Question

1. consider the diagram of a semicircle shown.

a. explain in words how to determine the perimeter of a semicircle.
b. using “d” to represent the diameter of the circle, write an algebraic expression that will result in the perimeter of a semicircle.
c. write another algebraic expression to represent the perimeter of a semicircle using r to represent the radius of a semicircle.
d. find the perimeter of the semicircle. let π ≈ 3.14.
(diagrams: first diagram is a semicircle with diameter labeled d and radius labeled r; second diagram is a semicircle with diameter labeled 17 in)

Explanation:

Part a

The perimeter of a semicircle (also called the circumference of the semicircle) is the sum of the length of the curved part (half of the circumference of the full circle) and the length of the straight diameter. First, find half of the circumference of the full circle using the formula for the circumference of a circle \( C = \pi d \) (or \( C = 2\pi r \)), then take half of that (\( \frac{1}{2}\pi d \) or \( \pi r \)) and add the length of the diameter (\( d \)) to it.

Step 1: Recall the circumference of a full circle

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

Step 2: Find the length of the curved part of the semicircle

The length of the curved part of the semicircle is half of the circumference of the full circle. So, the length of the curved part is \( \frac{1}{2}\times\pi d=\frac{\pi d}{2} \).

Step 3: Add the diameter to the curved part length

The perimeter of the semicircle is the sum of the curved part and the diameter (\( d \)). So the algebraic expression for the perimeter \( P \) of the semicircle is \( P=\frac{\pi d}{2}+d \).

Step 1: Recall the relationship between diameter and radius

We know that the diameter \( d = 2r \), where \( r \) is the radius.

Step 2: Substitute \( d = 2r \) into the formula from part b or derive from the circumference formula

We can also derive it from the circumference formula of the full circle \( C = 2\pi r \). The length of the curved part of the semicircle is half of the circumference of the full circle, so it is \( \frac{1}{2}\times2\pi r=\pi r \). The perimeter of the semicircle is the sum of the curved part (\( \pi r \)) and the diameter (\( 2r \)). So the perimeter \( P=\pi r + 2r \).

Answer:

To determine the perimeter of a semicircle, calculate half of the circumference of the full - circle (using the formula for the circumference of a circle, \( C=\pi d\) or \(C = 2\pi r\)) and then add the length of the diameter of the circle.

Part b