Question

A stop sign is a regular octagon whose sides are each 10 inches long and whose apothems are each 12 inches long. Find the area of a stop sign.

Short answer

The area of the stop sign is 480 square inches.

Step-by-step solution

Understanding the Problem

To find the area of a regular octagon, which is a polygon with 8 equal sides, we need certain measurements. We know that each side of the octagon is 10 inches and the apothem, which is the line from the center to the midpoint of one of its sides, is 12 inches.

Formula for Area of a Regular Polygon

The formula for the area of a regular polygon is \( A = \frac{1}{2} \times P \times a \) where \( P \) is the perimeter of the polygon and \( a \) is the apothem. In this case, the polygon is the octagon.

Calculate the Perimeter

Since the octagon has 8 equal sides each of length 10 inches, its perimeter \( P \) is calculated by multiplying the side length by the number of sides: \( P = 8 \times 10 = 80 \) inches.

Substitute Values into the Area Formula

Now substitute the values of the perimeter and apothem into the area formula: \( A = \frac{1}{2} \times 80 \times 12 \).

Simplify to Find the Area

Simplifying the expression, we calculate: \[ A = \frac{1}{2} \times 80 \times 12 = 40 \times 12 = 480 \text{ square inches} \]. Thus, the area of the stop sign is 480 square inches.

Key concepts

Understanding a Regular Octagon

A regular octagon is a geometric shape characterized by having eight equal-length sides and eight equal angles. This symmetry makes regular octagons a member of the regular polygon family. Regular polygons have sides that are all the same length and angles that are all identical. This uniformity simplifies calculations related to their geometry, including finding their area.

Regular octagons are prominent in architecture and design due to their pleasing symmetry and balance. They can be seen in various contexts, such as stop signs and tiles. The properties of a regular octagon can be used to calculate significant values like its perimeter and area. To determine these measurements, you need to know the length of a side or, in some cases, additional dimensions like the apothem.

Calculating the Area of Polygons

The area of a polygon is the total space enclosed within its sides. For regular polygons, like a regular octagon, the area can be calculated using a specific formula. This formula is particularly useful when a simple shape, such as a square, cannot represent the polygon. The formula for the area of a regular polygon is:

• \( A = \frac{1}{2} \times P \times a \)

where \( P \) is the perimeter of the polygon, and \( a \) is the apothem.

The perimeter is straightforward to calculate by multiplying the length of one side by the total number of sides. The apothem can be thought of as a helpful measurement; it is the shortest distance from the center of the polygon to the middle of one of its sides. Plugging these values into the formula gives us the area, which provides an insight into how much space the shape covers on a surface.

Understanding and Using the Apothem

The apothem plays a crucial role in calculating the area of regular polygons. It is a line segment from the center of the polygon to the midpoint of one of its sides, and it is always perpendicular to that side. In a regular octagon, the apothem is essential for determining the size of the areas formed by dividing the octagon into triangles which can be used in area calculations.

In the formula \( A = \frac{1}{2} \times P \times a \), the apothem (denoted by \( a \)) helps bridge the perimeter, a straightforward measure, with the area. Because of the symmetry of regular polygons, calculating the apothem may involve using special geometric principles, such as trigonometry for those undertaking more detailed studies.

Practically, knowing the apothem allows for precise configurations of the interior of the shape, especially when it is part of intricate designs like floor patterns or complex shapes like octagonal buildings. Understanding this measure ensures accurate results every time you calculate the area for regular polygons.