2005 AMC 10B Problem 8

P1-10~2 minPrint

Circles Quadrilaterals & Polygon Areas Cut the figure into known shapes (triangles, rectangles, sectors)

An $8$ -foot by $10$ -foot bathroom floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

Show solution without answering

Key insight

On one tile, the four white quarter circles fit together into one whole circle of radius 1/2. So each tile has 1 minus pi/4 shaded, and there are 80 tiles.

Solution

Every tile looks the same, so work out one tile and then multiply.

1. White part of one tile. There are four quarter circles, each with radius $\frac{1}{2}$ . Four quarters make one whole circle, so the white area is the area of one circle of radius $\frac{1}{2}$ :
$\pi \cdot \left(\frac{1}{2}\right)^2 = \frac{\pi}{4}.$
2. Shaded part of one tile. The tile is $1$ foot by $1$ foot, so its area is $1$ . The shaded part is what is not white: $1 - \frac{\pi}{4}$ .
3. Number of tiles. The floor is $8$ by $10$ , so it holds $8 \cdot 10 = 80$ tiles.
4. Total shaded area.
$80 \cdot \left(1 - \frac{\pi}{4}\right) = 80 - 20\pi.$

The answer is $\boxed{\textbf{(A)}\ 80-20\pi}$ .

Why this works

With a repeating pattern, study a single copy and multiply by the number of copies. Inside the tile, the four matching quarter circles combine into one full circle, so you only need the circle area formula. When the white shape is the easy one, find it and subtract from the total.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Look at a single tile. The four white pieces are quarter circles of the same radius $\frac{1}{2}$ , so together they have the area of one whole circle:
$\pi \left(\frac{1}{2}\right)^2 = \frac{\pi}{4}.$

The tile has area $1$ , so the shaded part of one tile is $1 - \frac{\pi}{4}$ .

The floor holds $8 \cdot 10 = 80$ tiles, giving a total shaded area of
$80\left(1 - \frac{\pi}{4}\right) = 80 - 20\pi.$

The answer is $\boxed{\textbf{(A)}\ 80-20\pi}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 11: Circles
Introduction to Geometry, Chapter 8: Quadrilaterals
Introduction to Geometry, Chapter 4: Perimeter and Area (§4.2, §4.3)
Introduction to Geometry, Chapter 12: Circles and Angles