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MATH / A decorative arrangement of floor tiles forms concentric circles, as shown in…
Problem
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problem
A decorative arrangement of floor tiles forms concentric circles, as shown in the figure to the right. The smallest circle has a radius of 2 feet, and each successive circle has a radius 2 feet longer. All the lines shown intersect at the center and form 12 congruent central angles. What is the area of the shaded region? Express your answer in terms of . [asy]
fill((2,0)--(4,0)..scale(4)*dir(15).. scale(4)*dir(30)--scale(2)*dir(30)..scale(2)*dir(15)..cycle,gray(.7));
draw(Circle((0,0),2),linewidth(1));
draw(Circle((0,0),4),linewidth(1));
draw(Circle((0,0),6),linewidth(1));
draw(Circle((0,0),8),linewidth(1));
draw((0,0)--scale(8)*dir(30),linewidth(1));
draw((0,0)--scale(8)*dir(60),linewidth(1));
draw((0,0)--scale(8)*dir(90),linewidth(1));
draw((0,0)--scale(8)*dir(120),linewidth(1));
draw((0,0)--scale(8)*dir(150),linewidth(1));
draw((0,0)--scale(8)*dir(180),linewidth(1));
draw((0,0)--scale(8)*dir(210),linewidth(1));
draw((0,0)--scale(8)*dir(240),linewidth(1));
draw((0,0)--scale(8)*dir(270),linewidth(1));
draw((0,0)--scale(8)*dir(300),linewidth(1));
draw((0,0)--scale(8)*dir(330),linewidth(1));
draw((0,0)--scale(8)*dir(0),linewidth(1));
[/asy]
Plain-text mathematical notation (without MathML)
A decorative arrangement of floor tiles forms concentric circles, as shown in the figure to the right. The smallest circle has a radius of 2 feet, and each successive circle has a radius 2 feet longer. All the lines shown intersect at the center and form 12 congruent central angles. What is the area of the shaded region? Express your answer in terms of π. [asy] fill((2,0)--(4,0)..scale(4)*dir(15).. scale(4)*dir(30)--scale(2)*dir(30)..scale(2)*dir(15)..cycle,gray(.7)); draw(Circle((0,0),2),linewidth(1)); draw(Circle((0,0),4),linewidth(1)); draw(Circle((0,0),6),linewidth(1)); draw(Circle((0,0),8),linewidth(1)); draw((0,0)--scale(8)*dir(30),linewidth(1)); draw((0,0)--scale(8)*dir(60),linewidth(1)); draw((0,0)--scale(8)*dir(90),linewidth(1)); draw((0,0)--scale(8)*dir(120),linewidth(1)); draw((0,0)--scale(8)*dir(150),linewidth(1)); draw((0,0)--scale(8)*dir(180),linewidth(1)); draw((0,0)--scale(8)*dir(210),linewidth(1)); draw((0,0)--scale(8)*dir(240),linewidth(1)); draw((0,0)--scale(8)*dir(270),linewidth(1)); draw((0,0)--scale(8)*dir(300),linewidth(1)); draw((0,0)--scale(8)*dir(330),linewidth(1)); draw((0,0)--scale(8)*dir(0),linewidth(1)); [/asy]
Original LaTeX notation
A decorative arrangement of floor tiles forms concentric circles, as shown in the figure to the right. The smallest circle has a radius of 2 feet, and each successive circle has a radius 2 feet longer. All the lines shown intersect at the center and form 12 congruent central angles. What is the area of the shaded region? Express your answer in terms of $\pi$. [asy] fill((2,0)--(4,0)..scale(4)*dir(15).. scale(4)*dir(30)--scale(2)*dir(30)..scale(2)*dir(15)..cycle,gray(.7)); draw(Circle((0,0),2),linewidth(1)); draw(Circle((0,0),4),linewidth(1)); draw(Circle((0,0),6),linewidth(1)); draw(Circle((0,0),8),linewidth(1)); draw((0,0)--scale(8)*dir(30),linewidth(1)); draw((0,0)--scale(8)*dir(60),linewidth(1)); draw((0,0)--scale(8)*dir(90),linewidth(1)); draw((0,0)--scale(8)*dir(120),linewidth(1)); draw((0,0)--scale(8)*dir(150),linewidth(1)); draw((0,0)--scale(8)*dir(180),linewidth(1)); draw((0,0)--scale(8)*dir(210),linewidth(1)); draw((0,0)--scale(8)*dir(240),linewidth(1)); draw((0,0)--scale(8)*dir(270),linewidth(1)); draw((0,0)--scale(8)*dir(300),linewidth(1)); draw((0,0)--scale(8)*dir(330),linewidth(1)); draw((0,0)--scale(8)*dir(0),linewidth(1)); [/asy]
level
Level 3
type
Geometry
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