14. Surface Area of 3D Shapes

Question 1

Calculate the surface area of the cuboid shown.

Select the correct answer from the list below:

A: 343\text{cm}^2

B: 294\text{cm}^2

C: 98\text{cm}^2

D: 122\text{cm}^2

CORRECT ANSWER:     D: 122\text{cm}^2

WORKED SOLUTION:

A cuboid has 6 flat, rectangular faces, and we will need the areas of all of them. Specifically, there are 3 pairs of faces since the front and back faces are the same, the top and bottom faces are the same, and the left and right faces are the same.

Firstly, the front face is a rectangle with height 4cm and width 3, so it has area

3\times4=12\text{cm}^2 .

Therefore, the back face also has area

3\times4=12\text{cm}^2 .

The face on the right has length 7cm and height 4mm, so it has area

7\times4=28\text{cm}^2 .

Therefore, the left face also has area

7\times4=28\text{cm}^2 .

The face on the top has length 7cm and width 3cm, so it has area

7\times3=21\text{cm}^2 .

Therefore, the bottom face also has area

7\times3=21\text{cm}^2 .

Thus, adding all these values together, we get the total surface area to be

Question 2

Calculate the surface area of a cube with height 7cm.

Select the correct answer from the list below:

A: 294 cm^2

B: 49cm^2

C: 294 cm^3

D: 49 cm^3

CORRECT ANSWER:     A: 294 cm^2

WORKED SOLUTION:

We know that a cube has 6 faces that are all the same (squares), so we only need to find the area of one and then multiply it by 6.

Because the height of the cube is 7cm, we know that each side of any given face will be 7cm, so will have area

7\times7=49\text{cm}^2 .

Now, all we need to do is multiply this by 6

49\times6=294\text{cm}^2 .

Question 3

A pyramid is made up of a square base with area 36\text{cm}^2 and four identical triangles with height 10cm.

Workout the surface area of this pyramid.

Select the correct answer from the list below:

A: 156\text{cm}^2

B: 216\text{cm}^2

C: 254\text{cm}^2

D: 96\text{cm}^2

CORRECT ANSWER:     A: 156\text{cm}^2

WORKED SOLUTION:

From the information we are given, we can create a diagram like this:

So, because we need the surface area, we need to find the area of the four triangle sides. To find the area of a triangle we do \dfrac{1}{2}\times\text{base}\times\text{height}. But we don’t have the base length of the triangle, so we need to find that first.

To find the length of the triangles’ base we need to use the area of the pyramid’s base. Because we have a square, we just need to square root this.

\text{length}=\sqrt{\text{area}} .
\sqrt{36}=6\text{cm} .

So, our triangles have a base length of 6 cm, which we can now put into our formula.

\text{triangle area }=\frac{1}{2}\times 6\times 10=3\times 10=30\text{cm}^2 .

And, because we have four of these, we need to multiply by 4

30\times 4=120\text{cm}^2 .

And finally, to find the total area, we need to add this to the area of the base

120+36=156\text{cm}^2 .

Question 4

A new corn silo is being built of corrugated steel; the design is shown below.

Given that the silo consists of a bottomless and topless cylinder, and a hemisphere, find how much corrugated steel is needed. Give your answer to 2 decimal places.

Select the correct answer from the list below:

A: 320.44\text{m}^2

B: 213.55\text{m}^2

C: 376.99\text{m}^2

D: 471.22\text{m}^2

CORRECT ANSWER:     A: 320.44\text{m}^2

WORKED SOLUTION:

Starting with the hemisphere, we can see that it has a radius of 3m. So, putting this into our formula for the surface area of a sphere, we get

\text{Surface area of sphere}=4\pi r^2=4\pi 3^2=4\pi\times9=36\pi \text{m}^2 .

However, this gives us the surface area of a sphere, but we have a hemisphere. We need to divide by 2.

\text{Surface area of hemisphere}=36\pi \div2=18\pi \text{m}^2 .

Now we need to find the area of the curved face of the cylinder. To do this, we need to find the circumference of the cylinder.

\text{Circumference }=\pi\times\text{diameter}=\pi\times6=6\pi\text{m} .

And to find the surface area of the cylinder we need to multiply the circumference by the height

6\pi\times14=84\pi\text{m}^2.

And now, to find the total surface area, we need to add the cylinder and hemisphere

84\pi+18\pi=102\pi=320.44\text{m}^2.

Question 5

Too identical cones with radius 3cm are stuck together by the base. Give than the height of this new shape is 8cm, find its surface area.

Give your answer to 2 decimal places.

Select the correct answer from the list below:

A: 87.56\text{cm}^2

B: 122.52\text{cm}^2

C: 150.8\text{cm}^2

D: 94.25\text{cm}^2

CORRECT ANSWER:     D: 94.25\text{cm}^2

WORKED SOLUTION:

A drawing of this shape will look a little something like this:

So, to find the area of this shape we need to find the area of the curved part of a cone. To do this, we need to use the formula \pi\times r\times l, where l is the slanted height. To do this, it might be easier to split this into the original two cones.

One cone will look like this

But we needed the slanted height, which this doesn’t have. Luckily though, because the radius and the height form a right-angle triangle, we can use Pythagoras!

c^2=a^2+b^2
c^2=3^2+4^2
c^2=9+16
c^2=25
c=\sqrt{25}
c=5\text{cm}

And now we can put this slanted height into our formula

But our original shape had two cones, so we need to multiply this by 2