# Stacked cubes

What is the total surface area of this shape?

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Three cubes have been glued together as shown in the diagram.

The side length of the smallest cube is 3 cm.

The side length of the medium cube is 5 cm.

The side length of the largest cube is 8 cm.

What is the total surface area of the figure?

*This problem is taken from the World Mathematics Championships*

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**Adding up the areas of the faces**

The area of each face of each cube is found by squaring the side length.

The areas of the faces of the shape, in square centimetres, are shown on the diagram to the right.

Remember that we can't see all of the faces!

The total area is:$64 + 64+64+64+64+(64-25)+25$

$+25+25+25+(25-9)+9+9+9+9+9$

$ = 64\times5 + 39 + 25\times4 + 16 + 9\times5$

$ = 640\div2 + 39 + 100 + 16 + 45$

$ = 320 + 139 + 61 $

$= 520$

So the total surface area is $520$ cm$^2$.

Beginning with the surface area of each cube

The area of each face of each cube is found by squaring the side length.

Each cube has 6 faces, so the surface area of each cube was originally $6\times\text{side length}^2$, so the total surface area was originally $6\times8^2+6\times5^2+6\times3^2$.

However, where the largest cube joins the medium cube, the area of one face of the medium cube is lost from both cubes. Similarly, where the meduim cube joins the smallest cube, the area of one face of the smallest cube is lost from both cubes.

So the total surface area is:$$\begin{align}6\times8^2+6\times5^2+6\times3^2-2\times5^2-2\times3^2&=6\times8^2+4\times(3^2+5^2)\\

&=6\times64 + 4\times(9+25)\\

&=360+24 + 4\times34\\

&=360+24+120+16\\

&=480+40\\

&=520\end{align}$$ So the total surface area is $520$ cm$^2$.