Surface Area of 3D Shapes

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 4mm and width 2.5mm, so it has area

 

4\times2.5=10\text{ mm}^2

 

Therefore, the back face also has area 10\text{ mm}^2.

 

The face on the right has length 6mm and height 4mm, so it has area

 

6\times4=24\text{ mm}^2

 

Therefore, the left face also has area 24\text{ mm}^2

 

The face on the top has length 6mm and width 2.5mm, so it has area

 

6\times2.5=15\text{ mm}^2

 

Therefore, the bottom face also has area 15\text{ mm}^2.

 

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

 

10+10+24+24+15+15=98\text{ mm}^2

We know the whole surface area is 120\text{cm}^2 and we also know the radius. To work out the slant height, we need to first work out what the curved surface area is. In other words, we need to subtract the surface area of the base of the cone (since that’s the only other face) from 120 to get the curved surface area. The base is a circle, so its area is:

 

\pi \times 3^2=9\pi\text{ cm}^2

 

Subtracting this from the total we have

 

120-9\pi=91.725...

 

This is written as a decimal to give an idea of how big it is, but when you put it into your calculator you should either use the ANS key to store the value, or you should directly type in 120-9\pi. If you don’t, it could affect your final answer.

 

Now, this must the area of the curved face, and the formula for the area of the curved face is given to us: \pi rl = 3\pi l (since we know r=3). So, setting this formula equal to the value we worked out, we get

 

3\pi l=120-9\pi

 

Then, to find the slant height, we will divide both sides by 3\pi to get

 

\dfrac{120-9\pi}{3\pi}=9.7\text{ cm (1dp)}.

We need to firstly work out the surface area of the sphere. So, with r=8.5, we get

 

\text{surface area }=4\times\pi\times(8.5)^2=907.920...\text{ m}^2

 

Now, since each pot covers ten square metres, we must divide this value by 10 to see how many pots he will require:

 

907.920... \div 10=90.720...

 

Since he can’t buy exactly 90.720… pots, he will have to buy 91 to cover the whole sphere. Therefore, the cost of all his paint will be

 

91\times 9.60=\pounds873.60

 

That’s a lot of paint.

The area of the two triangular faces is,

\text{area}=2\times \dfrac{1}{2}\times 6\times 8 = 48\text{ cm}^2

 

The area of the rectangular base is,

\text{area}=6\times 11 = 66 \text{ cm}^2

 

To find the area of the two sides we have to first find the slanted height of the prism:

 

\text{AB}=\sqrt{8^2+3^2} =\sqrt{73}

 

Hence, the area of the two sides are:

\text{area}=2\times 11\times \sqrt{73}

 

Total surface area:

\text{surface area }=48+66+\sqrt{73}= 301.97\text{ cm}^2

The area of the square base is,

 

\text{area}=12\times12=144\text{cm}^2

 

Finding the length of the midpoint of DC to E:

 

\sqrt{10^2+6^2} = \sqrt{136}

 

Hence the area of the 4 triangular faces is,

 

4\times\dfrac{1}{2}\times12\times\sqrt{136}= 279.89\text{cm}^2

Total surface area:

 

\text{surface area}=279.89+144= 423.89\text{ cm}^2