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Why the volume formulas are what they are, and how to build a bisector and a parallel with tools that cannot measure.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you find out where the volume formulas come from rather than only how to use them: Cavalieri's principle, which says that two solids with matching cross-sections hold the same amount however different they look, and the one third that a pyramid or a cone takes of the prism or cylinder around it. Then you build figures with a straight edge and a compass, and justify each construction by saying what the compass has made equal, rather than by pointing at the drawing.
You can find the area of a circle and of a polygon, and you know that a prism's volume is its base area times its height. Everything below is that one sentence, plus a reason for the factor of one third and a reason why a solid may lean without changing size.
Cross-section: the flat shape you get by slicing a solid horizontally at some height.
Cavalieri's principle: two solids of the same height whose cross-sections have equal areas at every level have equal volumes.
Prism: a solid with the same cross-section all the way up; a cylinder is a prism with a circular one.
Pyramid and cone: a base tapering to a single point — one third of the prism or cylinder on the same base with the same height.
Oblique: leaning, rather than standing square. An oblique prism has the same volume as the upright one beside it.
Exact form: a volume left as $36\pi$ rather than rounded.
Cavalieri's principle is the foundation: if two solids stand between the same two levels and their cross-sections at every height have equal areas, they hold the same amount. That is why a leaning stack of coins holds as much as an upright one, and why an oblique prism uses the same $\text{base} \times \text{height}$ as an upright one. A pyramid or cone holds exactly one third of the prism or cylinder on the same base with the same height, so a square pyramid with base side $6$ and height $10$ holds $\frac{1}{3} \times 36 \times 10 = 120$, and a cone inside a cylinder of $450$ holds $150$. A sphere of radius $r$ holds $\frac{4}{3}\pi r^3$, so radius $3$ gives $36\pi$; a hemisphere is half of that.
Another way: picture
Two stacks of the same coins side by side, one upright and one pushed over into a lean. At every height the two slices are identical circles, so the two stacks hold the same amount.
Another way: story
Fill a cone with water and pour it into the cylinder of the same base and height. It takes three cones exactly — which is the whole reason for the one third, and the only part of these formulas that has to be shown rather than argued.
Halving instead of thirding. A cone is one third of its cylinder, not one half. Three cones of water fill the cylinder, and it is worth watching that once so the number stops being arbitrary.
Cubing only part of the radius. In $\frac{4}{3}\pi r^3$ with $r = 3$, the $r^3$ is $27$ and the volume is $36\pi$. Cube the whole radius before anything else touches it.
Thinking a leaning solid holds less. An oblique prism has exactly the volume of the upright one with the same base and height: the slices are the same size all the way up, just moved sideways. That is the whole content of Cavalieri's principle.
The prism on the same base with the same height holds $36 \times 10 = 360$.
Base area times height.
A pyramid is one third of it: $360 \div 3 = 120$.
$r^3 = 3 \times 3 \times 3 = 27$.
Cube the whole radius.
$\frac{4}{3} \times 27 = 36$.
The volume is $36\pi$, and a hemisphere of the same radius is $18\pi$.
Exact, with the $\pi$ kept.
A cone is one third of its cylinder.
$600 \div 3 = 200$.
A pyramid has a square base of side $8$ and height $9$. What is its volume?
answer
A cylinder holds $177$ cubic centimetres. A cone with the same base and the same height holds how much?
answer
A sphere has radius $3$. Its volume is $k\pi$: what is $k$?
Answer:
Two solids are both $7$ cm tall, and at every height between the two ends their horizontal cross-sections have equal area. What follows?
A swimming pool is $28$ m long and $12$ m wide. Its floor slopes evenly from $1$ m deep at the shallow end to $3$ m deep at the other. How many cubic metres of water does it hold when full?
Answer:
You know that a perpendicular bisector is the line at right angles to a segment through its middle, that an angle bisector cuts an angle in half, and that a triangle with three equal sides is equilateral. Constructing them is those definitions carried out with two instruments that can do only two things.
Straight edge: draws the line through two points. It has no marks on it, so it cannot measure.
Compass: draws every point at a fixed distance from a centre. That fixed distance is the only thing a compass guarantees, and every construction below rests on it.
Arc: the part of a circle a compass actually draws.
Equidistant: the same distance from two things. The perpendicular bisector of $AB$ is exactly the set of points equidistant from $A$ and $B$.
Justification: the reason a construction produces what it claims, said in terms of equal lengths — not because it looks right.
A compass guarantees one thing: every point of the arc it draws is the same distance from its centre. Every construction is that fact used carefully. To bisect a segment $AB$, draw arcs of equal radius from $A$ and from $B$; each crossing point is the same distance from both, so the line through them is the perpendicular bisector. To bisect an angle, draw one arc from the vertex to mark a point on each arm, then equal arcs from those two marks: their meeting point is equally far from both arms, so the ray to it splits the angle. To build an equilateral triangle on $AB$, draw arcs of radius $AB$ from $A$ and from $B$ — the meeting point is $AB$ from both, so all three sides are equal. And a regular hexagon appears when the radius is stepped round a circle, because each step makes an equilateral triangle with two radii, so each central angle is $60^\circ$ and six of them fill the turn.
Another way: picture
A segment $AB$ with two arcs of the same radius, one centred at $A$ and one at $B$, crossing above the segment and below it. The line through the two crossings meets $AB$ at its midpoint and at a right angle.
Another way: steps
Ask three questions of any construction. What is the compass keeping equal? Which point is therefore equidistant from what? And what does that force about the line drawn through it?
Changing the compass opening halfway through. The two arcs from the ends of a segment must have the same radius as each other, or their meeting point is not equidistant and the line through it is not the bisector. Set the opening once and leave it.
Justifying a construction by its picture. A drawing that looks right proves nothing. The justification is always a statement about equal lengths: this point is the same distance from $A$ as from $B$, so it is on the bisector.
Thinking the compass measures angles. It does not; it measures distance. The reason the radius of a circle steps round it exactly six times is that each step makes an equilateral triangle with two radii — an argument about lengths that happens to produce $60^\circ$.
The two arcs have the same radius, so each crossing point is that distance from $A$ and the same distance from $B$.
The compass keeps the radius fixed.
A point equidistant from $A$ and $B$ lies on the perpendicular bisector, and two such points determine it.
So the line through the two crossings is the perpendicular bisector.
No measuring, and nothing read off a drawing.
Each step joins two points of the circle that are one radius apart, so with the two radii to them it makes a triangle with three equal sides.
An equilateral triangle has angles of $60^\circ$, so each step turns $60^\circ$ at the centre.
$360 \div 60 = 6$: the radius fits exactly six times, giving a regular hexagon.
An argument about lengths, which produced an angle.
Equal radii make the meeting point the same distance from both marks on the arms.
That is what puts it on the bisector, so the ray from the vertex through it halves the angle.
$A$ is at $(5, 4)$ and $B$ at $(11, 6)$, and the line through them is called $ab$. Construct the perpendicular bisector of $AB$: name the point on it that is equally far from $A$ and $B$ **m**, and name the bisector itself **b**.
This task has no paper form; do it on a device.
$A$ is at $(3, -1)$, $B$ at $(4, 1)$ and the line through them is called $ab$. A third point $C$ sits at $(6, 1)$, off that line. Construct the line through $C$ parallel to $ab$, and name it **g**.
This task has no paper form; do it on a device.
An angle of $154^\circ$ is to be cut into two angles of $77^\circ$ with a compass and a straight edge. Put the steps in order.
Number the steps in order (write the number in the box):
To build an equilateral triangle on a segment $AB$ of length $12$, you draw an arc of radius $12$ centred at $A$ and another of radius $12$ centred at $B$, and join both ends to where they meet. Why does that give an equilateral triangle?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A pyramid has a square base of side $3$ and height $9$. What is its volume?
answer
$A$ is at $(-4, 0)$ and $B$ at $(4, 2)$, and the line through them is called $ab$. Construct the perpendicular bisector of $AB$: name the point on it that is equally far from $A$ and $B$ **m**, and name the bisector itself **b**.
This task has no paper form; do it on a device.
You can justify and use the volume formulas for prisms, pyramids, cones and spheres, and construct a perpendicular bisector and a parallel line. Explain why a leaning stack holds as much as an upright one, and what a compass actually guarantees about the arc it draws.
8. Your turn: a cone whose cylinder holds $600$, step 2
20. Your turn: why do the two arcs in the angle bisector need equal radii?, step 2