Question

Nature What is the circumference of a bird's nest if its diameter is 7 inches? (Lesson 11-5)

Short answer

The circumference of the bird's nest is approximately 21.99 inches.

Step-by-step solution

Understand the Formula

The formula for the circumference of a circle is given by \( C = \pi \times d \), where \( C \) is the circumference, \( \pi \) (pi) is approximately 3.14159, and \( d \) is the diameter of the circle.

Insert Known Values

Substitute the given diameter into the formula. Here, the diameter \( d = 7 \) inches, so the formula becomes \( C = \pi \times 7 \).

Perform the Calculation

Calculate the circumference using the substituted values: \( C = 3.14159 \times 7 \). This equals approximately \( 21.99113 \).

Round the Result

Round the result to two decimal places to make it more readable and practical: \( C \approx 21.99 \) inches.

Key concepts

Geometry Formulas

In the world of geometry, formulas act as the backbone that helps us solve various problems involving shapes and sizes. One essential formula is for finding the circumference of a circle. The circumference is the distance around a circle, similar to the perimeter of polygonal shapes. To grasp this concept, remember that the formula for the circumference is given by:

• \( C = \pi \times d \)
• Alternatively, if the radius \( r \) is known rather than the diameter, the formula becomes \( C = 2 \pi \times r \).

Where:

• \( C \) stands for the circumference.
• \( \pi \) is a constant approximated to 3.14159.
• \( d \) is the diameter of the circle.
• \( r \) is the radius, which is half of the diameter.

These formulas make it easy to calculate the circumference, as long as one has the measurements of either the diameter or radius. It turns abstract measurements into tangible figures using constant mathematical principles.

Circle Measurements

Measuring circles can sometimes feel complex, but understanding a few basic terms makes it much simpler. The two main measurements one often deals with are the diameter and the radius. The "diameter" of a circle is the distance across the circle through its center. If you were to take a string and lay it flat from one edge of a circle straight across to the other side, passing through the center, this length is your diameter. In our bird's nest example, this was 7 inches. The "radius" is the distance from the center of the circle to any point on its edge. It's exactly half the length of the diameter. Therefore, if a circle's diameter is 7 inches, its radius is 3.5 inches. Grasping these measurements is crucial because they are used in different formulas to find other circle attributes, like the area or the circumference. When presented with either measurement, you can determine the other with ease:

• Diameter = 2 × Radius
• Radius = Diameter ÷ 2

Applying Pi

Once we grasp the measurements of a circle, it's time to apply pi (\( \pi \)). Pi is a mathematical constant representing the ratio of a circle's circumference to its diameter. The beauty of \( \pi \) is that it's a never-ending, non-repeating number, making it an irrational number.Although its infinite nature could be a bit hard to comprehend, \( \pi \) is often rounded to a more manageable 3.14159 for calculations. While this isn't the exact value, it's precise enough for most practical purposes. In our original exercise about calculating the circumference of a bird's nest, we applied \( \pi \) in this calculation: \( C = \pi \times 7 \). Round it, and you get \( C \approx 21.99 \) inches.This example highlights how \( \pi \) is crucial in any computation involving circles. Whether determining the circumference as in this instance or finding the area of a circle using \( A = \pi \times r^2 \), applying \( \pi \) bridges measurements to meaningful outcomes.