A circle has a radius of 3mm. A sector of the circle has a central angle of 250°. Find the area of the sector.



To find the area of a sector of a circle, we use the formula:

\[ \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]

where \( \theta \) is the central angle in degrees, and \( r \) is the radius of the circle.

Given:
- Radius \( r = 3 \) mm
- Central angle \( \theta = 250^\circ \)

We substitute these values into the formula:

\[ \text{Area of sector} = \frac{250^\circ}{360^\circ} \times \pi (3 \, \text{mm})^2 \]

First, calculate \( \pi (3 \, \text{mm})^2 \):

\[ \pi (3 \, \text{mm})^2 = \pi \times 9 \, \text{mm}^2 = 9\pi \, \text{mm}^2 \]

Now, substitute this into the sector area formula:

\[ \text{Area of sector} = \frac{250}{360} \times 9\pi \, \text{mm}^2 \]

Simplify the fraction:

\[ \frac{250}{360} = \frac{25}{36} \]

So,

\[ \text{Area of sector} = \frac{25}{36} \times 9\pi \, \text{mm}^2 \]

Multiply:

\[ \text{Area of sector} = \frac{25 \times 9\pi}{36} \, \text{mm}^2 \]

\[ \text{Area of sector} = \frac{225\pi}{36} \, \text{mm}^2 \]

Simplify the fraction:

\[ \text{Area of sector} = \frac{225}{36} \pi \, \text{mm}^2 \]

\[ \text{Area of sector} = \frac{25}{4} \pi \, \text{mm}^2 \]

So, the area of the sector is:

\[ \text{Area of sector} = 6.25\pi \, \text{mm}^2 \]

Approximating \( \pi \approx 3.14 \):

\[ \text{Area of sector} \approx 6.25 \times 3.14 \, \text{mm}^2 \]

\[ \text{Area of sector} \approx 19.625 \, \text{mm}^2 \]

Therefore, the area of the sector is approximately \( 19.625 \, \text{mm}^2 \).

Would you like more details on the steps? Do you have any questions?

Here are 5 additional questions you might be interested in:
1. How do you convert degrees to radians?
2. What is the formula for the circumference of a circle?
3. How do you find the length of an arc in a circle?
4. What is the relationship between a sector and a segment of a circle?
5. How can you find the area of a segment in a circle?

**Tip:** Remember that the area of a sector can also be found using radians: \(\text{Area of sector} = \frac{1}{2} r^2 \theta \) when \(\theta\) is in radians.

Learn how to calculate the area of a sector of a circle with a radius of 3mm and a central angle of 250 degrees using the sector area formula and degrees to radians conversion. Suitable for advanced high school students and anyone interested in circle geometry.

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