Question

The area of a regular octagon is \(392.4\) square meters, and an apothem is \(10.9\) meters long. a. Find the perimeter. b. Find the length of one side.

Short answer

The perimeter is 72 meters, and the length of one side is 9 meters.

Step-by-step solution

Understand the Formula for the Area of a Regular Octagon

The area \( A \) of a regular polygon can be calculated using the formula \( A = \frac{1}{2} \times ext{Perimeter} \times ext{Apothem} \). For an octagon, this formula applies directly where the apothem and area are known.

Rearrange the Formula to Find the Perimeter

Rearrange the formula for area to solve for the perimeter (\( P \)): \( A = \frac{1}{2} \times P \times ext{Apothem} \). Therefore, \( P = \frac{2A}{ ext{Apothem}} \).

Substitute Known Values to Find the Perimeter

Substitute \( A = 392.4 \) and \( ext{Apothem} = 10.9 \) into the equation for the perimeter: \( P = \frac{2 imes 392.4}{10.9} = 72 \; \text{meters} \).

Calculate the Length of One Side

Since the perimeter (\( P \)) is the total of all side lengths, divide the perimeter by the number of sides in the octagon to find the length of one side: \( ext{Side Length} = \frac{P}{8} = \frac{72}{8} = 9 \; \text{meters} \).

Key concepts

Regular Octagon

A regular octagon is a specific type of polygon consisting of eight equal-length sides and eight identical angles. It is important to distinguish it from random octagons, which may have differing side lengths and angles. Regular means uniformity in both measurements, making geometric calculations, like area and perimeter, more straightforward.
The octagon is a prevalent shape in architecture and design due to its symmetrical properties.

Area Calculation

To calculate the area of a regular polygon like the octagon, one commonly used method involves both the perimeter and the apothem. The general formula for the area of a regular polygon is given by:
\[ A = \frac{1}{2} \times P \times \text{Apothem} \]
This equation implies that you need two key parameters: the perimeter and the apothem of the octagon. The area calculation becomes easier if these values are known, as it avoids elaborate manual measurements. In this exercise, the area of the octagon is provided, which simplifies the task of finding other values.

Perimeter of a Polygon

A polygon's perimeter is the total distance around it, calculated by summing all side lengths. In the case of a regular octagon, this involves multiplying one side's length by eight. If the side's length is unknown, use the area formula to find it.

• Rearrange the area formula: \[ P = \frac{2A}{\text{Apothem}} \]
• Plug in known values for area and apothem to find the perimeter.

This rearrangement leverages the provided area to directly compute the perimeter. Knowing the perimeter is crucial, as it easily leads to the length of individual sides.

Apothem

The apothem is a line from the center to the midpoint of one of the sides of a regular polygon. It is perpendicular to that side. In regular polygons, the apothem is equal for all sides, providing an important geometric tool.
The role of the apothem in calculations is pivotal in regular octagons.

• It acts as a height in the area equation for regular polygons.
• Knowing the apothem helps determine both perimeter and side lengths when area is known.

In this instance, the apothem simplifies the complex geometrical space inside the octagon by breaking it into manageable calculations.

Polygon Side Length

The side length of a regular polygon like an octagon is quickly determined once the perimeter is known. Each side is an equal portion of the total perimeter. To find the side length:

• Calculate perimeter first if not given.
• Divide the perimeter by the number of sides. For an octagon, this is eight.

This method ensures you effortlessly find the length of each side without direct measurement. Accurate side lengths are beneficial beyond this exercise, aiding in designs and further mathematical computations involving the octagon.