2023 AMC 8 Problems/Problem 12

1 Problem
2 Solution 1
3 Solution 2
4 Video Solution by Math-X (How to do this question under 30 seconds)
5 Video Solution by CoolMathProblems
6 Video Solution (A Clever Explanation You’ll Get Instantly)
7 Video Solution (HOW TO THINK CREATIVELY!!!)
8 Video Solution (Animated)
9 Video Solution by Magic Square
10 Video Solution by SpreadTheMathLove
11 Video Solution by Interstigation
12 Video Solution by harungurcan
13 Video Solution by Dr. David
14 Video Solution by WhyMath
15 See Also

Problem

The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?

$[asy] // Diagram by TheMathGuyd size(6cm); draw(circle((3,3),3)); filldraw(circle((2,3),2),lightgrey); filldraw(circle((3,3),1),white); filldraw(circle((1,3),1),white); filldraw(circle((5.5,3),0.5),lightgrey); filldraw(circle((4.5,4.5),0.5),lightgrey); filldraw(circle((4.5,1.5),0.5),lightgrey); int i, j; for(i=0; i<7; i=i+1) { draw((0,i)--(6,i), dashed+grey); draw((i,0)--(i,6), dashed+grey); } [/asy]$

$\textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{11}{36} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{19}{36} \qquad \textbf{(E)}\ \frac{5}{9}$

Solution 1

First, the total area of the radius $3$ circle is simply just $9\cdot \pi$ when using our area of a circle formula.

Now from here, we have to find our shaded area. This can be done by adding the areas of the $\frac{1}{2}$ -radius circles and add; then, take the area of the $1$ radius circles and subtract that from the area of the $2$ radius circle to get our resulting complex shape area. Adding these up, we will get $3\cdot \frac{1}{4} \pi + 4 \pi -\pi - \pi = \frac{3}{4} \pi + 2 \pi = \frac{11\cdot \pi}{4}$ .

So, our answer is $\frac {\frac{11}{4} \pi}{9 \pi} = \boxed{\textbf{(B)}\ \frac{11}{36}}$ .

~apex304 Minor edits by ~NXC

Minor edits by ~Shriyans Chowdhury

Solution 2

Pretend each circle is a square. The large shaded circle is a square with area $16~\text{units}^2$ , and the two white circles inside it each have areas of $4~\text{units}^2$ , which adds up to $8~\text{units}^2$ . The three small shaded circles become three squares with area $1~\text{units}^2$ , and add up to $3~\text{units}^2$ . Adding the areas of the shaded circles (19) and subtracting the areas of the white circles (8), we get $11~\text{units}^2$ . Since the largest white circle in which all these other circles are becomes a square that has area $36~\text{units}^2$ , our answer is $\boxed{\textbf{(B)}\ \dfrac{11}{36}}$ .

-claregu LaTeX (edits -apex304, CoOlPoTaToEs, KGINSPECTORBOI, SlimeKnight, rickastleylover99)

Video Solution by Math-X (How to do this question under 30 seconds)

https://youtu.be/Ku_c1YHnLt0?si=stUHQ9nHZZE_x-CC&t=1852

~Math-X

Video Solution by CoolMathProblems

https://youtu.be/9WP3LQaMIVg?feature=shared&t=128

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/zntZrtsnyxc?si=nM5eWOwNU6HRdleZ&t=765 ~hsnacademy

Video Solution (HOW TO THINK CREATIVELY!!!)

https://youtu.be/5wpEBWZjl6o

~Education the Study of everything

Video Solution (Animated)

https://youtu.be/5RRo6pQqaUI

~Star League (https://starleague.us)

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=4590

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=UWoUhV5T92Y

Video Solution by Interstigation

https://youtu.be/DBqko2xATxs&t=1137

Video Solution by harungurcan

https://www.youtube.com/watch?v=oIGy79w1H8o&t=1154s

~harungurcan

Video Solution by Dr. David

https://youtu.be/2Ih7F0XHmls

Video Solution by WhyMath

https://youtu.be/ZOi0faHzBR4

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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All AJHSME/AMC 8 Problems and Solutions

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