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Difficulty: MediumCircles

A flat metal washer is manufactured by stamping a circular hole with a diameter of 4 mm4 \text{ mm} out of the exact center of a solid circular disk with a diameter of 12 mm12 \text{ mm}. What is the area of the remaining flat face of the washer in square millimeters? (Enter only the numerical coefficient of π\pi in your answer. For example, if the area is 50π50\pi, enter 50).

Answer: 32

Answer

32
The area of the remaining metal washer face is found by calculating the area of the large outer circle (36π36\pi) and subtracting the area of the smaller inner circle hole (4π4\pi), which leaves a remaining area of 32π32\pi. Since the question asks for the coefficient of π\pi, the correct numerical entry is 32.

Step-by-Step Solution

1
Calculate the radius of the outer disk.
R=12÷2=6 mmR = 12 \div 2 = 6 \text{ mm}
The area formula for a circle requires the radius, which is half of the diameter.
2
Calculate the area of the outer disk.
Areaouter=π×62=36π mm2\text{Area}_{\text{outer}} = \pi \times 6^2 = 36\pi \text{ mm}^2
This represents the total surface area before the center hole is removed.
3
Calculate the radius of the inner hole.
r=4÷2=2 mmr = 4 \div 2 = 2 \text{ mm}
The radius of the removed section is needed to determine how much area to subtract.
4
Calculate the area of the inner hole.
Areainner=π×22=4π mm2\text{Area}_{\text{inner}} = \pi \times 2^2 = 4\pi \text{ mm}^2
This is the area of the metal that is stamped out.
5
Subtract the inner area from the outer area.
36π−4π=32π mm236\pi - 4\pi = 32\pi \text{ mm}^2
The remaining face area of the washer is the difference between the full disk and the removed hole.
6
Identify the coefficient of π\pi.
32
The stem specifically asks for the numerical coefficient only.

Key Concept

Calculating the area of an annulus (the region between two concentric circles) by subtracting the area of the inner circle from the area of the outer circle.
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Topics

Circles

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