Question

The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $2?$

$\dfrac{1}{3}\pi + \dfrac{\sqrt{3}}{2}$

$\dfrac{2}{3}\pi$

$\dfrac{2}{3}\pi + \dfrac{\sqrt{3}}{4}$

$\dfrac{2}{3}\pi + \dfrac{\sqrt{3}}{3}$

$\dfrac{2}{3}\pi + \dfrac{\sqrt{3}}{2}$

Solution:

Since the base $2$ equals two radii, the radius is $1.$ The trefoil is made of four equilateral triangles and four circular segments, which reassemble into four $60^\circ$ sectors of a circle of radius $1.$ Their total area is $4 \cdot \dfrac{60}{360}\pi (1)^2 = \dfrac{2}{3}\pi.$

Thus, the correct answer is B.

2005 AMC 10A Problem 12

Solution:

Problem 12 in Other Years