/*! 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 15 Find the area of each polygon. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 15

Find the area of each polygon. A regular hexagon with perimeter 72

Short Answer

Expert verified
The area of the given hexagon is 187.2 square units.

Step by step solution

01

Determine the Sidelength

Calculate the sidelength (s) of the hexagon. The perimeter of a regular hexagon is six times the sidelength. Hence, divide the given perimeter by 6. For a hexagon with a perimeter of 72, we have sidelength s = 72 / 6 = 12.
02

Calculate the Area

With the sidelength determined in step 1, we can now calculate the area of the hexagon using the formula: A = \(\frac{3\sqrt{3}}{2}\) s虏. Substituting s = 12 into the formula, we get: A = \(\frac{3\sqrt{3}}{2}\) * 12虏 = 187.2 square units.

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.

Calculating Area
When it comes to calculating the area of polygons, each shape has its own unique formula. For a regular hexagon, this process involves a specific approach because all sides and angles are equal. Here's a breakdown:
  • The formula used for the area of a regular hexagon is: \[A = \frac{3\sqrt{3}}{2} s^2\]where \(s\) is the length of one side of the hexagon. This formula is derived from the fact that a hexagon can be divided into 6 equilateral triangles.
  • For an equilateral triangle, the area is given by \(\frac{\sqrt{3}}{4} s^2\), and since a hexagon has six of these triangles, the area becomes the hexagon's area formula.
  • Plug in the side length into the hexagon area formula to find your value.
This hexagonal formula highlights the beauty of geometry by showing relationships between side measurements and angles. Be sure to find the side length accurately for precise calculations.
Perimeter of Polygons
To understand the perimeter of polygons, it's crucial to know what perimeter involves. Perimeter refers to the total length around a polygon. For any regular polygon:
  • The perimeter can be calculated by multiplying the length of one side by the total number of sides.
  • For example, in a regular hexagon, which has six equal sides, the formula for finding the perimeter is: \[P = 6 \times s\]where \(s\) is the side length.
  • This formula simplifies finding perimeter by using multiplication instead of individual measurements.
Understanding the perimeter helps in various real-life applications like fencing a garden or framing. It's a handy concept that simplifies measurement tasks by breaking them into manageable calculations, especially for regular shapes.
Geometry Problems
Geometry problems often involve understanding various properties and relationships of shapes to solve practical and theoretical issues. In the case of regular hexagons:
  • They have equal side lengths and angles, making them symmetrical and aesthetically pleasing.
  • Breaking down complex shapes into simpler units (like splitting a hexagon into triangles) helps solve geometry problems efficiently.
  • Utilizing known formulas not only saves time but also ensures accuracy. For instance, using area and perimeter formulas, one can quickly solve issues of design or space calculation.
  • Geometry is vastly used in fields like architecture, engineering, and art. Solving for areas and perimeters is not just academic but also practical in these professional settings.
Geometry problems enhance critical thinking by providing challenges that require creative and logical solutions. They form the foundation of many design and spatial reasoning tasks encountered daily.

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

Find the area of each figure. A parallelogram with a \(45^{\circ}\) angle and sides 6 and 10

Use a calculator or the trigonometry table on page 311 to find the area of each figure to the nearest tenth. An isosceles triangle with a \(32^{\circ}\) vertex angle and a base of \(8 \mathrm{cm}\)

Draw any circle \(O\) and any circle \(P .\) Construct a circle whose area equals the sum of the areas of circle \(O\) and circle \(P\).

Find the circumference of a circle inscribed in a rhombus with diagonals \(12 \mathrm{cm}\) and \(16 \mathrm{cm}\).

In \(\triangle L M N,\) altitude \(\overline{L K}\) is 12 cm long. Through point \(J\) of \(\overline{L K}\) a line is drawn parallel to \(\overline{M N}\), dividing the triangle into two regions with equal areas. Find \(L J\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

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.