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Chapter 7: Problem 23

Find the length of the diameter of a circle whose circumference is \(\frac{5}{8} \pi\) in.

Short Answer

Expert verified
The diameter is \( \frac{5}{8} \) inch.

Step by step solution

01

Recall the formula for the circumference of a circle

The formula for the circumference of a circle is \( C = \pi \times d \), where \( C \) is the circumference and \( d \) is the diameter.
02

Substitute the given circumference into the formula

Given that the circumference \(C = \frac{5}{8} \pi\) inches, substitute this value into the formula \pi \times d = \frac{5}{8} \pi
03

Solve for the diameter

To isolate \(d\), divide both sides of the equation \( \pi \times d = \frac{5}{8} \pi \) by \(\pi\): \d = \frac{5}{8}\ inch.

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Key Concepts

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

geometry
Geometry is a branch of mathematics that deals with shapes, sizes, and properties of space. It encompasses concepts such as points, lines, angles, surfaces, and solids. When we talk about circles in geometry, they are defined as a set of points in a plane that are equidistant from a given point called the center.
This equidistant length from the center to any point on the circle is called the radius.
Another important concept is the diameter, which is twice the length of the radius. These basic properties of circles are fundamental to solving many geometric problems.
circumference formula
To find the circumference of a circle, we use the formula: \( C = \pi \times d \)
  • \( C \) is the circumference, which is the distance around the circle.
  • \( \pi \) (pi) is a constant approximately equal to 3.14159.
  • \( d \) is the diameter of the circle.
This formula allows us to calculate the circumference if we know the diameter.
Understanding this formula is vital as it can also be rearranged to find the diameter if the circumference is known.
In the given exercise, this rearrangement is needed to solve for the diameter when the circumference is provided.
diameter calculation
Let's delve into calculating the diameter using the provided circumference. For the given problem, the circumference (C) is \frac{5}{8} \pi inches. Using our formula \( C = \pi \times d \), we substitute C with \frac{5}{8} \pi to get:
\(\ \pi \times d = \frac{5}{8} \pi \).
To solve for the diameter (\( d \)), we need to isolate d. This is done by dividing both sides of the equation by \( \pi \):
\( d = \frac{5}{8} \ inch \).
Therefore, the diameter of the circle is \( \frac{5}{8} \ inches \). It's simple: once we know the circumference and use the formula correctly, finding the diameter becomes straightforward.
circle properties
Circles have several key properties that are useful in geometry.
  • The radius (r) is the distance from the center of the circle to any point on its edge.
  • The diameter (d) is twice the radius: \( d = 2r \).
  • The circumference (C) is the total distance around the circle and is given by \( C = \pi \times d \) or \( C = 2 \pi \times r \).
  • The area of the circle is also important, calculated by the formula \( A = \pi \times r^2 \).
Understanding these properties helps simplify many geometrical problems involving circles, like the one in the exercise. Recognizing these core relationships, especially how the radius and diameter connect to the circumference, is crucial for solving related problems efficiently.

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Most popular questions from this chapter

The area of a sector of a circle is \(50 \pi \mathrm{cm}^{2} .\) If the arc of the sector is \(45^{\circ},\) find the diameter of the circle.

A regular polygon has perimeter \(144 \mathrm{cm}\) and apothem \(18 \mathrm{cm} .\) Find its area.

In Exercises \(10-13,\) find the approximate circumference and area of each circle with the given radius or diameter using the calculator to approximate the answer to the nearest hundredth. \(d=12.00 \mathrm{mi}\)

Find the lengths of the radius and the apothem of a square whose side measures 10 in.

Brahmagupta was an East Indian mathematician who lived during the seventh century. He discovered a formula for the area of a quadrilateral inscribed within a circle. As we learned in Chapter \(6,\) this means the vertices of the quadrilateral lie on the circle. \(A=\sqrt{(s-a)(s-b)(s-c)(s-d)},\) where \(a, b, c,\) and \(d\) are the lengths of the sides of the quadrilateral and \(s=\frac{a+b+c+d}{2}\) This formula looks similar to Heron's formula for the area of a triangle. Discuss the similarities and differences. Show a numerical example of how the formula works. Research more about the life of Brahmagupta.
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