15. Frustums - Cardiff Tutor Company

Question 1

Below is a frustum of a cone. Given that the original cone was 15cm tall, find the volume of the frustum. Give your answer to 3 significant figures.

Select the correct answer from the list below:

A: $393\text{cm}^3$

B: $427\text{cm}^3$

C: $353\text{cm}^3$

D: $612\text{cm}^3$

CORRECT ANSWER: C: $353\text{cm}^3$

WORKED SOLUTION:

The volume of a cone is $\dfrac{1}{3}\pi r^2 h$ . To find the volume of the frustum we need the volume of the whole cone and the volume of the missing portion. To calculate the latter, we will need the radius of the top of the frustum (aka the base of the missing portion), and to find this we will use similarity.

The whole cone and the missing portion are similar. To find the scale factor between the dimensions of these two shapes, we must divide the height of one by the height of the other. The height of the missing portion is $15-8=7\text{cm}$ , so the scale factor is $7\div15$ . Therefore, the radius of the top face of the frustum is $\dfrac{7}{15}\times5=\dfrac{35}{15}=\dfrac{7}{3}$ . So, we get

$\text{Volume of Whole Cone} =\frac{1}{3}\pi\times5^2\times15=125\pi$ $\text{Volume of Smaller Cone} =\frac{1}{3}\pi\times\left(\frac{7}{3}\right)^2\times7=\frac{343}{27}\pi$

Finally, to 3 significant figures, we get

$\text{Volume of Frustum} = 125\pi-\frac{343}{27}\pi=353\text{cm}^3$

Question 2

The following square based frustum is part of a square based pyramid with an original height of $18$ cm.

Given that the apex is directly above the centre, find its volume to 3 significant figures.

Select the correct answer from the list below:

A: $470\text{cm}^3$

B: $701\text{cm}^3$

C: $642\text{cm}^3$

D: $315\text{cm}^3$

CORRECT ANSWER: A: $470\text{cm}^3$

WORKED SOLUTION:

The volume of a square based $\dfrac{1}{3}\l^2 h$ , where $l$ is the length of the base. To find the volume of the frustum we need the volume of the whole pyramid and the volume of the missing portion. To calculate the latter, we will need the length of the top of the frustum (aka the base of the missing portion), and to find this we will use similarity.

The whole pyramid and the missing portion are similar. To find the scale factor between the dimensions of these two shapes, we must divide the length of the base by the length of the top. The length of the top is 6cm and the length of the base is 10cm, so the scale factor is $6\div10=\frac{3}{5}$ . Therefore, the height of the missing pyramid is $\dfrac{3}{5}\times18=\dfrac{54}{5}$ . So, we get

$\text{Volume of Whole Pyramid} =\frac{1}{3}\times10^2\times18=600$ $\text{Volume of Smaller Pyramid} =\frac{1}{3}\times6^2\times\frac{54}{5} =\frac{648}{5}$

Finally, to 3 significant figures, we get

\text{Volume of Frustum} = 600-\frac{648}{5} =470\text{cm}^3

Question 3

The frustum below is part of a larger cone with height 8cm.

Find the curved surface area of the frustum.

Select the correct answer from the list below:

A: $521\text{cm}^2$

B: $255\text{cm}^2$

C: $167\text{cm}^2$

D: $331\text{cm}^2$

CORRECT ANSWER: D: $331\text{cm}^2$

WORKED SOLUTION:

The area of the curved part of a cone is given by $\pi rl$ , where $l$ is the ‘slant height’. To find curved surface area of the frustum we need the curved surface area of the whole cone and the curved surface area of the missing portion. To calculate the latter, we will need the radius of the top of the frustum (aka the base of the missing portion), and to find this we will use similarity.

The whole cone and the missing portion are similar. To find the scale factor between the dimensions of these two shapes, we must divide the height of one by the height of the other. The height of the missing portion is $8-3=5\text{cm}$ , so the scale factor is $5\div8$ . Therefore, the radius of the top face of the frustum is $\dfrac{5}{8}\times12=\dfrac{60}{8}=\dfrac{15}{2}$ . We now need to use Pythagoras to find the slant height. It is helpful to look at the frustum head on.

And now we can find the curve surface areas.

$\text{Curved Surface Area of Whole Cone} =\pi\times12\times14.4=\frac{864}{5}\pi$ $\text{Curved Surface Area of Smaller Cone} =\pi\times\frac{15}{2}\times9.01=\frac{2703}{40}\pi$

Finally, to 3 significant figures, we get

\text{Volume of Frustum} = \frac{864}{5}\pi -\frac{2703}{40}\pi =331\text{cm}^2

Question 4

The frustum below is part of a larger cone with height 19cm.

Find the total surface area of the frustum.

Select the correct answer from the list below:

A: $1780\text{cm}^2$

B: $2030\text{cm}^2$

C: $1250\text{cm}^2$

D: $1890\text{cm}^2$

CORRECT ANSWER: A: $1780\text{cm}^2$

WORKED SOLUTION:

The whole cone and the missing portion are similar. To find the scale factor between the dimensions of these two shapes, we must divide the radius of one by the radius of the other, $\dfrac{6}{15}=\dfrac{2}{5}$ . Therefore, because the total height is 19, the height of the small cone $\dfrac{2}{15}\times19=\dfrac{38}{5}$ . We now need to use Pythagoras to find the slant height. It is helpful to look at the frustum head on.

And now we can find the curve surface areas.

$\text{Curved Surface Area of Whole Cone} =\pi\times15\times24.2=363\pi$ $\text{Curved Surface Area of Smaller Cone} =\pi\times6\times9.68=\frac{2452}{25}\pi$

The curved surface area is therefore:

\text{Curved Surface Area of Frustum } =363\pi-\frac{2452}{25}\pi=\frac{7623}{25}\pi

Now we need to find the area of the top and base, but that is simple because they are just circles.

$\text{Area of Base } =\pi\times15^2=225\pi$
$\text{Area of Top } =\pi\times6^2=36\pi$

Finally, we just need to add these three pieces together and give our answer to 3 significant figures,

\text{Surface Area of Frustum} =\frac{7623}{25}\pi+225\pi + 36\pi=1780\text{cm}^2

Question 5

Two identical square based pyramids with original height 23cm are cut into frustums and stacked as so:

Find the total surface area of the shape.

Give your answer to 3 significant figures.

Select the correct answer from the list below:

A: $634\text{cm}^2$

B: $598\text{cm}^2$

C: $520\text{cm}^2$

D: $326\text{cm}^2$

CORRECT ANSWER: A: $634\text{cm}^2$

WORKED SOLUTION:

Because the two pieces are identical, we can just think about the surface area of one of these, which will look like this:

We’re going to need the area of the base, so let’s find that first:

\text{Area of Base }= 8^2=64\text{cm}^2

The four sides are identical trapeziums. We find the area of a trapezium by doing $\dfrac{(a+b)h}{2}$ , where $a$ and $b$ are the parallel sides (top and bottom of the frustum). We know the length on the bottom, but we need to find the length at the top.

The whole pyramid and the missing portion are similar. To find the scale factor between the dimensions of these two shapes, we must divide the height of the missing pyramid by the height of the whole pyramid. The height of the missing pyramid is $23-10=13\text{cm}$ . So, the scale factor is $13\div23=\frac{13}{23}$ . Therefore, the top length of the missing pyramid is $\dfrac{13}{23}\times8=\dfrac{104}{23}$ .

We can also use this scale factor to help us find the height of the trapezium. First of all, if we think about a right-angle triangle inside the whole pyramid, we can find the total slanted height.

Note how the triangle base will be half the length of the square.

\text{Slanted Height} =\sqrt{4^2+23^2}=23.3\text{cm}

And now, if we apply the scale factor to this triangle slanted height, it will give us the height of the trapezium.

Trapezium Height = Full Pyramid Slant – Missing Pyramid Slant

We can find the slant of the missing pyramid by multiplying the full pyramid slant by the scale factor.

$\text{Missing Pyramid Slant } = 23.3\times\frac{13}{23}=13.2\text{cm}$ $\text{Trapezium Height } =23.3-13.2=10.1\text{cm}$

And now we can find the area of the trapezium.

\text{Trapezium Area } =\frac{(8+\frac{104}{23})}{2}\times10.1=63.2\text{cm}^2

Now, the shape is made up of 8 trapeziums and 2 lots of the base:

$\text{8 Trapeziums } =8\times63.2=506\text{cm}^2$
$\text{8 Squares} =2\times64=128\text{cm}^2$
$\text{Total Surface Area 3.s.f. } =506+128=634 {cm}^2$