Problem 89 - Prealgebra

Question

A circle has a diameter of \(8.56\) inches. Using \(\pi \approx 3.14\), find the circumference of the circle, correct to the nearest tenth of an inch.

Step-by-Step Solution

Answer

The circumference of the circle is approximately 26.9 inches.

Step 1: Understanding the Formula

To find the circumference of a circle, we use the formula \(C = \pi \times d\), where \(C\) is the circumference and \(d\) is the diameter of the circle. In this exercise, the diameter is given as \(8.56\) inches.

Step 2: Applying the Formula

Now, substitute the given diameter \(d = 8.56\) inches into the formula \(C = \pi \times d\). This gives: \[ C = 3.14 \times 8.56 \].

Step 3: Performing the Calculation

Multiply \(3.14\) by \(8.56\) to find the circumference. Doing the multiplication, we get: \[ C \approx 26.8624 \] inches.

Step 4: Rounding the Result

Round the calculated circumference \(26.8624\) to the nearest tenth. This gives \(26.9\) inches.

Key Concepts

Understanding Diameter

The diameter of a circle is a straight line that passes through the center of the circle and connects two points on its edge. It's essentially the longest distance across the circle.
Think of it as slicing a pizza perfectly through the middle, from one edge to the other.
Some key points about the diameter:

• The diameter is always twice as long as the radius, the distance from the center to any point on the edge.
• If you know the radius of a circle, you can find the diameter by multiplying the radius by 2.
• The diameter helps us in calculating the circumference, which is the distance around the circle.

Understanding the diameter is crucial when working with problems involving circles, as it's a fundamental part of many formulas, including the circumference formula.

Exploring Pi (π)

Pi, denoted by the symbol \(\pi\), is a special mathematical constant that represents the ratio of a circle's circumference to its diameter. This means if you divide the circumference by the diameter of any circle, you'll get the same number: pi.
Pi is approximately equal to 3.14, but it extends infinitely without repeating, making it an irrational number.
Some interesting facts about pi:

• Pi is used in various calculations involving circles, such as finding the area and circumference.
• Even though we often use 3.14 for simplicity, pi is an infinite decimal that goes on forever without a pattern.
• It's celebrated every year on Pi Day, March 14th (3/14), because its approximate value starts with 3.14.

In problems involving circles, pi is essential for accurate calculations, hence why it features prominently in the formula for circumference: \(C = \pi \times d\).

The Nature of a Circle

A circle is a shape with all points in a plane that are equidistant from a fixed center point. When you look at a wheel or a dinner plate, you're looking at circles.
They are perfectly symmetrical and offer fascinating properties and concepts.
Here are some important properties of circles:

• Every circle has a center point from which every edge point has the same distance, defined as the radius.
• The distance around a circle is known as the circumference, calculated as \(C = \pi \times d\), where \(d\) is the diameter.
• The circle's area, or space within its outline, can be calculated using \(A = \pi \times r^2\), where \(r\) is the radius.

The circle's simple yet complex nature makes it an intriguing subject in mathematics, and understanding it provides the foundation for solving many geometric problems.