Frottoir musique concrete
Frottoir musique concrete
A circle is an important geometric shape in our world and is very relevant to our everyday experiences. In this lesson, we will explore one aspect of the circle, the segment. In addition, we will also discuss a few other vocabulary words related to the circle.
Do you like pizza? Most people do. Some people even like to make their own pizza. Let's image that you just took a circular pizza out of the oven, and you cut it into two equal parts. You would have made two segments. Each half has two parts: a straight line and a curved part. Each of those parts have names: a chord and an arc, respectively.
Before we begin our discussion about segments, let's review some important vocabulary related to the circle. Here are some reminders:
- The diameter (D) is two times the length of the radius (r); that means D = 2 r.
- The radius of a circle is the line segment formed from the center of the circle to any point on the circumference.
As you can see in the diagram, the point O is the center of the circle, and line segment AB is the diameter. There are also three radii: line segments OB, OA, and OC.
- A chord is a line segment whose endpoints lie on the circumference of the circle. Remember that the circumference is the total distance around the circle (all of the crust of your pizza). In other words, the chord is the line formed when one point on the circumference is connected to another point on the circumference (or crust). Note also that the diameter of a circle is the biggest chord.
- An arc is any curved part on the circumference of the circle. When you have a slice of pizza and eat the crust, you're having the arc.
- A segment is the section of a circle enclosed by a chord and an arc. Therefore, those halves of the pizza are segments. If you eat one half, you would have eaten a semicircle (half of a circle), which is the biggest segment of a circle.
Since a circle has an infinite number of points on the circumference, there are many possibilities for a chord and, hence, many possibilities for segments.
When a circle is divided into two segments of different areas, the biggest segment is called the major segment and the smaller segment is called the minor segment.
Area of a Segment
Now, if you order pizza, how does it usually come? Yes, pre-sliced. Those slices look like triangles with a curve, right? Those are sectors. A sector of a circle is the section enclosed by two radii (the two sides of the slice) and an arc (the crust).
We can divide a sector so that it looks like a triangle and a segment.
We divided the sector so that a triangle is formed and a segment is formed. If we were to consider the total area of that sector, we could say that the total area of the sector is made up of the area of the triangle and the area of the segment. Some people don't like the crust of the pizza. So when they get a slice, they cut off the crust. However, their full slice includes the crust, plus the part that is left. Therefore, we can say that a sector is made up of the triangle and the segment (the crust).
Using that example, we can find the area of a segment using the formula:
Area of Segment = Area of Sector - Area of Triangle
First, we find the area of the sector, which would be like finding the area of your slice of pizza. Next, we would find the area of the triangle (the part left after you cut off the crust). If we know the area of the full slice of pizza and we know the area of the triangle, then what is left must be the area of the crust. This is the same principle used to find the area of the segment represented by the crust. The formulas for each are as follows:
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Remember that r represents the radius, b represents the base of the triangle, h represents the height, and theta represents the measure of the sector in degrees. In the case of the triangle, sometimes we are not given the height; therefore, we have to use a second formula:
Area of a Triangle = (1/2)absinC
Here C is the angle. In this case, a and b are both radii of the circle. Let's make this clearer with some examples.
Examples of Finding the Area of a Segment
We will first use an example finding the area of a segment using this area of a triangle formula: (1/2)absinC.
In this example, we have sector AOB. AB is the chord that bounds the segment (the shaded portion). We want to find the area of the shaded portion. We first find the area of the whole sector (the whole slice).
Area of sector = (120/360) 3.14 4 4 = 16.75m^2
Then, we find the area of the triangle:
Area of triangle = (1/2) 4 4 sin120 = 6.93m^2
What we have left after we subtract is the area of the segment.
16.75 - 6.93 = 9.82m^2
Our next example will use the formula for the area of a triangle as (1/2) base height.
Since the triangle ROM is a right triangle, it makes our calculations a bit easier because the height becomes obvious. Although the angle is 90 degrees in this case, our answer would still be the same if we were to use the (1/2)abSinC formula.
Area of sector = (90/360) 3.14 6 6 = 28.26m^2
Area of triangle = (1/2) 9 4 = 18m^2
Area of segment = 28.26 - 18 = 10.26m^2
Using a pizza to help you understand the parts of a circle can be very helpful. Half of a pizza represents a semicircle. Each semicircle has a diameter (the straight part of the pizza), radius, and chord, and each semicircle has an arc (the crust around the pizza). Each diameter may also be divided into two radii.
A circle may be divided into two or more segments. A segment is the region of a circle bounded by a chord and an arc. If the circle is divided into two segments, the bigger portion is called the major segment and the smaller portion is called the minor segment. The semi-circle is the biggest segment of any circle. A sector is the region bounded by two radii and an arc.
In order to find the area of a segment, we first have to find the area of the sector where the segment lies. Then, we divide the sector into two parts: a triangle and a segment. The formula for the area of a segment is:
Area of a Segment = Area of a sector - Area of the triangle
Remember the formula for each portion:
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