Mensuration of Plane Figures: Exercises and Solutions for Geometry Students, Exercises of Mathematics

Download Mensuration of Plane Figures: Exercises and Solutions for Geometry Students and more Exercises Mathematics in PDF only on Docsity!

OBJECTIVES:

1. Review and master how to find the area of any plane and composite plane figures

INTRODUCTION

Solid Mensuration deals primarily with the various solids. The formulas developed in this text are used extensively in railway engineering, in road and bridge construction, in chemical and physical analyses, and in a large variety of commercial and engineering projects.

Suggested Steps in Computing Solid Mensuration Problems

1. Draw an appropriate figure on which all dimensions are shown.
2. Write down all formulas by means of which the unknown quantities are to be found.
3. Be sure that your work is arranged so that it can be followed at any time by yourself or another person.
4. In many problems, it is helpful to employ literal quantities to denote numerical values in carrying on the work.
5. In completing a problem, it is necessary finally to replace these literal quantities by the numbers they represent.

MENSURATION OF PLANE FIGURES

A large part of the work in Solid mensuration has to do with the computation of surface areas and the volumes of solids. In this connection it is frequently necessary to pass a plane through a solid to form a plane section, find the area of this section, and multiply it by the length of a line. Thus it is important for you to be thoroughly familiar with the mensuration of the standard plane figures. For this reason you should carefully review the following list of formulas relating to plane figures.

Activity 1.

The section of a certain solid consists of a semicircle, a rectangle and a triangle, as shown. The altitude of the rectangle is three times the radius of the semicircle, the altitude of the triangle is twice the same radius, and the area of the triangle is 20. Find the area of the figure.

Activity 2.

A city block is in the form of a parallelogram whose shorter diagonal AB is perpendicular to side BC , as shown in figure

3. The shorter sides represent streets and the longer sides represent avenues. If the distance between the avenues is 400 ft and the length of each street is 500 ft., find the area of the block.

Exercises:

Solve the following problems. Show your solution in a separate sheet of paper

1. Find the area of the largest circle which can be cut from a square of edge 4 inches. What is the area of the materials wasted?
2. The plane area shown in the figure consists of an isosceles trapezoid and a segment of a circle. If the non-parallel sides are tangent to the segment at points A and B, find the area of the composite figure.
3. Find the area of the rectilinear figure shown, if it is the difference between two isosceles trapezoids whose corresponding sides are parallel.

Figure 4Figure 4

Answers Key:

Activity 1.

Solution. Let , and denote the area of the triangle, rectangle, and semicircle, respectively. Using formulas listed in this article under the appropriate headings, we find in terms of r Semicircle is half of a circle, so

Givens:

h = 2x ft b = 2x ft

Givens: b =3x h = 2x

Givens: Diameter: d = 2x Radius is half of diameter: r = x

Solution: Substitute the givens to the formula to find x

Therefore,

Solution: Substitute the givens to the formula

Since the value of x=10ft, we have,

Solution: Substitute the givens to the formula

Since the value of x=10ft, we have,

To find the area of the figure , we will add all the 3 planes formed.

Acitivity 2.

Solution. From B drop the perpendicular BD to line AC. Since BD by construction is the distance between the avenues, we have, BD = 400

Applying the Pythagorean Theorem to right triangle BDC , we obtain,

√

√ = 300

Since the sides of angle DBC are respectively perpendicular to the sides of angle BAC , then angle DBC = angle BAC. Hence right triangle DBC and BAC are similar. Consequently, we can get the value of side AB by

Considering BC as base and AB as altitude of the parallelogram, we have for its area using the formula: A = bh