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Problem of the Day #40: Painting a Face April 28, 2011

Posted by Alex in : potd , trackback

Albert wishes to paint his face a very special color, RGB #FF6EC7. He does not want to waste a single drop of paint, but he also does not want to leave a spot of his face unpainted. Luckily, Albert can find the volume of paint (of thickness 2) he needs to get because he has found a way to represent his face geometrically.

Let a circle with center $O$ and radius $8$ represent the face. Let octagon $ABCDEFGH$ be a regular octagon inscribed in circle $O$. Let $I$, $J$, $K$, and $L$ be the midpoints of $\stackrel{\frown{}}{AB}$, $\stackrel{\frown{}}{BC}$, $\stackrel{\frown{}}{CD}$, $\stackrel{\frown{}}{DE}$, respectively. Albert’s hair consists of four circles, each marked by two radii:

  1. $IA$ and $IB$
  2. $JB$ and $JC$
  3. $KC$ and $KD$
  4. $LD$ and $LE$

The two eyes are the largest circles inside $\triangle{}AOB$ and $\triangle{}DOE$, respectively, that do overlap with any hair. Albert’s mouth is quadrilateral $AZEG$, where Z is the midpoint of $\overline{OG}$. This geometric representation of Albert’s face does not contain any other facial features.

Given that Albert does not want to paint his eyes, his hair, or his mouth, calculate the volume of paint needed.

Diagram, almost drawn to scale.

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