# 2007 AMC 8 Problems/Problem 23

## Problem

What is the area of the shaded pinwheel shown in the $5 \times 5$ grid? $[asy] filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black); int i; for(i=0; i<6; i=i+1) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } [/asy]$ $\textbf{(A)}\: 4\qquad\textbf{(B)}\: 6\qquad\textbf{(C)}\: 8\qquad\textbf{(D)}\: 10\qquad\textbf{(E)}\: 12$

## Solution

The area of the square around the pinwheel is 25. The area of the pinwheel is equal to $\text{the square } - \text{ the white space.}$ Each of the four triangles have a base of 3 units and a height of 2.5 units, and so their combined area is 15 units squared. Then the unshaded space consists of the four triangles with total area of 15, and there are four white corner squares. Therefore the area of the pinwheel is $25-(15+4)$ which is $\boxed{\textbf{(B) 6}}$

## Solution 2

Notice that the pinwheel is symmetrical, thus we only have to find the area of one of the sections and multiply that by four. Using Pick’s Theorem, b/2+i-1, we can count 3 lattice points on the border, so b/2 is 1.5. The number of interior points is 1. So the area of one section of the pinwheel is 1.5. Multiplying that by 4 yields $\boxed{\textbf{(B) 6}}$

~YJC64002776

## Video Solution

https://youtu.be/KOZBOvI9WTs -Happytwin

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