Problem

Assuming that the area enclosed by the large and small circles in the figure is 237 square inches and the difference between the radii of the large and small circles is 5 inches, what are the diameters of the large and small circles? Note that $π=3.16$ is used in this problem.

---

$$ $$

$$ $$

$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$

$$ $$

$$ $$

$$ $$

$$ $$

$$ $$

$$ $$

$$ $$

$$ $$

$$ $$

$$ $$

---

Solution

Let $x$ be the diameter of the large circle and $y$ be the diameter of the small circle. When $k$ is the difference between the radii of the large and small circles, and $S$ is the area enclosed by the large and small circles, it follows that

$$S=π((\frac{x}{2})^2-(\frac{y}{2})^2)=π(\frac{x-y}{2})(\frac{x+y}{2}) —[1],$$

$$\frac{x-y}{2}=k —[2],$$

$$\frac{x+y}{2}=\frac{x-y}{2}+y=k+y,$$

$$\therefore S=πk(k+y) —[1]’.$$

From the problem statement,

$$S=237 —[3], k=5 —[4], π=3.16 —[5].$$

Substituting [3], [4] and [5] into [1]’, we see that

$$3.16×5×(5+y)=237,$$

$$5×(5+y)=75,$$

$$ y=10 —[6].$$

From [2], [4] and [6],

$$x=y+2k=10+2×5=20.$$

(Answer) $x=20$ inches, $y=10$ inches.

---

Reference

Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), pp.59-60; pp.285-286.