solid geometry - Circle Related

SOURCE:COMPETITION
Number of Problems: 19. : (Book)

Problem 19

Three semicircles of radius $1$ are constructed on diameter $overline{AB}$ of a semicircle of radius $2$ . The centers of the small semicircles divide $overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?

$extbf{(A) } pi - sqrt{3} qquad extbf{(B) } pi - sqrt{2} qquad extbf{(C) } frac{pi + sqrt{2}}{2} qquad extbf{(D...$

A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?

$(mathrm {A}) 200+25pi quad (mathrm {B}) 100+75pi quad (mathrm {C}) 75+100pi quad (mathrm {D}) 100+100pi quad (ma...$

10.

A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?

$mathrm{(A) } frac{3sqrt{5}}{2} qquad mathrm{(B) } frac{7}{2} qquad mathrm{(C) } sqrt{15} qquad mathrm{(D) ...$

11.

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

$ext{(A)} pi qquad ext{(B)} 1.5pi qquad ext{(C)} 2pi qquad ext{(D)} 3pi qquad ext{(E)} 3.5pi$

''>''

$ext{(A)} pi qquad ext{(B)} 1.5pi qquad ext{(C)} 2pi qquad ext{(D)} 3pi qquad ext{(E)} 3.5pi$

12.

An $8$ -foot by $10$ -foot floor is tiles with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

$mathrm{(A)} 80-20pi qquadmathrm{(B)} 60-10pi qquadmathrm{(C)} 80-10pi qquadmathrm{(D)} 60+10pi ...$

13.

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?

$mathrm{(A) } frac{3sqrt{2}}{pi}qquad mathrm{(B) } frac{3sqrt{3}}{pi}qquad mathrm{(C) } sqrt{3}qquad math...$

14.

A semicircle of diameter $1$ sits at the top of a semicircle of diameter $2$ , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.

$mathrm{(A) } frac{1}{6}pi-frac{sqrt{3}}{4}qquad mathrm{(B) } frac{sqrt{3}}{4}-frac{1}{12}piqquad mathrm{(C) ...$

15.

Circles $A$ , $B$ , and $C$ are externally tangent to each other and internally tangent to circle $D$ . Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$ . What is the radius of circle $B$ ?

$mathrm{(A) } frac{2}{3} qquad mathrm{(B) } frac{sqrt{3}}{2} qquad mathrm{(C) } frac{7}{8} qquad mathrm{(D) ...$

16.

17.

An equilateral triangle has side length $6$ . What is the area of the region containing all points that are outside the triangle but not more than $3$ units from a point of the triangle?

$mathrm{(A)} 36+24sqrt{3}qquadmathrm{(B)} 54+9piqquadmathrm{(C)} 54+18sqrt{3}+6piqquadmathrm{(D)} left(2sqrt{...$

18.

A square of side length $1$ and a circle of radius $dfrac{sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?

$extbf{(A)} dfrac{pi}{3}-1 qquad extbf{(B)} dfrac{2pi}{9}-dfrac{sqrt{3}}{3} qquad extbf{(C)} dfrac{pi}{18} ...$

19.

Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?

$mathrm{(A)} 3-2sqrt2qquadmathrm{(B)} 2-sqrt2qquadmathrm{(C)} 4(3-2sqrt2)qquadmathrm{(D)} frac12(3-sqrt2)qqua...$