Question

Name each part of pentagon PENTA. Find the perimeter of a regular heptagon whose sides are \(8.1\) meters long.

Short answer

The perimeter of the heptagon is 56.7 meters.

Step-by-step solution

Understand the Problem

We are tasked with finding the perimeter of a regular heptagon. A regular heptagon is a seven-sided polygon where each side has the same length.

Determine the Formula for Perimeter

The perimeter of a polygon is calculated by summing the lengths of all its sides. For any regular polygon with side length \(s\) and \(n\) sides, the perimeter \(P\) is given by the formula: \[ P = n \times s \]

Identify Known Values

In this problem, the heptagon has sides of length \(8.1\) meters. Since it's a heptagon, it has \(7\) sides. Therefore, \(n = 7\) and \(s = 8.1\).

Calculate the Perimeter

Substitute the values of \(n\) and \(s\) into the perimeter formula: \[ P = 7 \times 8.1 = 56.7 \] Thus, the perimeter of the heptagon is \(56.7\) meters.

Key concepts

Regular Polygons

A regular polygon is a flat shape with straight sides. The word "regular" means that all the sides are of equal length, and all the angles are equal in size.
These shapes can vary in the number of sides, such as triangles, squares, and hexagons.
In mathematics, regular polygons are quite interesting because they have symmetrical properties and follow specific geometric rules. Understanding how regular polygons work is essential for calculations like finding the perimeter or area.
Here are some key aspects of regular polygons:

• Equal sides: Every side has the same length.
• Equal angles: All internal angles are equal.
• Symmetry: Regular polygons are symmetrical, often in multiple ways.

These properties make regular polygons very predictable, which simplifies calculations like determining their perimeter.

Perimeter Calculation

Calculating the perimeter of a polygon is an essential concept in geometry. The perimeter is the total length around a shape.
It's like taking a walk around the edge of a field to know its boundary length.
For a regular polygon, finding its perimeter is straightforward: simply multiply the number of sides by the length of one side.The formula for the perimeter, especially in regular polygons, is:\[ P = n \times s \]Where:

• \(P\) is the perimeter.
• \(n\) is the number of sides.
• \(s\) is the length of one side.

These calculations are straightforward and save time, especially when dealing with many-sided shapes like heptagons or octagons.

Heptagon Properties

A heptagon is a specific type of polygon with exactly seven sides.
In a regular heptagon, not only are all sides equal, but each internal angle is equal too. This symmetry is what gives it the label "regular."
Heptagons have some unique properties that set them apart from polygons with fewer sides.Here are some details about regular heptagons:

• Seven equal sides and seven equal angles.
• Each interior angle in a regular heptagon is approximately 128.57 degrees. You can calculate this by using the formula: \(\frac{(n-2)\times180}{n}\) where \(n\) is the number of sides.
• The perimeter, as calculated earlier, can be found using the formula \(P = 7 \times s\), where \(s\) is the side length.

Recognizing these properties gives you a foundation for exploring more about heptagons and applying these concepts to solve related problems.