An arc of a circle subtends an angle of 54° at the center. If the arc is 9cm long, calculate the circumference of the circle.
Understand the Problem
The question describes an arc of a circle that subtends a 54-degree angle at the center and has a length of 9cm. It asks us to find the circumference of the entire circle.
Answer
The circumference of the circle is $60$ cm.
Answer for screen readers
The circumference of the circle is $60$ cm.
Steps to Solve
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Set up a proportion
The length of the arc is proportional to the angle it subtends at the center. We can set up a proportion comparing the arc length to the circumference and the angle to the total degrees in a circle.
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Write the proportion
The proportion will look like this: $$ \frac{\text{arc length}}{\text{circumference}} = \frac{\text{angle}}{360^\circ} $$
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Plug in the given values
We know that the arc length is $9$ cm and the angle is $54^\circ$. Let $C$ be the circumference. Plugging in the values, we get: $$ \frac{9}{C} = \frac{54}{360} $$
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Solve for the circumference $C$ Cross-multiply to solve the equation: $$ 9 \cdot 360 = 54 \cdot C $$ $$ 3240 = 54C $$ Divide both sides by 54: $$ C = \frac{3240}{54} $$ $$ C = 60 $$
The circumference of the circle is $60$ cm.
More Information
The circumference of any circle is $2\pi r$, where $r$ is the radius of the circle.
Tips
A common mistake is to forget that the total angle in a circle is $360^\circ$. Another common mistake is to incorrectly set up the proportion. Make sure the arc length corresponds to the angle, and the circumference corresponds to $360^\circ$.
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