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1. Cross sections and solids of rotation

Warm-up

Three quick questions from earlier lessons.

Discover

A loaf of bread shaped like a cylinder is sliced straight across. What shape is each slice?

A circle
A rectangle

Key idea

Solids from slices and from rotations.

Not to scale

A cross section is the shape where a plane cuts a solid. Slices parallel to the base of a prism or cylinder copy the base. Rotating a rectangle about a side gives a cylinder; a right triangle about a leg gives a cone; a semicircle about its diameter gives a sphere.

Worked example

A square pyramid.

Horizontal slice: a smaller square

Parallel to the base.

Vertical slice through the apex: a triangle

Through the apex.

Your turn, guided

Your turn: how many faces does a triangular prism have?

5

2 triangles + 3 rectangles.

Practice ★

A cylinder of radius 4 is sliced by a plane parallel to its base. What shape is the cross section?
1. A circle of radius 4
2. A rectangle
3. An ellipse

A 9 by 6 rectangle is rotated about one of its sides. Which solid does it sweep out?
1. A cylinder
2. A cone
3. A sphere

Watch out!

Practice ★★

A cone of height 3 is sliced by a vertical plane through its apex. What shape is the cross section?
1. An isosceles triangle
2. A circle
3. A rectangle

Exit ticket

Three questions to check you have got it.

A cylinder of radius 4 is sliced by a plane parallel to its base. What shape is the cross section?
1. A circle of radius 4
2. A rectangle
3. An ellipse
A cylinder of radius 2 is sliced by a plane parallel to its base. What shape is the cross section?
1. A circle of radius 2
2. A rectangle
3. An ellipse
A cylinder of radius 5 is sliced by a plane parallel to its base. What shape is the cross section?
1. A circle of radius 5
2. A rectangle
3. An ellipse

Summary card

Cross sections and rotations

Parallel slices copy the base; rectangle → cylinder, right triangle → cone, semicircle → sphere.

Watch out: the angle of the slice decides the shape.

2. Scaling solids: areas and volumes

Warm-up

Three quick questions from earlier lessons.

Discover

A 1 cm cube is enlarged to a 2 cm cube. How many small cubes fit inside the big one?

Not to scale

Key idea

Lengths, areas and volumes scale differently.

Under a scale factor

k

: lengths ×

k

, areas (including surface area) ×

k^2

, volumes ×

k^3

. Conversely, a volume ratio

V

gives scale factor

\sqrt[3]{V}

Worked example

A statue is copied at scale 3.

Paint needed: 9 times as much

Areas × 9.

Material needed: 27 times as much

Volumes × 27.

Your turn, guided

Your turn: volumes in ratio 64 : 1. The scale factor.

4

∛64.

Practice ★

A solid has volume 25 cm³. It is enlarged with scale factor 2. Find the new volume.
A solid has volume 5 cm³. It is enlarged with scale factor 3. Find the new volume.

A solid has surface area 49 cm². It is enlarged with scale factor 3. Find the new surface area.

Watch out!

Practice ★★

Two similar solids have volumes in the ratio $27 : 1$ . Find the ratio of their heights (scale factor).
Two similar solids have volumes in the ratio $125 : 1$ . Find the ratio of their heights (scale factor).

Exit ticket

Three questions to check you have got it.

A solid has volume 25 cm³. It is enlarged with scale factor 2. Find the new volume.
A solid has volume 5 cm³. It is enlarged with scale factor 3. Find the new volume.
A solid has volume 9 cm³. It is enlarged with scale factor 2. Find the new volume.

Summary card

Scaling solids

Length $k$ , area $k^2$ , volume $k^3$ .

Watch out: cube roots undo volume ratios.

3. Volume of spheres; density

Warm-up

Three quick questions from earlier lessons.

Discover

A ball of radius $r$ fits exactly in a cylinder of radius $r$ and height $2r$ . The ball fills $\tfrac{2}{3}$ of the cylinder. What is the ball's volume?

Not to scale

$\tfrac{4}{3}\pi r^3$
$4\pi r^2$

Key idea

Spheres, and how heavy a solid is.

Sphere:

V = \tfrac{4}{3}\pi r^3

(surface area

4\pi r^2

). Density = mass ÷ volume, so mass = density × volume.

Worked example

A steel ball of radius 3 cm, density 7.8 g/cm³.

V = 36\pi \approx 113.1

cm³

(4/3)π · 27.

m \approx 7.8 \times 113.1 \approx 882

Mass = density × volume.

Your turn, guided

Your turn: mass 200 g, volume 50 cm³. The density.

4

g/cm³

200 ÷ 50.

Practice ★

Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale

A block has mass 526 g and volume 27 cm³. Find its density (g/cm³).

Watch out!

Practice ★★

Find the volume of a hemisphere (half a ball) of radius 3 cm (exact, in terms of π).
Find the volume of a hemisphere (half a ball) of radius 5 cm (exact, in terms of π).

Exit ticket

Three questions to check you have got it.

Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale
Find the volume of a sphere of radius 8 cm (exact, in terms of π).Not to scale

Summary card

Spheres and density

$V = \tfrac{4}{3}\pi r^3$ ; density = mass/volume.

Watch out: $r^3$ for volume, $r^2$ for area.

4. Converting units of measurement

Warm-up

Three quick questions from earlier lessons.

Discover

A table is 2 meters long. About how many centimeters is that?

200
0.02

Key idea

Smaller units: more of them.

To convert, use the rate between the units (1 ft = 12 in, 1 yd = 3 ft, 1 m = 100 cm, 1 kg = 1000 g). Converting to a smaller unit, multiply; to a larger unit, divide.

Worked example

Convert 54 inches to feet.

54 \div 12 = 4.5

Feet are larger: divide.

Your turn, guided

Your turn: 3 kg in grams.

3000

× 1000.

Practice ★

How many inches are in 5 feet? (1 foot = 12 inches)
How many inches are in 2 feet? (1 foot = 12 inches)

How many meters are 600 centimeters? (1 m = 100 cm)

Watch out!

Practice ★★

A rope is 11 yards and 2 feet long. How long is it in feet? (1 yard = 3 feet)
A rope is 10 yards and 2 feet long. How long is it in feet? (1 yard = 3 feet)

Exit ticket

Three questions to check you have got it.

How many inches are in 5 feet? (1 foot = 12 inches)
How many inches are in 2 feet? (1 foot = 12 inches)
How many inches are in 7 feet? (1 foot = 12 inches)

Summary card

Converting units

To a smaller unit: multiply; to a larger unit: divide.

Watch out: check that the answer makes sense in size.

Answers: Modeling with geometry

1. Cross sections and solids of rotation

A circle of radius 4

A cylinder

An isosceles triangle

A circle of radius 4
A circle of radius 2
A circle of radius 5

2. Scaling solids: areas and volumes

$200$
$135$

$441$

$200$
$135$
$72$

3. Volume of spheres; density

$\frac{500 \pi}{3}$
$972 \pi$

$\frac{526}{27}$

$18 \pi$
$\frac{250 \pi}{3}$

$\frac{500 \pi}{3}$
$972 \pi$
$\frac{2048 \pi}{3}$

4. Converting units of measurement

$60$
$24$

$35$
$32$

$60$
$24$
$84$