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Magazine Chapter 7: Problem 17
Find the apothen, perimeter and area to the nearest tenth: regular decagon with radius 10 .
Short Answer
Expert verified
Apothem is 9.5, perimeter is 62.0, and area is 294.5.
Step by step solution
01
Understand the properties of a regular decagon
A regular decagon has 10 equal sides and 10 equal angles. The radius given is the distance from the center of the decagon to a vertex.
02
Calculate the side length of the decagon
The side length \( s \) of a regular decagon with a radius \( r \) is calculated using the formula: \[ s = 2r \cdot \sin\left(\frac{\pi}{n}\right) \]where \( n \) is the number of sides, which is 10 in this case. Substituting the values we get:\[ s = 2 \cdot 10 \cdot \sin\left(\frac{\pi}{10}\right) \approx 6.2 \]
03
Calculate the apothem (central height)
The apothem \( a \) of a regular decagon is calculated using the formula:\[ a = r \cdot \cos\left(\frac{\pi}{n}\right) \]Substitute \( r = 10 \) and \( n = 10 \):\[ a = 10 \cdot \cos\left(\frac{\pi}{10}\right) \approx 9.5 \]
04
Calculate the perimeter of the decagon
The perimeter \( P \) of the decagon is simply the side length multiplied by the number of sides:\[ P = 10 \times s = 10 \times 6.2 = 62.0 \]
05
Calculate the area of the decagon
The area \( A \) of a regular polygon is given by:\[ A = \frac{1}{2} \times P \times a \]Substitute \( P = 62.0 \) and \( a = 9.5 \):\[ A = \frac{1}{2} \times 62.0 \times 9.5 \approx 294.5 \]
06
Round each value to the nearest tenth
The calculated values are already rounded to the nearest tenth: Apothem = 9.5, Perimeter = 62.0, Area = 294.5.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Apothem
The apothem of a regular polygon, like our decagon, plays a vital role in calculating its area. It acts as the line segment from the center to the midpoint of one of the sides, perpendicular to the side. For a regular decagon, the apothem can be found using trigonometry.
The formula for calculating the apothem \( a \) is:
The formula for calculating the apothem \( a \) is:
- \( a = r \cdot \cos\left(\frac{\pi}{n}\right) \)
- \( a = 10 \cdot \cos\left(\frac{\pi}{10}\right) \approx 9.5 \)
Perimeter
The perimeter is the total length around the decagon. For a regular polygon, this involves multiplying the side length by the number of sides. First, we need the side length \( s \), which we calculated using:
- \( s = 2r \cdot \sin\left(\frac{\pi}{n}\right) \)
- \( P = 10 \times s = 10 \times 6.2 = 62.0 \)
Area Calculation
Calculating the area of a regular decagon involves understanding its apothem and perimeter. Using the formula:
- \( A = \frac{1}{2} \times P \times a \)
- \( A = \frac{1}{2} \times 62.0 \times 9.5 \approx 294.5 \)
Trigonometry
Trigonometry helps to discover lengths and angles within geometric shapes. For a regular decagon, it is crucial in determining the side length and apothem. We often use the sine and cosine functions:
- \( s = 2r \cdot \sin\left(\frac{\pi}{n}\right) \)
- \( a = r \cdot \cos\left(\frac{\pi}{n}\right) \)
Geometry Formulas
Geometry provides us with numerous formulas beneficial for solving problems related to polygons, like the regular decagon. Fundamental formulas help calculate perimeter, areas, and apothem efficiently:
- Perimeter: \( P = n \times s \)
- Area of a regular polygon: \( A = \frac{1}{2} \times P \times a \)
- Apothem using Cosine: \( a = r \cdot \cos\left(\frac{\pi}{n}\right) \)
- Side length using Sine: \( s = 2r \cdot \sin\left(\frac{\pi}{n}\right) \)
