147$ m. Estimate its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume), and express it in millions of cubic meters.

**Solution of Exercise 70.8.**

$V = \frac13 \times 230^2 \times 147 = \frac13 \times 52\,900 \times 147
= 52\,900 \times 49 = 2\,592\,100$ m$^3$ — about $2.6$ million cubic meters.

**Exercise 70.9 ★★★.**

A cone-shaped funnel of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $6$ cm and height $12$ cm, apex down, is filled with water up to [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of its *height* (measured from the apex).

1. What [fraction](https://one-course.com/books/math/1/en/chapter/63-fractions-and-powers#def-g9-fractions-fraction) of the funnel’s [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) is filled?
2. If instead it is filled with [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of its *[volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume)* of water, show that the water height $d$ satisfies $\left(\frac{d}{12}\right)^3 = \frac12$ , and give $d$ to the millimeter ( $\sqrt[3]{0.5} \approx 0.7937$ ).

**Solution of Exercise 70.9.**

*1.* The water forms a [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) scaled by $k = \frac12$ (apex down, [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the height), 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 funnel’s: one eighth, much less than [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)!

*2.* Water up to height $d$ forms a [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) scaled by $k = \frac{d}{12}$, of [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) $k^3$ times the funnel’s. [Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) means $k^3 = \frac12$, i.e. $\left(\frac{d}{12}\right)^3 = \frac12$. Then $\frac{d}{12} = \sqrt[3]{0.5} \approx 0.7937$, so $d \approx 9.5$ cm: the second [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) occupies only the top $2.5$ cm — the wide part of the [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) holds most of the water.

## 70.5 Problem: Archimedes’ tombstone

**Problem 70.1.**

Weekend problem — the sphere is two thirds of its cylinder (twice over), the 1:2:3 stack, and the square–cube law that forbids giants

Archimedes proved many theorems, but one made him so proud that he asked for its *figure* to be carved on his tomb: a sphere nested in its tightest [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism). A century later the Roman writer Cicero, searching the brambles near Syracuse, recognized the grave “by the sphere and the [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism)”. This problem retrieves what the carving encodes — a double two-thirds miracle — then follows [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) and surfaces ([Theorem 70.1](#thm-g9-solids-volumes), [Theorem 70.6](#thm-g9-solids-scaling)) to a law that governs giants, ants and cooling planets.

**Part I — The carving decoded.** A sphere of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $r$ sits exactly inside a [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism): same [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes), height $2r$.

1. Compute 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) in terms of $r$ .
2. Compute the ratio of the sphere’s [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) to the [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) ’s. Why might Archimedes have liked that the answer contains no $\pi$ and no $r$ ?
3. Now the surfaces: compare the sphere’s surface [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) with the [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) ’s *total* surface (lateral part plus the two lids). What ratio appears — again?
4. The empty space between sphere and [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) has [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) $2\pi r^3 - \frac43\pi r^3$ . Show that this leftover exactly equals the [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) of *two [cones](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def)* of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $r$ and height $r$ .
5. A basketball of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $12$ cm is sold in the tightest cylindrical [box](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) . Compute both [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) (to the nearest $100$ cm $^3$ ) and the percentage of the [box](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) that is empty.

**Part II — Bowls, molds and moons.**

6. A hemispherical bowl has [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $10$ cm. Compute its [capacity](https://one-course.com/books/math/1/en/chapter/12-money-and-measures#def-g2-measure-units) , in cm $^3$ and in liters (to the deciliter).
7. The 1:2:3 stack: a [cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) , a hemisphere and a [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) , all of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $10$ cm and height $10$ cm. Compute the three [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) and verify the legendary proportion $1 : 2 : 3$ . (Archimedes would have appreciated this one, too.)
8. A chocolate sphere of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ cm is melted into a cylindrical mold of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ cm. What height does the chocolate reach? (Exact [fraction](https://one-course.com/books/math/1/en/chapter/63-fractions-and-powers#def-g9-fractions-fraction) , then to the millimeter.)
9. The Earth’s [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) is about $6\,371$ km, the Moon’s about $1\,737$ km. Compute the ratio of the radii, then — with [Theorem 70.6](#thm-g9-solids-scaling) — the ratio of the [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) : how many Moons would fit into a hollow Earth, by [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) ?
10. From the same theorem: what is the Earth–Moon ratio of *surface [areas](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area)* ? And in general, when a balloon’s [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) doubles, what happens to its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) and to its surface? State the *square–cube law* : as a shape [scales](https://one-course.com/books/math/1/en/chapter/49-proportionality#def-g7-prop-scale) up, [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) outrun surfaces.

**Part III — The square–cube law rules the world.**

11. Galileo’s argument against giants: imagine a human scaled up by a factor $10$ , same proportions. By what factor does the weight grow (weight follows [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) )? By what factor does the [cross-section](#prop-g9-solids-sections) of the bones grow (an [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) )? By what factor, then, does the *pressure* on each square centimeter of bone grow — and what happens to the giant?
12. The same law in reverse explains ant heroics: strength follows muscle [cross-section](#prop-g9-solids-sections) (an [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) ), weight follows [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) . For an animal $100$ times smaller in every direction, by what factors do weight and strength shrink, and by what factor does the strength-to-weight ratio improve?
13. Heat is produced by the body’s [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) and lost through its surface. Show that for a sphere the surface-to-volume ratio is $\frac{3}{r}$ , and compute it for $r = 1$ and $r = 2$ . Which cools faster, a mouse or a bear — and why do babies need hats in winter?
14. Two spherical oranges of radii $4$ cm and $5$ cm are sold at the same price. Compute the ratio of their [volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) : how much more orange does the big one give for the same money?
15. Finale: describe precisely what the tombstone figure encodes — the two “two thirds” of questions 2 and 3 — and answer Cicero in one sentence: why would a mathematician choose, over every conquest of his engineering genius, a sphere in a [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) ?

**Solution of Problem 70.1.**

**1.** $V_{\text{cyl}} = \pi r^2 \times 2r = 2\pi r^3$.

**2.** $\dfrac{V_{\text{sphere}}}{V_{\text{cyl}}}
= \dfrac{\frac43 \pi r^3}{2 \pi r^3} = \dfrac23$. The $\pi$ and the $r^3$ cancel: the proportion is *universal* — true for a marble and for a planet. A relation between shapes, not between numbers: exactly the kind of truth worth carving in stone.

**3.** Sphere: $4\pi r^2$. [Cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism): lateral $2\pi r \times 2r = 4\pi r^2$, plus two lids $2 \times \pi r^2$: total $6\pi r^2$. Ratio: $\frac{4\pi r^2}{6\pi r^2} = \frac23$ — the same two thirds, for surfaces. (And a bonus: the sphere’s [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) exactly equals the [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism)’s lateral [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area).)

**4.** Leftover: $2\pi r^3 - \frac43\pi r^3 =
\frac23 \pi r^3$. Two [cones](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def) of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $r$ and height $r$: $2 \times \frac13 \pi r^2 \times r = \frac23 \pi r^3$. Equal.

**5.** Ball: $\frac43 \pi \times 12^3 \approx 7\,200$ cm$^3$; [box](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def): $2\pi \times 12^3 \approx 10\,900$ cm$^3$. Empty: one third of the [box](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), about $33\,\%$ — guaranteed by question 2, whatever the ball’s size.

**6.** [Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) a sphere: $\frac12 \times \frac43 \pi \times 10^3 = \frac23 \pi \times
1\,000 \approx 2\,094$ cm$^3 \approx 2.1$ L.

**7.** [Cone](https://one-course.com/books/math/1/en/chapter/62-pyramids-and-cones#def-g8-solids-def): $\frac13 \pi \times 10^2 \times 10 =
\frac{1\,000\pi}{3} \approx 1\,047$ cm$^3$. Hemisphere: $\frac{2\,000\pi}{3} \approx 2\,094$ cm$^3$. [Cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism): $1\,000\pi \approx 3\,142$ cm$^3$. Ratios: $\frac{1000\pi}{3} : \frac{2000\pi}{3} : \frac{3000\pi}{3}
= 1 : 2 : 3$ exactly.

**8.** $\frac43 \pi \times 27 = \pi \times 9 \times h$ gives $h = \frac43 \times 3 = 4$ cm exactly.

**9.** Radii: $\frac{6\,371}{1\,737} \approx 3.67$. [Volumes](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) [scale](https://one-course.com/books/math/1/en/chapter/49-proportionality#def-g7-prop-scale) as the [cube](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ([Theorem 70.6](#thm-g9-solids-scaling)): $3.67^3 \approx 49$. About fifty Moons fit in the Earth.

**10.** Surfaces [scale](https://one-course.com/books/math/1/en/chapter/49-proportionality#def-g7-prop-scale) as the square: $3.67^2 \approx
13.5$. Doubling a balloon’s [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) multiplies its [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) by $2^3 = 8$ but its surface by only $2^2 = 4$: as things grow, [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume) (weight, content, heat produced) outruns surface (skin, material, heat lost) — the square–cube law.

**11.** Weight: $\times 10^3 = 1\,000$. Bone [cross-section](#prop-g9-solids-sections): $\times 10^2 = 100$. Pressure — weight per [area](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-area) of bone: $\times \frac{1000}{100} = 10$. Bones built for human pressure receive ten times more: the giant’s skeleton snaps under its own weight. Giants are geometrically impossible; large animals need disproportionately thick bones (compare an elephant’s legs with a gazelle’s).

**12.** Weight: $\div 100^3 = 10^6$. Strength (muscle [cross-section](#prop-g9-solids-sections)): $\div 100^2 = 10^4$. Strength-to-weight: $\times \frac{10^6}{10^4} = 100$. The ant lifting fifty times its weight is not a super-athlete — it is merely small; a human shrunk to ant size could do likewise.

**13.** $\dfrac{S}{V} = \dfrac{4\pi r^2}{\frac43\pi r^3}
= \dfrac{3}{r}$: for $r = 1$ the ratio is $3$, for $r = 2$ it is $1.5$ — halving the size doubles the surface available per unit of heat-producing [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume). Small bodies bleed heat: the mouse must eat constantly, the bear can hibernate, and the baby — small sphere, large $\frac Sr$ — needs the hat.

**14.** $\left(\frac54\right)^3 = \frac{125}{64} \approx
1.95$: the big orange holds nearly *twice* the fruit for the same price. Buy [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes).

**15.** The carving says: sphere $= \frac23$ of the [cylinder](https://one-course.com/books/math/1/en/chapter/53-areas-and-volumes#def-g7-areas-prism) in [volume](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#def-g6-measure-volume), *and* $\frac23$ of it in total surface — two exact, universal proportions linking the roundest [solid](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) to the simplest one. Cicero’s answer: because a theorem is the only monument that neither armies nor centuries erode — Archimedes’ machines burned with Syracuse, but the two thirds are still exactly two thirds.