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SOURCE:COMPETITION Number of Problems: 19. FOR PRINT ::: (Book)
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.
The outer circle has radius , and thus area . The little circles have area each; since there are 7, their total area is . Thus, our answer is .
An -foot by -foot floor is tiles with square tiles of size foot by foot. Each tile has a pattern consisting of four white quarter circles of radius 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?
There are 80 tiles. Each tile has shaded. Thus:
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?
Let be the length of a side of the equilateral triangle and let be the radius of the circle.
In a circle with a radius the side of an inscribed equilateral triangle is .
So .
The perimeter of the triangle is
The area of the circle is
So:
A semicircle of diameter sits at the top of a semicircle of diameter , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
Let denote the area of region in the figure above.
The shaded area is equal to the area of the smaller semicircle minus the area of a sector of the larger circle plus the area of a triangle formed by two radii of the larger semicircle and the diameter of the smaller semicircle .
The area of the smaller semicircle is .
Since the radius of the larger semicircle is equal to the diameter of the smaller semicircle, the triangle is an equilateral triangle and the sector measures .
The area of the sector of the larger semicircle is .
The area of the triangle is .
So the shaded area is .
Circles , , and are externally tangent to each other and internally tangent to circle . Circles and are congruent. Circle has radius and passes through the center of . What is the radius of circle ?
Let be the center of circle for all and let be the tangent point of . Since the radius of is the diameter of , the radius of is . Let the radius of be and let . If we connect , we get an isosceles triangle with lengths . Then right triangle has legs and hypotenuse . Solving for , we get .
Also, right triangle has legs , and hypotenuse . Solving,
So the answer is .
An equilateral triangle has side length . What is the area of the region containing all points that are outside the triangle but not more than units from a point of the triangle?
The region described contains three rectangles of dimensions , and three degree arcs of circles of radius . Thus the answer is
A square of side length and a circle of radius share the same center. What is the area inside the circle, but outside the square?
The radius of circle is . Half the diagonal of the square is . We can see that the circle passes outside the square, but the square is NOT completely contained in the circle Therefore the picture will look something like this:
Then we proceed to find: 4 * (area of sector marked off by the two radii - area of the triangle with sides on the square and the two radii).
First we realize that the radius perpendicular to the side of the square between the two radii marking off the sector splits in half. Let this half-length be . Also note that because it is half the sidelength of the square. Because this is a right triangle, we can use the Pythagorean Theorem to solve for
Solving, and . Since , is an equilateral triangle and the central angle is . Therefore the sector has an area .
Now we turn to the triangle. Since it is equilateral, we can use the formula for the area of an equilateral triangle which is
Putting it together, we get the answer to be
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?
Draw some of the radii of the small circles as in the picture below.
Out of symmetry, the quadrilateral in the center must be a square. Its side is obviously , and therefore its diagonal is . We can now compute the length of the vertical diameter of the large circle as . Hence , and thus .
Then the area of the large circle is . The area of four small circles is . Hence their ratio is: