2005 AMC 10A Problems/Problem 12
Problem
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 ?
Solution
The area of the trefoil is equal to the area of an equilateral triangle with side length , plus the area of
segments. Each segment has area equal to that of a
sector with radius
, minus the area of an equilateral triangle with side length
.
As there are segments, the area of the equilateral triangle with side length
will be multiplied by
, and this is equivalent to the area of an equilateral triangle with side length
(since the area scale factor is the square of the length scale factor). Accordingly, the area of the equilateral triangle with side length
will exactly cancel out, and we are left only with
times the area of a
sector with radius
.
Thus the answer is .
See also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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