Problem 10 - Basic Engineering Mathematics

Question

Calculate the area of a regular octagon, if each side is \(5 \mathrm{~cm}\) and the width across the flats is \(12 \mathrm{~cm}\).

Step-by-Step Solution

Answer

The area of the regular octagon is 120 cm².

Step 1: Understand the Properties of a Regular Octagon

A regular octagon is an eight-sided polygon with all sides and angles being equal. To find its area, we need the measurement of its side length and apothem or the width across the flats (perpendicular distance from one side to the opposite side).

Step 2: Identify the Given Values

We are given that each side of the octagon is 5 cm, and the distance across the flats is 12 cm. These are key measurements needed to determine the area of the octagon.

Step 3: Determine the Apothem

The apothem of a regular octagon can be calculated using the distance across the flats, which is given as 12 cm. For a regular octagon, the apothem (\(a\)) is half the distance across the flats, so \(a = \frac{12}{2} = 6\) cm.

Step 4: Apply the Area Formula for a Regular Octagon

The area \(A\) of a regular octagon can be calculated using the formula: \[ A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \]First, find the perimeter: since each side is 5 cm, the perimeter is \(8 \times 5 = 40\) cm. Now substitute the perimeter and apothem into the formula: \[ A = \frac{1}{2} \times 40 \times 6 = 120 \text{ cm}^2 \]

Step 5: Calculate the Final Area

Substituting the known values, the area of the octagon is 120 cm², confirming the final calculation is correct.

Key Concepts

Regular Octagon

A regular octagon is a fascinating shape in geometry, primarily because of its equal sides and angles. Imagine an octagon as a stop sign, where every side is identical, giving it a symmetry that's both aesthetically pleasing and mathematically intriguing. All eight interior angles of a regular octagon are equal, each measuring 135 degrees. This evenness allows for straightforward calculations when determining the properties of the shape. It’s important when solving geometry problems to know its components, as understanding these can make solving for area and other properties much simpler.

Area Calculation

The area of a regular octagon can be found using a specific formula that involves both its perimeter and its apothem. Calculating the area is crucial when you want to know how much space the shape occupies. The formula used here is: \[ A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \] Where:

• "Perimeter" refers to the total length around the octagon.
• "Apothem" is the distance from the center to the midpoint of one of its sides.

To find the area:

• First, determine the perimeter by multiplying the side length by eight since there are 8 sides.
• Multiply this perimeter by the apothem.
• Finally, take half of the resulting product to find the area.

This method shows why understanding both perimeter and apothem are crucial in area calculations.

Apothem

The apothem is a key element in determining a polygon's area and is particularly vital when working with regular octagons. In essence, the apothem is the shortest distance from the center of the octagon to one of its sides, making an angle of 90 degrees with the side. Think of it as a radius-like line, but instead of stretching outward to the corner of a side, it meets the middle of any side perpendicularly.
For a regular octagon, the apothem can be calculated as half the width across the flats, or sometimes directly with the side length and using trigonometry. In this exercise, the apothem helps simplify calculations because it reduces the complexity involved with angles and provides a consistent point of calculation for the entire polygon. Calculating the apothem with known measures of width across the flats is straightforward and simplifies further area calculations, making it essential to understand its role in geometry.