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How do you find the perimeter of a quarter circle?

Understand the Problem

The question is asking for the method to calculate the perimeter of a quarter circle, which involves understanding that the perimeter consists of both the arc length and the straight edges forming the right angle.

Answer

The perimeter of a quarter circle is given by $P = r \left( \frac{\pi}{2} + 2 \right)$.
Answer for screen readers

The perimeter of a quarter circle is given by:

$$ P = r \left( \frac{\pi}{2} + 2 \right) $$

Steps to Solve

  1. Identify Components of the Perimeter

The perimeter of a quarter circle consists of the length of the arc plus the lengths of the two straight edges. We need to calculate these components separately.

  1. Calculate the Arc Length

The formula for the arc length of a circle is given by:

$$ L = r \cdot \theta $$

For a quarter circle, $\theta$ (in radians) is $\frac{\pi}{2}$. Therefore, the arc length can be calculated as:

$$ L = r \cdot \frac{\pi}{2} $$

  1. Calculate the Length of the Straight Edges

The two straight edges forming the right angle are both equal to the radius $r$. Therefore, the total length of the straight edges is:

$$ \text{Straight Edges Length} = r + r = 2r $$

  1. Sum Up the Perimeter Components

Now, we add the arc length and the straight edges lengths to find the total perimeter $P$:

$$ P = L + (2r) $$ Substituting for $L$ from step 2, we have:

$$ P = r \cdot \frac{\pi}{2} + 2r $$

  1. Combine the Terms

We can factor out $r$ from both terms:

$$ P = r \left( \frac{\pi}{2} + 2 \right) $$

The perimeter of a quarter circle is given by:

$$ P = r \left( \frac{\pi}{2} + 2 \right) $$

More Information

The formula shows that the perimeter is dependent on the radius $r$ of the circle. This concept can be used to find the perimeter for any quarter circle by simply substituting in the value of $r$. The presence of $\pi$ indicates the connection to circular geometry, and the two straight segments reflect the nature of the quarter circle as part of a full circle.

Tips

  • Confusing Angle Measurements: Make sure to use radians when calculating the arc length. Many might mistakenly use degrees.
  • Forgetting to Add Both Straight Edges: Remember that both straight edges contribute to the perimeter and should both be included in the total calculation.
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