A Regular Octagon Has An Apothem Measuring 10 In, And A Perimeter Of 66.3 In.What Is The Area Of The Octagon, Rounded To The Nearest Square Inch?A. 88 In. $^2$B. 175 In. $^2$C. 333 In. $^2$D. 700 In. $^2$
Understanding the Problem
A regular octagon is a polygon with eight equal sides and eight equal angles. Given the apothem (the distance from the center of the octagon to one of its vertices) and the perimeter (the total length of all sides), we need to find the area of the octagon. The apothem measures 10 inches, and the perimeter is 66.3 inches.
Key Concepts and Formulas
To solve this problem, we need to understand the relationship between the apothem, the perimeter, and the area of a regular polygon. The key concepts and formulas involved are:
- Apothem: The distance from the center of the polygon to one of its vertices.
- Perimeter: The total length of all sides of the polygon.
- Area: The amount of space inside the polygon.
- Regular Polygon Formula: The area of a regular polygon can be calculated using the formula: A = (n * s^2) / (4 * tan(π/n)), where n is the number of sides, s is the length of one side, and π is a mathematical constant approximately equal to 3.14.
- Apothem Formula: The apothem of a regular polygon can be calculated using the formula: a = s / (2 * tan(Ï€/n)), where a is the apothem, s is the length of one side, and n is the number of sides.
Calculating the Side Length
To find the area of the octagon, we first need to calculate the length of one side. We can use the apothem formula to find the side length:
a = s / (2 * tan(Ï€/n))
Rearranging the formula to solve for s, we get:
s = 2 * a * tan(Ï€/n)
Substituting the given values, we get:
s = 2 * 10 * tan(Ï€/8)
Using a calculator to evaluate the expression, we get:
s ≈ 2 * 10 * 0.41421356
s ≈ 8.28 inches
Calculating the Area
Now that we have the side length, we can use the regular polygon formula to find the area:
A = (n * s^2) / (4 * tan(Ï€/n))
Substituting the values, we get:
A = (8 * 8.28^2) / (4 * tan(Ï€/8))
Using a calculator to evaluate the expression, we get:
A ≈ (8 * 68.2624) / (4 * 0.41421356)
A ≈ 333.33 square inches
Rounding to the nearest square inch, we get:
A ≈ 333 square inches
Conclusion
The area of the regular octagon is approximately 333 square inches, rounded to the nearest square inch.
Answer
The correct answer is:
C. 333 in. $^2$
Understanding the Problem
A regular octagon is a polygon with eight equal sides and eight equal angles. Given the apothem (the distance from the center of the octagon to one of its vertices) and the perimeter (the total length of all sides), we need to find the area of the octagon. The apothem measures 10 inches, and the perimeter is 66.3 inches.
Key Concepts and Formulas
To solve this problem, we need to understand the relationship between the apothem, the perimeter, and the area of a regular polygon. The key concepts and formulas involved are:
- Apothem: The distance from the center of the polygon to one of its vertices.
- Perimeter: The total length of all sides of the polygon.
- Area: The amount of space inside the polygon.
- Regular Polygon Formula: The area of a regular polygon can be calculated using the formula: A = (n * s^2) / (4 * tan(π/n)), where n is the number of sides, s is the length of one side, and π is a mathematical constant approximately equal to 3.14.
- Apothem Formula: The apothem of a regular polygon can be calculated using the formula: a = s / (2 * tan(Ï€/n)), where a is the apothem, s is the length of one side, and n is the number of sides.
Calculating the Side Length
To find the area of the octagon, we first need to calculate the length of one side. We can use the apothem formula to find the side length:
a = s / (2 * tan(Ï€/n))
Rearranging the formula to solve for s, we get:
s = 2 * a * tan(Ï€/n)
Substituting the given values, we get:
s = 2 * 10 * tan(Ï€/8)
Using a calculator to evaluate the expression, we get:
s ≈ 2 * 10 * 0.41421356
s ≈ 8.28 inches
Calculating the Area
Now that we have the side length, we can use the regular polygon formula to find the area:
A = (n * s^2) / (4 * tan(Ï€/n))
Substituting the values, we get:
A = (8 * 8.28^2) / (4 * tan(Ï€/8))
Using a calculator to evaluate the expression, we get:
A ≈ (8 * 68.2624) / (4 * 0.41421356)
A ≈ 333.33 square inches
Rounding to the nearest square inch, we get:
A ≈ 333 square inches
Q&A
Q: What is the formula for calculating the area of a regular polygon?
A: The area of a regular polygon can be calculated using the formula: A = (n * s^2) / (4 * tan(π/n)), where n is the number of sides, s is the length of one side, and π is a mathematical constant approximately equal to 3.14.
Q: How do I calculate the side length of a regular polygon?
A: To find the side length of a regular polygon, you can use the apothem formula: a = s / (2 * tan(Ï€/n)), where a is the apothem, s is the length of one side, and n is the number of sides.
Q: What is the apothem of a regular polygon?
A: The apothem of a regular polygon is the distance from the center of the polygon to one of its vertices.
Q: How do I calculate the area of a regular octagon?
A: To find the area of a regular octagon, you can use the regular polygon formula: A = (n * s^2) / (4 * tan(π/n)), where n is the number of sides, s is the length of one side, and π is a mathematical constant approximately equal to 3.14.
Q: What is the relationship between the apothem and the perimeter of a regular polygon?
A: The apothem and the perimeter of a regular polygon are related by the formula: a = s / (2 * tan(Ï€/n)), where a is the apothem, s is the length of one side, and n is the number of sides.
Conclusion
The area of a regular octagon can be calculated using the regular polygon formula: A = (n * s^2) / (4 * tan(π/n)), where n is the number of sides, s is the length of one side, and π is a mathematical constant approximately equal to 3.14. The apothem and the perimeter of a regular polygon are related by the formula: a = s / (2 * tan(π/n)), where a is the apothem, s is the length of one side, and n is the number of sides.
Answer
The correct answer is:
C. 333 in. $^2$