Mrs. Frank loves the clock in her classroom because it has the school colors, green and purple. The shape of the clock is a regular dodecagon with a radius of 14 centimeters. |Centered on the clock's face is a green cirele of radius 9 cm. If the region outside the circle is purple, which color has more area?
Mrs. Frank loves the clock in her classroom because it has the school colors, green and purple. The shape of the clock is a regular dodecagon with a radius of 14 centimeters. |Centered on the clock's face is a green cirele of radius 9 cm. If the region outside the circle is purple, which color has more area?
Mrs. Frank loves the clock in her classroom because it has the school colors, green and purple. The shape of the clock is a regular dodecagon with a radius of 14 centimeters. |Centered on the clock's face is a green cirele of radius 9 cm. If the region outside the circle is purple, which color has more area?
Transcribed Image Text:**Problem 7:**
Mrs. Frank loves the clock in her classroom because it has the school colors, green and purple. The shape of the clock is a regular dodecagon with a radius of 14 centimeters. Centered on the clock’s face is a green circle of radius 9 cm. If the region outside the circle is purple, which color has more area?
**Explanation of Diagram:**
The image shows a clock with two main components:
1. **Outer Shape:** A regular dodecagon (12 sides) with a radius of 14 centimeters.
2. **Inner Circle:** A circle with a radius of 9 centimeters centered on the face of the clock.
The question requires finding which of the two colors, green (representing the inner circle) or purple (representing the outer region of the dodecagon), has more area.
### Detailed Steps for Calculation:
1. **Calculate the area of the dodecagon:**
- A regular dodecagon can be divided into 12 isosceles triangles. The area \(A\) of each triangle can be given by:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{side length} \times \text{apothem}
\]
- The total area \(A_{\text{dodecagon}}\) will then be:
\[
A_{\text{dodecagon}} = 12 \times A_{\text{triangle}}
\]
- For simplicity, the approximation formula for the area of a regular dodecagon with radius \(r\) is:
\[
A_{\text{dodecagon}} = 3r^2
\]
2. **Calculate the area of the green circle:**
- The area of a circle is given by:
\[
A_{\text{circle}} = \pi r^2
\]
### Using the given values:
1. **Area of the dodecagon:**
- Given radius \(r = 14 \, \text{cm}\):
\[
A_{\text{dodecagon}} = 3 \times (14^2) = 3 \times 196 = 588 \, \text{cm}^2
\]
2. **Area of the circle:**
-
Definition Definition Two-dimentional plane figure composed of a finite number of straight line segments connected to form a closed chain or circuit. A polygonal circuit's segments are known as its edges or sides, and the points where two edges meet are known as its vertices or corners.
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![**Problem 7:**
Mrs. Frank loves the clock in her classroom because it has the school colors, green and purple. The shape of the clock is a regular dodecagon with a radius of 14 centimeters. Centered on the clock’s face is a green circle of radius 9 cm. If the region outside the circle is purple, which color has more area?
**Explanation of Diagram:**
The image shows a clock with two main components:
1. **Outer Shape:** A regular dodecagon (12 sides) with a radius of 14 centimeters.
2. **Inner Circle:** A circle with a radius of 9 centimeters centered on the face of the clock.
The question requires finding which of the two colors, green (representing the inner circle) or purple (representing the outer region of the dodecagon), has more area.
### Detailed Steps for Calculation:
1. **Calculate the area of the dodecagon:**
- A regular dodecagon can be divided into 12 isosceles triangles. The area \(A\) of each triangle can be given by:
\[
A_{\text{triangle}} = \frac{1}{2} \times \text{side length} \times \text{apothem}
\]
- The total area \(A_{\text{dodecagon}}\) will then be:
\[
A_{\text{dodecagon}} = 12 \times A_{\text{triangle}}
\]
- For simplicity, the approximation formula for the area of a regular dodecagon with radius \(r\) is:
\[
A_{\text{dodecagon}} = 3r^2
\]
2. **Calculate the area of the green circle:**
- The area of a circle is given by:
\[
A_{\text{circle}} = \pi r^2
\]
### Using the given values:
1. **Area of the dodecagon:**
- Given radius \(r = 14 \, \text{cm}\):
\[
A_{\text{dodecagon}} = 3 \times (14^2) = 3 \times 196 = 588 \, \text{cm}^2
\]
2. **Area of the circle:**
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd23b2e37-8ff8-4dcd-9ea0-dfa5d98ae8ea%2F8c0456db-1dfe-4d37-ae2b-28af75565b0b%2Fh32jdgb_processed.png&w=3840&q=75)
