A circle of radius 10 cm has an arc length of 25.6 cm. Calculate the angle subtended at the center of the circle by the arc ( your answer must be in degrees )

To find the angle subtended at the center of the circle by the arc, we can use the formula that relates the arc length, radius, and central angle:

\[
\theta = \frac{L}{r}
\]

Where:
- \(\theta\) is the angle subtended by the arc (in radians),
- \(L\) is the length of the arc,
- \(r\) is the radius of the circle.

Given:
- \(L = 25.6 \, \text{cm}\)
- \(r = 10 \, \text{cm}\)

Substituting the values into the formula:

\[
\theta = \frac{25.6}{10} = 2.56 \, \text{radians}
\]

To convert this angle from radians to degrees, we use the conversion factor:

\[
\theta (\text{degrees}) = \theta (\text{radians}) \times \frac{180^\circ}{\pi}
\]

Now, calculate the angle in degrees:

\[
\theta (\text{degrees}) = 2.56 \times \frac{180^\circ}{\pi} \approx 146.66^\circ
\]

So, the angle subtended at the center of the circle by the arc is approximately **146.66 degrees**.

Would you like further details or have any questions? Here are some related questions for further practice:

1. How would the angle change if the radius were doubled but the arc length remained the same?
2. What would be the arc length if the angle subtended at the center was 90 degrees?
3. How do you convert an angle from degrees to radians?
4. If the angle subtended at the center is 180 degrees, what is the length of the arc?
5. What is the formula to find the circumference of the circle?

**Tip:** Remember that the full angle in a circle is \(360^\circ\), so any arc's corresponding central angle will be a fraction of this, depending on the arc length and radius.

Learn how to calculate the angle subtended at the center of a circle with radius 10 cm by an arc of length 25.6 cm. This problem involves using the arc length formula and converting radians to degrees. Suitable for high school students studying geometry.

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