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Geometry and Trigonometry / Circles Difficulty: Hard

A circle has center $O$ , and points $A$ and $B$ lie on the circle. The measure of arc $A B$ is $45^{\circ}$ and the length of arc $A B$ is $3$ inches. What is the circumference, in inches, of the circle?

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Explanation

Choice D is correct. It’s given that the measure of arc $A B$ is $45 degree$ and the length of arc $A B$ is $3 inches$ . The arc measure of the full circle is $360 degree$ . If $x$ represents the circumference, in inches, of the circle, it follows that $\frac{45 °}{360 °} = \frac{3 inches}{x inches}$ . This equation is equivalent to $\frac{45}{360} = \frac{3}{x}$ , or $\frac{1}{8} = \frac{3}{x}$ . Multiplying both sides of this equation by $8 x$ yields $1 left parenthesis x right parenthesis equals 3 left parenthesis 8 right parenthesis$ , or $x equals 24$ . Therefore, the circumference of the circle is $24 inches$ .

Choice A is incorrect. This is the length of arc $A B$ .

Choice B is incorrect and may result from multiplying the length of arc $A B$ by $2$ .

Choice C is incorrect and may result from squaring the length of arc $A B$ .