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## Monday, 12 May 2014

### Sunday Afternoon Maths XII Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

#### Circles

Let $$4x$$ be the side length of the square. This means that the radius of the red circle is $$2x$$ and the radius of a blue circle is $$x$$. Therefore the area of the red circle is $$4\pi x^2$$.
The area of one of the blue squares is $$\pi x^2$$ so the blue area is $$4\pi x^2$$. Therefore the two areas are the same.
##### Extension
Is the red or blue area larger?

#### Largest Triangle

As our shape is a triangle, the 4cm and 5cm sides must be adjacent. Call the angle between them be $$\theta$$.
The area of the triangle is $$\frac{1}{2}\times 4\times 5 \times \sin{\theta}$$ or $$10\sin{\theta}$$. This has a maximum value when $$\theta=90^\circ$$, so the largest triangle has and area of 10cm2 and looks like:
##### Extension
What is the largest area triangle with a perimeter of 12cm?

#### Unit Octagon

Name the regions as follows:
$$E$$ is a 1×1 square. Placed together, $$A$$, $$C$$, $$G$$ and $$I$$ also make a 1×1 square. $$B$$ is equal to $$H$$ and $$D$$ is equal to $$F$$.
Therefore $$B+E+F=A+C+D+G+H+I$$. Therefore The hatched region is $$C$$ larger than the shaded region. The area of $$C$$ (and therefore the difference) is $$\frac{1}{4}$$
##### Extension
What is the difference between the shaded and the hatched regions in this dodecagon?

#### 1 comment:

1. Those red and blue circles - in a way we should know it instantly, without pi or x... because the area of any shape scales up by 9 when its length goes up by 3. Just been thinking about this in relation to Pythagoras theorem:
http://seekecho.blogspot.fr/2014/05/the-proof-see.html