/*! 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 211 (a) Find an expression involving... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 211

(a) Find an expression involving "sin \(\frac{2 \pi}{n} "\) for the ratio \(\frac{\text { area of inscribed regular } n \text { -gon }}{\text { area of circumscribed circle }}\). (b) Find an expression involving "tan \(\frac{\pi}{n} "\) for the ratio \(\frac{\text { area of circumscribed regular } n \text { -gon }}{\text { area of inscribed circle }}\)

Short Answer

Expert verified
(a) \( \frac{n}{2\pi} \sin\left(\frac{2\pi}{n}\right) \); (b) \( \frac{n}{\pi} \tan\left(\frac{\pi}{n}\right) \).

Step by step solution

01

Understanding the Situation in (a)

We need to find an expression for the ratio \( \frac{\text{Area of an inscribed regular } n \text{-gon}}{\text{Area of a circumscribed circle}} \). The regular \( n \)-gon is inscribed in a circle of radius \( R \).
02

Area of Inscribed Regular n-gon (a)

The area \( A \) of a regular \( n \)-gon with a side length \( s \) inscribed in a circle of radius \( R \) is given by: \[ A = \frac{n}{2} R^2 \sin\left(\frac{2\pi}{n}\right) \]This formula comes from dividing the \( n \)-gon into \( n \) isosceles triangles formed by the center and two adjacent vertices.
03

Area of Circumscribed Circle (a)

The area of the circle circumscribing the \( n \)-gon with radius \( R \) is:\[ \pi R^2 \]
04

Forming the Expression for Ratio (a)

Now, the ratio is:\[\frac{\text{Area of inscribed regular } n \text{-gon}}{\text{Area of circumscribed circle}} = \frac{\frac{n}{2} R^2 \sin\left(\frac{2\pi}{n}\right)}{\pi R^2} = \frac{n}{2\pi}\sin\left(\frac{2\pi}{n}\right) \]
05

Understanding the Situation in (b)

For part (b), we are looking for an expression for the ratio \( \frac{\text{Area of circumscribed regular } n \text{-gon}}{\text{Area of an inscribed circle}} \), where the \( n \)-gon is circumscribing a circle of radius \( r \).
06

Area of Circumscribed Regular n-gon (b)

The area \( A \) of a regular \( n \)-gon that circumscribes a circle of radius \( r \) is given by:\[ A = nr^2 \tan\left(\frac{\pi}{n}\right) \]This is derived by considering the \( n \)-gon to be composed of \( n \) tangential triangles to the inscribed circle.
07

Area of Inscribed Circle (b)

The area of the circle inscribed in the \( n \)-gon with radius \( r \) is:\[ \pi r^2 \]
08

Forming the Expression for Ratio (b)

The desired ratio is:\[\frac{\text{Area of circumscribed regular } n \text{-gon}}{\text{Area of inscribed circle}} = \frac{n r^2 \tan\left(\frac{\pi}{n}\right)}{\pi r^2} = \frac{n}{\pi} \tan\left(\frac{\pi}{n}\right) \]

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.

Regular Polygon
A regular polygon is a fascinating geometric figure where all sides and angles are equal. These polygons can take various shapes, such as triangles, squares, or pentagons, depending on the number of sides (denoted as \( n \)). One of the special features of a regular polygon is that it can perfectly fit inside and outside a circle.
  • Inscribed Polygon: This is a regular polygon that fits snugly inside a circle, touching the circle at each vertex. This circle is known as the circumscribed circle of the polygon.
  • Circumscribed Polygon: Conversely, this is a regular polygon that has a circle snugly fitted inside it, touching it on each side. This circle is called the inscribed circle of the polygon.
Understanding the nature of these polygons helps in solving problems related to geometry because it involves understanding concepts like angles, lines, and circles. By learning the properties of regular polygons, calculating areas and other measures becomes easier.
Trigonometry
Trigonometry is an essential branch of mathematics that deals with angles, circles, and waves. In problems involving regular polygons, trigonometry is fundamental because it gives us the tools to calculate side lengths, angles, and areas using functions like sine (\( \sin \)) and tangent (\( \tan \)).
  • For inscribed polygons, the area can be found using a formula involving the sine function. This results from splitting the polygon into isosceles triangles and calculating their individual areas before summing them.
  • For circumscribed polygons, the tangent function is used, which helps in determining the area composed of tangential triangles.
Trigonometry simplifies complex geometric configurations by breaking them down into manageable calculations. It transforms geometric figures into numerical expressions, making it easier to compare their sizes, like finding ratios of different areas.
Inscribed and Circumscribed Circles
The concept of inscribed and circumscribed circles is vital in understanding the geometric relationships between polygons and circles.
  • An inscribed circle is the largest circle that fits inside a polygon, touching all its sides. It is crucial when finding the area of polygons that encapsulate this circle since the radius of the inscribed circle is utilized in such calculations.
  • A circumscribed circle, on the other hand, is the smallest circle that fits around a polygon, touching all its vertices. Its importance lies in deriving the area of a polygon that sits within this circle.
These circles help in simplifying the computation of complex polygonal areas, particularly when the circle and polygon share a center. By understanding the properties of these circles, one can easily determine the geometric mean and area ratios that involve these shapes. They facilitate the conversion of geometric figures into trigonometric problems, making it more straightforward to apply mathematical techniques.

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

(a)(i) Sketch a unit 2 D-cube as follows. Starting with two unit \(1 \mathrm{D}\) -cubes one directly above the other. Then join up each vertex in the upper 1D-cube to the vertex it corresponds to in the lower 1 D-cube (directly beneath it). (ii) Label each vertex of your sketch with coordinates \((x, y)(x, y=0\) or 1) so that the lower \(2 \mathrm{D}\) -cube has the equation " \(y=0\) " and the upper 2D-cube has the equation " \(y=1 "\). (b)(i) Sketch a unit 3D-cube, starting with two unit \(2 \mathrm{D}\) -cubes \(-\) one directly above the other. Then join up each vertex in the upper \(2 \mathrm{D}\) -cube to the vertex it corresponds to in the lower \(2 \mathrm{D}\) -cube (directly beneath it). (ii) Label each vertex of your sketch with coordinates \((x, y, z)\) (where each \(x, y, z=0\) or 1 ) so that the lower \(2 \mathrm{D}\) -cube has the equation \(" z=0 "\) and the upper \(2 \mathrm{D}\) -cube has the equation \(" z=1 "\). (c)(i) Now sketch a unit \(4 \mathrm{D}\) -cube in the same way \(-\) starting with two unit 3D-cubes, one "directly above" the other. (ii) Label each vertex of your sketch with coordinates \((w, x, y, z)\) (where each \(w, x, y, z=0\) or 1\()\) so that the lower 3 D-cube has the equation \(" z=0\) " and the upper 3 D-cube has the equation " \(z=1\) ".

(a) Construct a right angled triangle that explains the standard formula for the distance from \(P=(a, b)\) to \(Q=(d, e)\). (b) Use part (a) to derive the standard formula for the distance from \(P=\) \((a, b, c)\) to \(Q=(d, e, f)\)

Consider the cube with edges of length 2 running parallel to the coordinate axes, with its centre at the origin \((0,0,0),\) and with opposite corners at (1,1,1) and (-1,-1,-1) . The \(x-, y-,\) and \(z\) -axes, and the \(x y\) -, \(y z-,\) and \(z x\) -planes cut this cube into eight unit cubes \(-\) one sitting in each octant. (i) Let \(A=(0,0,1), B=(1,0,0), C=(0,1,0), W=(1,1,1) .\) Describe the solid \(A B C W\) (ii) Let \(D=(-1,0,0), X=(-1,1,1) .\) Describe the solid \(A C D X\). (iii) Let \(E=(0,-1,0), Y=(-1,-1,1) .\) Describe the solid \(A D E Y\). (iv) Let \(Z=(1,-1,1) .\) Describe the solid \(A E B Z\). (v) Let \(F=(0,0,-1)\) and repeat steps (i)-(iv) to obtain the four mirror image solids which lie beneath the \(x y\) -plane. (vi) Describe the solid \(A B C D E F\) which is surrounded by the eight identical solids in (i)-(v).

You are given a pyramid \(A B C D\) with all three faces meeting at \(A\) being right angled triangles with right angles at \(A .\) Suppose \(\underline{A B}=b,\) \(\underline{A C}=c, \underline{A D}=d\) (a) Calculate the areas of \(\triangle A B C, \triangle A C D, \triangle A D B\) in terms of \(b, c, d\). (b) Calculate the area of \(\triangle B C D\) in terms of \(b, c, d\). (c) Compare your answer in part (b) with the sum of the squares of the three areas you found in part (a).

Suppose a regular icosahedron (Problem 189) has edges of length 2. Position vertex \(A\) at the 'North pole', and let \(B C D E F\) be the regular pentagon formed by its five neighbours. (a)(i) Calculate the exact angle between the two faces \(A B C\) and \(A C D\). (ii) How many identical regular icosahedra can one fit together, without overlaps, around a single edge? (b) Let \(\mathcal{C}\) be the circumcircle of \(B C D E F,\) and let \(O\) be the circumcentre of this regular pentagon. (i) Prove that the three edge lengths of the right-angled triangle \(\triangle B O A\) are the edge lengths of the regular hexagon inscribed in the circle \(\mathcal{C}\), of the regular 10 -gon inscribed in the circle \(\mathcal{C},\) and of the regular 5 -gon inscribed in the circle \(\mathcal{C}\). (ii) Calculate the distance separating the plane of the regular pentagon \(B C D E F,\) and the plane of the corresponding regular pentagon joined to the 'South pole'
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.