Vicious Circles And Savage Shapes

Geometry Level 1

The following figure consists of a large semicircle of diameter $20\text{ cm}$ , a smaller full circle of diameter $10\text{ cm}$ , a semicircle of diameter $5\text{ cm}$ , a quadrant of radius $5 \text{ cm}$ , and a right-angled triangle of sides $20\text{ cm}$ and $10\text{ cm}$ . Find the shaded area in $\text{cm}^2$ .

Give your answer to 1 decimal place.

125.4 145.4 185.8 190.6

1 solution

Wee Xian Bin
Jul 9, 2016

Relevant wiki: Length and Area - Composite Figures

Dissect the figure into the following:

Then notice that:

Therefore:

$A+C+D+E=(\frac{1}{2})(\pi)(10)^2+(\frac{1}{2})(\pi)(\frac{10}{2})^2-(\frac{1}{2})(20)(10)=96.35$

$B=(\frac{1}{4})(\frac{1}{2})(20)(10)=25$

$H=(\frac{10}{2})^2-(\frac{1}{4})(\pi)(\frac{10}{2})^2=5.37$

$F+G=(\frac{1}{2})(\pi)(\frac{10}{2})^2-(\frac{1}{2})(\pi)(\frac{10}{4})^2-H=24.08$

And as such $A+B+C+D+E+F+G=145.4$

Great use of the De Morgans law and the fact $A \cup B + A \cap B = A + B$ .

Jerry McKenzie - 4 years ago