SPI Mathematics · Mensuration 3D

Combined Solids and the Frustum

A joined solid's volume is the sum of its parts and its surface is only what shows outside; a frustum is the piece of a cone left when its top is cut off parallel to the base.

All Mensuration 3D notes

Why this matters

Eleven questions in the bank, the hardest page of the chapter: nine MODERATE, one HARD and one EASY. Ice-cream cones, toys, buildings with a conical roof, buckets and glasses: each needs the formulas of the last two pages put together, plus the frustum's own three.

Concept 1 of 3: Solids joined together

Glue a cone onto a half-ball and you get a toy. To fill it you need both volumes added. To paint it you paint only what you can see: the circle where the two pieces meet is hidden inside.

Definition

Volume of a joined solid $=$ the volumes of its parts added.
Surface $=$ only the outside. Faces glued together are hidden and are not counted.
Cone on a hemisphere: $\pi rl + 2\pi r^2$ . Cylinder with a conical top (a tent or a building): $2\pi rh + \pi rl$ , plus $\pi r^2$ only if the floor is asked for.
A cone whose vertical angle is a right angle has a half-angle of $45^\circ$ , so its height equals its radius and $l = r\sqrt{2}$ .

Cone on a hemisphere

V = \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3, \qquad S = \pi rl + 2\pi r^2

Worked example

A toy is a cone of radius 3 cm and height 4 cm standing on a hemisphere of the same radius. Find its volume and its surface area, in terms of π.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Mensuration 3D · Combined Solids and the Frustum

An icecream cone has hemispherical top. If the height of the cone is 9 cm and base radius is 2.5 cm, then the volume of icecream is

Counting the glued faces

A cone on a hemisphere shows the cone's curved surface and the hemisphere's curved surface:

\pi rl + 2\pi r^2

. Writing

3\pi r^2

for the hemisphere counts a flat face that is glued out of sight.

A right vertical angle is the whole angle

The vertical angle is measured across the tip. If it is

90^\circ

, each side makes

45^\circ

with the axis, so the height equals the radius.

Drill 3 more on solids joined together

Concept 2 of 3: The frustum: buckets and glasses

A bucket or a drinking glass is a cone with its tip cut off: two circles of different sizes joined by a sloping side. It has its own slant height, curved surface and volume formulas.

Definition

Slant height: $l = \sqrt{h^2 + (R - r)^2}$ , using the difference of the radii.
Curved surface $= \pi(R + r)l$ .
Volume $= \frac{1}{3}\pi h(R^2 + r^2 + Rr)$ .
A bucket is open at the top: its metal is the curved surface plus the smaller bottom circle, $\pi(R + r)l + \pi r^2$ .
Capacity in litres: divide cubic cm by 1000.

Frustum

l = \sqrt{h^2 + (R - r)^2}, \quad \text{CSA} = \pi(R + r)l, \quad V = \frac{1}{3}\pi h(R^2 + r^2 + Rr)

R, rradii of the larger and the smaller circular ends
hvertical height
lslant height

Worked example

A bucket has end radii 10 cm and 4 cm and is 8 cm high. Find its slant height, curved surface area and volume. (Use π = 3.14.)

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Mensuration 3D · Combined Solids and the Frustum

If the radii of the circular ends of a bucket of height 40 cm are of lengths 35 cm and 14 cm, then the volume of the bucket in cubic centimeters, is

Adding the radii in the slant height

The slant height uses the difference:

\sqrt{h^2 + (R - r)^2}

. The sum

R + r

belongs in the curved surface, not here.

Closing the top of a bucket

A bucket is open at the top. Its metal is the curved surface plus the smaller bottom circle only, not both circles.

Leaving the capacity in cubic cm

If the options are in litres, divide by 1000: 45 600 cubic cm is 45.6 litres.

Drill 4 more on the frustum: buckets and glasses

Concept 3 of 3: A cone cut parallel to its base

Slice the top off a cone with a cut parallel to the base. The little cone you remove is a smaller copy of the whole one, so the scaling rule from spheres works here too.

Definition

The small top cone is similar to the whole cone: its height and radius are the same fraction $k$ of the whole cone's.
Its volume is $k^3$ of the whole, so from a volume fraction take the cube root.
$k$ gives the small cone's height, measured from the tip. The cut is $H - kH$ above the base.

Similar top cone

\frac{h}{H} = \frac{r}{R} = k, \qquad \frac{V_{\text{top}}}{V} = k^3

Worked example

A cone 24 cm high is cut parallel to its base so that the small top cone has

\frac{1}{8}

of the volume. How high above the base is the cut?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Mensuration 3D · Combined Solids and the Frustum

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be

\frac{1}{27}

of the volume of the given cone, then at what height above the base is the section made?

Measuring from the wrong end

The cube root gives the height of the small cone, measured from the tip. The height of the cut above the base is the whole height minus that.

Square root for a volume fraction

A volume fraction needs the cube root.

\frac{1}{8}

of the volume means half the height, not

\frac{1}{\sqrt{8}}

Drill 1 more on a cone cut parallel to its base

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Solids joined together
Cone on a hemisphere
$V = \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3, \qquad S = \pi rl + 2\pi r^2$
The frustum: buckets and glasses
Frustum
$l = \sqrt{h^2 + (R - r)^2}, \quad \text{CSA} = \pi(R + r)l, \quad V = \frac{1}{3}\pi h(R^2 + r^2 + Rr)$
A cone cut parallel to its base
Similar top cone
$\frac{h}{H} = \frac{r}{R} = k, \qquad \frac{V_{\text{top}}}{V} = k^3$

Watch out for (7)

Counting the glued faces
→ Solids joined together
A right vertical angle is the whole angle
→ Solids joined together
Adding the radii in the slant height
→ The frustum: buckets and glasses
Closing the top of a bucket
→ The frustum: buckets and glasses
Leaving the capacity in cubic cm
→ The frustum: buckets and glasses
Measuring from the wrong end
→ A cone cut parallel to its base
Square root for a volume fraction
→ A cone cut parallel to its base

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