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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 prisms and cylinders

Warm-up

Three quick questions from earlier lessons.

Discover

A stack of identical coins forms a cylinder. If each coin has area $A$ and the stack is $h$ high, what is its volume?

$A \times h$
$A + h$

Key idea

Base area times height.

For any prism or cylinder:

V = B \cdot h

, with

B

the area of the base. Cylinder:

V = \pi r^2 h

Worked example

A cylinder with $r = 3$ , $h = 10$ .

Not to scale

B = 9\pi

Base area.

V = 90\pi \approx 282.7

Times height.

Your turn, guided

Your turn: a rectangular prism 4 by 5 by 6.

120

4 × 5 × 6.

Practice ★

Find the volume of a cylinder with radius 3 cm and height 4 cm (exact, in terms of π).Not to scale
Find the volume of a cylinder with radius 8 cm and height 11 cm (exact, in terms of π).Not to scale

A prism has a triangular base with base 2 cm and height 7 cm, and the prism is 7 cm long. Find its volume (cm³).Not to scale

Watch out!

Practice ★★

A cylinder of radius 5 cm has volume $175\pi$ cm³. Find its height (cm).Not to scale
A cylinder of radius 6 cm has volume $360\pi$ cm³. Find its height (cm).Not to scale

Exit ticket

Three questions to check you have got it.

Find the volume of a cylinder with radius 3 cm and height 4 cm (exact, in terms of π).Not to scale
Find the volume of a cylinder with radius 8 cm and height 11 cm (exact, in terms of π).Not to scale
Find the volume of a cylinder with radius 5 cm and height 5 cm (exact, in terms of π).Not to scale

Summary card

Prisms and cylinders

$V = Bh$ ; cylinder $V = \pi r^2 h$ .

Watch out: base area, not circumference.

4. Volume of pyramids and cones

Warm-up

Three quick questions from earlier lessons.

Discover

A cone and a cylinder have the same base and height. How many cones of water fill the cylinder?

Not to scale

Tilt them to compare. How many full cones of water fill the cylinder?

Key idea

Pointed solids hold one third.

For a pyramid or cone:

V = \tfrac{1}{3}Bh

. Cone:

V = \tfrac{1}{3}\pi r^2 h

Worked example

A square pyramid, base 6 m, height 10 m.

B = 36

Base area.

V = \tfrac{1}{3} \cdot 36 \cdot 10 = 120

m³

One third of Bh.

Your turn, guided

Your turn: a cone with $r = 3$ , $h = 4$ (in terms of π).

12\pi

⅓ · 9π · 4.

Practice ★

Find the volume of a cone with radius 6 cm and height 6 cm (exact, in terms of π).Not to scale
Find the volume of a cone with radius 6 cm and height 12 cm (exact, in terms of π).Not to scale

A pyramid has a square base of side 4 m and height 15 m. Find its volume (m³).Not to scale

Watch out!

Practice ★★

A cone and a cylinder have the same radius 5 and the same height. What fraction of the cylinder's volume is the cone's?

Boss problem. An ice cream cone has radius 3 cm and height 8 cm, topped with a hemisphere of ice cream of the same radius. Give exact answers in terms of π.

Exit ticket

Three questions to check you have got it.

Find the volume of a cone with radius 6 cm and height 6 cm (exact, in terms of π).Not to scale
Find the volume of a cone with radius 6 cm and height 12 cm (exact, in terms of π).Not to scale
Find the volume of a cone with radius 4 cm and height 9 cm (exact, in terms of π).Not to scale

Summary card

Pyramids and cones

$V = \tfrac{1}{3}Bh$ .

Watch out: do not forget the ⅓.

5. 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.

6. 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.

7. Percent error

Warm-up

Three quick questions from earlier lessons.

Discover

A 50 cm board is measured as 52 cm. Which describes the error?

A 4% error
A 2% error

Key idea

Percent error compares the error with the actual value.

Percent error

= \dfrac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100

. A tolerance of

p\%

allows values within

p\%

of the target, above or below.

Worked example

A 400 g bag with at most 5% error.

20

5% of 400.

From 380 g to 420 g

Range.

Your turn, guided

Your turn: actual 40, measured 38. The percent error.

5

2 ÷ 40 × 100.

Practice ★

A board is really 36 cm long, but Ana measures 40 cm. What is her percent error?
A board is really 73 cm long, but Ana measures 76 cm. What is her percent error?

A box really holds 400 cups, but a label says 520. What is the percent error?

Watch out!

Practice ★★

A bag should hold 600 g of rice, with at most 7% error. What is the smallest allowed mass (g)?
A bag should hold 400 g of rice, with at most 8% error. What is the smallest allowed mass (g)?

Exit ticket

Three questions to check you have got it.

A board is really 36 cm long, but Ana measures 40 cm. What is her percent error?
A board is really 73 cm long, but Ana measures 76 cm. What is her percent error?
A board is really 52 cm long, but Ana measures 53 cm. What is her percent error?

Summary card

Percent error

Error ÷ actual × 100.

Watch out: divide by the actual value, not the measured one.

Answers: Solid 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 prisms and cylinders

$36 \pi$
$704 \pi$

$49$

$7$
$10$

$36 \pi$
$704 \pi$
$125 \pi$

4. Volume of pyramids and cones

$72 \pi$
$144 \pi$

$80$

$\frac{1}{3}$

$42 \pi$

$72 \pi$
$144 \pi$
$48 \pi$

5. Volume of spheres; density

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

$\frac{526}{27}$

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

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

6. Converting units of measurement

$60$
$24$

$35$
$32$

$60$
$24$
$84$

7. Percent error

$\frac{100}{9}$
$\frac{300}{73}$

$30$

$558$
$368$

$\frac{100}{9}$
$\frac{300}{73}$
$\frac{25}{13}$