Question:
In the figure below, a square ABCD is inscribed in a circle. If the length of arc AB is 4\(\pi\) unit, what is the diameter of the circle?
In the figure below, a square ABCD is inscribed in a circle. If the length of arc AB is 4\(\pi\) unit, what is the diameter of the circle?
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For any regular n-sided polygon inscribed in a circle, it divides the circumference into n equal arcs. The central angle subtended by each side is \(360\^{}\circ / n\). For a square, this is \(360\^{}\circ / 4 = 90\^{}\circ\), which means each arc is a quarter of the circle.
Updated On: Oct 3, 2025
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Solution and Explanation
Step 1: Understanding the Concept:
When a square is inscribed in a circle, its vertices lie on the circumference of the circle. The four vertices of the square divide the circumference into four equal arcs. The total circumference of the circle is the sum of the lengths of these four arcs. The diameter of a circle is related to its circumference by the formula \(C = \pi d\).
Step 2: Key Formula or Approach:
1. Recognize that an inscribed square creates four equal arcs on the circle's circumference.
2. Calculate the total circumference by multiplying the length of one arc by 4.
3. Use the circumference formula, \(C = \pi d\), where \(C\) is the circumference and \(d\) is the diameter, to solve for \(d\).
Step 3: Detailed Explanation:
The vertices of the inscribed square ABCD divide the circle into four equal arcs: arc AB, arc BC, arc CD, and arc DA.
We are given that the length of arc AB is \(4\pi\) units.
Since all four arcs are equal, the total circumference \(C\) of the circle is four times the length of arc AB.
\[ C = 4 \times (\text{length of arc AB}) \] \[ C = 4 \times 4\pi = 16\pi \text{ units} \] The formula for the circumference of a circle is \(C = \pi d\), where \(d\) is the diameter.
We can set our calculated circumference equal to the formula to find the diameter.
\[ 16\pi = \pi d \] To solve for \(d\), we divide both sides by \(\pi\).
\[ d = \frac{16\pi}{\pi} = 16 \text{ units} \] Step 4: Final Answer:
The diameter of the circle is 16 units.
When a square is inscribed in a circle, its vertices lie on the circumference of the circle. The four vertices of the square divide the circumference into four equal arcs. The total circumference of the circle is the sum of the lengths of these four arcs. The diameter of a circle is related to its circumference by the formula \(C = \pi d\).
Step 2: Key Formula or Approach:
1. Recognize that an inscribed square creates four equal arcs on the circle's circumference.
2. Calculate the total circumference by multiplying the length of one arc by 4.
3. Use the circumference formula, \(C = \pi d\), where \(C\) is the circumference and \(d\) is the diameter, to solve for \(d\).
Step 3: Detailed Explanation:
The vertices of the inscribed square ABCD divide the circle into four equal arcs: arc AB, arc BC, arc CD, and arc DA.
We are given that the length of arc AB is \(4\pi\) units.
Since all four arcs are equal, the total circumference \(C\) of the circle is four times the length of arc AB.
\[ C = 4 \times (\text{length of arc AB}) \] \[ C = 4 \times 4\pi = 16\pi \text{ units} \] The formula for the circumference of a circle is \(C = \pi d\), where \(d\) is the diameter.
We can set our calculated circumference equal to the formula to find the diameter.
\[ 16\pi = \pi d \] To solve for \(d\), we divide both sides by \(\pi\).
\[ d = \frac{16\pi}{\pi} = 16 \text{ units} \] Step 4: Final Answer:
The diameter of the circle is 16 units.
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