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# Puzzles

## 2 December

Today's number is the area of the largest dodecagon that it's possible to fit inside a circle with area $$\displaystyle\frac{172\pi}3$$.

## Two semicircles

The diagram shows two semicircles.
$$CD$$ is a chord of the larger circle and is parallel to $$AB$$. The length of $$CD$$ is 8m. What is the area of the shaded region (in terms of $$\pi$$)?

## 1 December

What is area of the largest area rectangle which will fit in a circle of radius 10?

## Squared circle

Each side of a square has a circle drawn on it as diameter. The square is also inscribed in a fifth circle as shown.
Find the ratio of the total area of the shaded crescents to the area of the square.

## Dartboard

Concentric circles with radii 1, $$\frac{1}{2}$$, $$\frac{1}{3}$$, $$\frac{1}{4}$$, ... are drawn. Alternate donut-shaped regions are shaded.
What is the total shaded area?

## Circles

Which is largest, the red or the blue area?

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?

## Semi circle in a triangle

This right-angled triangle above has sides of lengths 12cm, 5cm and 13cm. The diameter of the semicircle lies on the 12cm side and the 13cm side is a tangent to the circle. What is the radius of the semi circle?

## Archive

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