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The diagram shows an outer circle of radius 8 cm and an inner circle of radius 5 cm.

Both circles have the same centre.
Find the perimeter of the shaded region.

The shaded region in this diagram is the segment formed between two radii of the outer circle (with radius 8 cm) subtended by a central angle of 140°.

To calculate the perimeter of the shaded region, we need to account for two parts:
1. The arc length of the outer circle (corresponding to the 140° angle).
2. The two straight-line segments from the center to the circumference of the outer circle (these are the radii of 8 cm each).

### Step 1: Find the arc length of the outer circle
The formula for the arc length \( L \) is:
\[
L = \frac{\theta}{360^\circ} \times 2\pi r
\]
where:
- \( \theta = 140^\circ \) is the central angle,
- \( r = 8 \, \text{cm} \) is the radius of the outer circle.

Substitute the values:
\[
L = \frac{140}{360} \times 2\pi \times 8 = \frac{7}{18} \times 16\pi \approx 19.56 \, \text{cm}
\]

### Step 2: Add the two radii
Each radius is 8 cm, and there are two of them. Therefore:
\[
2 \times 8 = 16 \, \text{cm}
\]

### Step 3: Total Perimeter
The total perimeter of the shaded region is the sum of the arc length and the two radii:
\[
\text{Perimeter} = 19.56 \, \text{cm} + 16 \, \text{cm} = 35.56 \, \text{cm}
\]

Thus, the perimeter of the shaded region is approximately **35.56 cm**.

Would you like more details or have any questions? Here are 5 related questions you might consider:

1. How would the perimeter change if the central angle were different?
2. Can we calculate the area of the shaded region instead of the perimeter?
3. What is the length of the arc for the inner circle subtended by the same angle?
4. How does the difference in radii affect the geometry of the shaded region?
5. How do we calculate the total circumference of a circle from its radius?

Tip: To solve arc length problems quickly, remember that it is simply a fraction of the full circumference determined by the central angle!

The diagram shows an outer circle of radius 8 cm and an inner circle of radius 5 cm. Both circles have the same centre. Find the perimeter of the shaded region.

Learn how to calculate the perimeter of a shaded region formed by two radii and an arc of a circle. This problem involves calculating the arc length for a circle with a radius of 8 cm, using a 140° central angle. Suitable for students in Grades 9-11.

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