21. The diagram shows a regular octagon. What is the ratio of the area of the shaded trapezium to the area of the whole octagon?
A $1: 4$
B $5: 16$
C $1: 3$
D $\sqrt{2}: 2$
E $3: 8$
21. A Let $a$ be the side length of the octagon and $b$ be as shown on the diagram. The square in the centre is $a$ by $a$, each rectangle is $a$ by $b$ and the triangles are each half of a $b$ by $b$ square. Applying Pythagoras' Theorem to a triangle shows that $a^{2}=2 b^{2}$. So the shaded area is $b^{2}+a b=b^{2}+\sqrt{ } 2 b^{2}=b^{2}(1+\sqrt{ } 2)$.

Similarly the total area of the figure is $a^{2}+4 a b+2 b^{2}=4 b^{2}+4 \sqrt{ } 2 b^{2}$ $=b^{2}(4+4 \sqrt{ } 2)$. Therefore the ratio required is $(1+\sqrt{ } 2):(4+4 \sqrt{ } 2)=1: 4$.