Math help- polygons and circles **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:** -

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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?

Holt Mcdougal Larson Pre-algebra: Student Edition 2012

1st Edition

ISBN:9780547587776

Author:HOLT MCDOUGAL

Publisher:HOLT MCDOUGAL

Chapter6: Ratio, Proportion, And Probability

Section: Chapter Questions

Problem 3CR

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Math help- polygons and circles

**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:**
-

Two-dimensional figure measured in terms of radius. It is formed by a set of points that are at a constant or fixed distance from a fixed point in the center of the plane. The parts of the circle are circumference, radius, diameter, chord, tangent, secant, arc of a circle, and segment in a circle.

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