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Chapter 9: Problem 63

Solve. Choose the figure that has the greater distance around. a. A square with side length 3 inches b. A circle with diameter 4 inches

Short Answer

Expert verified
The circle has the greater distance around.

Step by step solution

01

Calculate the perimeter of the square

To find the perimeter of a square, use the formula: \( P = 4s \), where \( s \) is the side length. Here, \( s = 3 \, \text{inches} \). So, \( P = 4 \times 3 = 12 \, \text{inches} \).
02

Calculate the circumference of the circle

To find the circumference of a circle, use the formula: \( C = \pi d \), where \( d \) is the diameter. Here, \( d = 4 \, \text{inches} \). So, \( C = \pi \times 4 \approx 12.57 \, \text{inches} \).
03

Compare the perimeter and circumference

Now, compare the perimeter of the square (12 inches) with the circumference of the circle (approximately 12.57 inches). The circle has a slightly greater distance around.

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Key Concepts

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

Perimeter
Perimeter is a fundamental concept in geometry that relates to the total distance around the edges of a two-dimensional shape, like a square or a rectangle. When calculating the perimeter of any polygon, you sum up the lengths of all its sides.

In the case of a square, which is a regular polygon with all four sides of equal length, the formula to calculate the perimeter is simple: \( P = 4s \), where \( s \) represents the side length.

For example, if a square has a side length of 3 inches, its perimeter is \( 4 imes 3 = 12 \) inches. Understanding how to calculate perimeter helps you grasp the essential aspect of measuring shapes and the spaces they enclose.
Circumference
Circumference measures the distance around a circle. Unlike polygons, circles have no sides, so their perimeter has a specific name - circumference.

The formula used to calculate the circumference of a circle is different than that for polygons. It is calculated as \( C = \pi d \), where \( d \) represents the diameter of the circle, which is the distance across the circle through its center.

For instance, for a circle with a diameter of 4 inches, the circumference is calculated as \( \pi imes 4 \approx 12.57 \) inches. This slight increase in the distance around the circle compared to a square makes understanding circumference important in distinguishing shapes.
Square
A square is a unique and regular quadrilateral, which means it has four equal sides and four equal angles.

All interior angles of a square are right angles, meaning each angle measures \( 90^\circ \). What makes squares special is their regularity, which allows for easy calculations and symmetry.

When you know one attribute of a square, like its side length, it's straightforward to determine other characteristics, such as the area (\

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Most popular questions from this chapter

Without actually converting, choose the most reasonable answer. The weight of an average man is about ___________. a. \(700 \mathrm{~kg}\) b. \(7 \mathrm{~kg}\) c. \(0.7 \mathrm{~kg}\) d. \(70 \mathrm{~kg}\)

Determine whether the measurement in each statement is reasonable. Josiah has a fever if a thermometer shows his tem- perature to be \(40^{\circ} \mathrm{F}\).

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In your own words, explain how to find the perimeter of any polygon.

Simplify. $$ 6 \cdot(8+2) $$
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