/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Find the lengths of the apothem ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 14

Find the lengths of the apothem and the radius of a regular hexagon whose sides have length \(6 \mathrm{cm} .\)

Short Answer

Expert verified
The apothem is \(3 \sqrt{3}\) cm, and the radius is 6 cm.

Step by step solution

01

Understand the Properties of a Regular Hexagon

A regular hexagon is composed of 6 equilateral triangles. Each side of these triangles is equal to the side length of the hexagon, which in this case is 6 cm. The apothem is the distance from the center of the hexagon to the midpoint of one of its sides, and it acts as the height of one of these equilateral triangles. The radius is the distance from the center to one of the vertices.
02

Calculate the Apothem

The apothem of an equilateral triangle can be calculated by using the formula \(a = \frac{s \sqrt{3}}{2}\), where \(s\) is the side length. For our hexagon, this becomes:\[a = \frac{6 \sqrt{3}}{2} = 3 \sqrt{3}\]},
03

Calculate the Radius

The radius of a regular hexagon is equal to the side length of the hexagon. Therefore, the radius \(r\) is:\[r = 6 \text{ cm}\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Apothem
The apothem of a regular hexagon is a valuable measure that contributes to its geometric understanding. In simple terms, the apothem is the line from the center of the hexagon to the midpoint of one of its sides. Think of it as the height when the hexagon is divided into individual triangles.

The apothem's length can be found using the formula for an equilateral triangle's height:
  • For a regular hexagon with side length \(s\), the apothem \(a\) is calculated as \(a = \frac{s \sqrt{3}}{2}\).
In our exercise, since the side length of the hexagon is 6 cm, the apothem is \(3\sqrt{3}\) cm.

This is a crucial measurement in problems involving hexagonal area calculations because
  • The apothem acts as the height of equilateral triangles that make up the hexagon.
  • It helps in determining the area through the formula: \((\frac{1}{2} \times \text{Perimeter} \times \text{Apothem})\).
Equilateral Triangle
At the heart of the regular hexagon's structure are equilateral triangles. A regular hexagon is essentially made up of six of these triangles. Each triangle has equal side lengths, matching the side length of the hexagon. This setup forms a beautiful symmetry that allows for easy calculations.

Equilateral triangles have several characteristics:
  • All sides are of equal length. In our exercise, each side is 6 cm.
  • The angles in an equilateral triangle are all 60 degrees, forming a consistently balanced shape.
  • The height or altitude of each triangle, which acts as the apothem of the hexagon, can be found using \(h = \frac{s \sqrt{3}}{2}\).
Understanding these triangles facilitates calculating other important hexagon features, such as its apothem and the area. These calculations rely on the relationship between side lengths and heights of these equilateral triangles.
Hexagon Radius
The radius of a regular hexagon is an intuitive concept. In a regular hexagon, the radius is just the distance from the center to any of its vertices. Because of the nature of a regular hexagon, this is straightforwardly equal to its side length.

In the given exercise, with a side length of 6 cm, we likewise have:
The radius \(r = 6\) cm.
This correlation between the side length and the radius simplifies many geometric calculations involving hexagons, as they are often used as interchangeably calculated values.
  • The radius plays a key role in constructing the hexagon and understanding its symmetrical properties.
  • It aids in calculations involving circumferential components, such as the circular area enclosing the hexagon.
In discussions about the hexagonal shape, this dimension seamlessly links the geometry of equilateral triangles to the radial symmetry of the hexagon.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Does a rectangle have (a) an incenter? (b) a circumcenter?

Find the distance from the circumcenter to each vertex of an equilateral triangle whose sides have the length \(10 .\)

Draw a triangle and, by construction, find its incenter.

A distributing company plans an Illinois location that would be the same distance from each of its principal delivery sites at Chicago, St. Louis, and Indianapolis. Use a construction method to locate the approximate position of the distributing company.

Prove: \(\quad\) If a circle is divided into \(n\) congruent arcs \((n \geq 3),\) the chords determined by joining consecutive endpoints of these arcs form a regular polygon.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.