Elementary Geometry for College Students

Daniel C. Alexander, Alexander Koeberlein

ISBN #9781439047903

5th Edition

2,198 Questions

20,921 Students Helped

Homework Questions

Summary

Learning Objectives

Key Concepts

Example Problems

Explanations

Common Mistakes

Summary

This section emphasizes understanding area as the measure of a plane region using square units. It introduces fundamental area postulates and shows how areas are computed for rectangles, parallelograms, and triangles by selecting an appropriate base and its corresponding altitude. Moreover, it illustrates the importance of proper unit conversion and the principle that the area of combined non-overlapping regions is the sum of the individual areas.

Learning Objectives

Explain the concept of area as the measure of a plane region using square units.

Describe and apply the area postulates, including the area-addition postulate and initial postulates.

Calculate the areas of common polygons such as rectangles, parallelograms, and triangles using base and altitude.

Convert and handle different measurement units when computing areas.

Key Concepts

CONCEPT

DEFINITION

Line Segment

A one-dimensional figure measured only in length using linear units such as inches, centimeters, etc.

Plane Region

A closed or bounded portion of a plane which is measured in square units.

Square Unit

The unit used to measure area; a square with each side of length 1 (e.g., 1 in², 1 cm²).

Area Postulate

The accepted idea that if two closed plane figures are congruent, then their areas are equal.

Area-Addition Postulate

States that if two regions do not overlap and share a common boundary, the area of their union equals the sum of their areas.

Base and Altitude

The base is a chosen side of a polygon (especially in triangles and parallelograms) and the altitude is the perpendicular distance from the base to the opposite side.

Example Problems

Example 1

Suppose that two triangles have equal areas. Are the triangles congruent? Why or why not? Are two squares with equal areas necessarily congruent? Why or why not?

Example 2

The area of the square is $12,$ and the area of the circle is 30. Does the area of the entire shaded region equal $42 ?$ Why or why not? (FIGURE CAN'T COPY)

Example 3

Consider the information in Exercise $2,$ but suppose you know that the area of the region defined by the intersection of the square and the circle measures $5 .$ What is the area of the entire colored region? (FIGURE CAN'T COPY)

Example 4

If $M N P Q$ is a rhombus, which formula from this section should be used to calculate its area? (FIGURE CAN'T COPY)

Example 5

In rhombus $M N P Q,$ how does the length of the altitude from $Q$ to $\overline{P N}$ compare to the length of the altitude from $Q$ to $M N ?$ Explain. (FIGURE CAN'T COPY)

Step-by-Step Explanations

QUESTION

How do you calculate the area of a rectangle with a length of 12 cm and a width of 7 cm?

STEP-BY-STEP ANSWER:

Step 1: Identify the dimensions. Here, length (or base) = 12 cm and width (or altitude) = 7 cm.
Step 2: Apply Postulate 21, which states that the area A of a rectangle is given by A = b × h.
Step 3: Multiply the two dimensions: 12 cm × 7 cm = 84 cm².
Final Answer: The area of the rectangle is 84 cm².

Area of a Rectangle

QUESTION

How do you calculate the area of a triangle with base 10 cm and corresponding altitude 7 cm?

STEP-BY-STEP ANSWER:

Step 1: Identify the base and its corresponding altitude. In this case, base = 10 cm and altitude = 7 cm.
Step 2: Use the area formula for a triangle: A = 1/2 × base × height.
Step 3: Substitute the given values: A = 1/2 × 10 cm × 7 cm.
Step 4: Multiply to get A = 35 cm².
Final Answer: The area of the triangle is 35 cm².

Area of a Triangle

Common Mistakes

Using linear units (like cm or in) instead of square units (cm² or in²) for area.
Forgetting to convert units when dimensions are provided in different units.
Misidentifying the appropriate altitude for a polygon, especially in parallelograms and triangles.
Assuming that area calculation methods for triangles can be directly applied to the triangle’s sides rather than its enclosed region.