Back To CourseGeometry: High School
15 chapters | 160 lessons
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
If you're like me, you think about pizza often. And with pizza, there's so much to consider. Thin crust or deep dish. Pepperoni or veggies. Red pepper flakes sprinkled on top or a ridiculous amount of red pepper flakes poured on top. Mmm, tasty and burning.
Now, most pizzas are circles. And circles are geometry. So, why not contemplate geometry while you eat pizza? It's still not healthy for your body, but at least it can be good for your brain!
That slice of pizza? That's called a sector. A sector is a part of a circle enclosed by two radii and the connecting arc. You can have a normal pizza slice sector, or you can have a gigantic pizza slice sector. The key is that it touches the center of the circle and is bound by the two radius lines.
All sectors have a central angle. This is the angle the sector subtends to the center of the circle. We know there are 360 degrees in a circle, so the central angle will be some subsection of that. In this slice, it's 45 degrees:
In this one, it's 90:
That's a special sector known as a quadrant. Get it? 'Quad-' means 4, and this is one-fourth of the circle. In our half-pizza slice below, it's 180 degrees. That's a special sector called a semicircle.
We can also look at it in radians instead of degrees. A radian is just a different way of measuring an angle. A radian is what you get when you take the radius of the circle and lay it on the circumference.
So, let's say you've got your normal-sized pizza slice, and you want to know its area. The area of a sector can be found in a couple of different ways, depending on what you know. You'll always need to know the radius. Remember, the radius is half the diameter. So, in a 12-inch pizza, the radius is 6 inches.
If we wanted the area of the entire circle, it's π*r2. For the semicircle? 1/2*π*r2, since it's half the circle. The principle of the area of a sector follows this same logic. We just take the circle area formula and multiply it by a fraction that represents our sector.
If you know the central angle, the area is (n/360)*π*r2, where n is the number of degrees in the central angle. So, let's say our sector has an angle of 23 degrees. Let's plug that into the formula for our slice with a 6-inch radius. Its area is (23/360)*π*62. That's 7.2 inches squared.
If we know the angle in radians, it's even simpler. It follows the same logic. We start with π*r2. A circle has a total angle of 2*π. So, if we call our angle theta, then the equivalent of n/360 is (theta)/(2*π). Plug that into the same formula: ((theta)/(2*π))*π*r2. That simplifies to ((theta)/2)*r2. So, if our angle is .4 radians, then we have (.4/2)*62. Again, we get 7.2 inches squared.
This works if we know the central angle. But what if we don't? We then need to know the arc length. The arc length is the distance along the arc, or circumference of the circle. We write this as lAB.
If you need to find the area of a sector using the arc length, that distance will be given to you. But know that you can figure it out if you have the central angle. We just take the circumference formula (2*π*r) and multiply that by n/360, so it's 2*π*r*(n/360). That looks familiar, doesn't it? It's the same as the area of a sector formula, just swapping the circumference for the area.
In radians, it's even simpler. Again, a radian is what you get when you take the radius of a circle and lay it on the circumference. So, it's directly related to the circumference. Therefore, the arc length in radians is r*C, where r is the radius, and C is the central angle in radians.
Ok, now let's find out the area of a sector using arc length. Again, this is handy if you're given the radius and arc length, but not the central angle. Here, the area of a sector is just 1/2*r*L, where r is the radius, and L is the arc length. How can you remember this? Just take your sector, or pizza slice, and turn it like this:
What does that look like? A triangle! And what's the area formula for a triangle? 1/2*base*height. That's looks a lot like 1/2*radius*arc length, doesn't it?
Let's go back to our pizza slice. Let's say the radius is 5 inches, and the arc length is 4 inches. Just plug that into 1/2*r*L, and we get 1/2*5*4. That's going to be 10 square inches.
In summary, we learned about sectors and arc lengths. A sector is basically your pizza slice, or the section of a circle bound by two radii and an arc, which is the part of the circumference between the radius lines.
If we know the radius and the central angle, or the angle formed by the radii, we can find the area of the sector by converting the area of a circle formula. If we're using degrees, it's n/360 (where n is the number of degrees) times pi times the radius squared. If we're using radians, it's just theta divided by 2, where theta is the central angle in radians, times the radius squared.
We then looked at arc lengths. You can find the arc length by converting the circumference formula. With a central angle in degrees, it's 2 times pi times the radius (that's the circumference formula) times n/360, where n is the central angle. With radians, it's just the radius times the angle, or r*C.
To find the area of a sector using the arc length, you find 1/2 times the radius times the arc length. This is very similar to the area of a triangle formula.
We also justified eating pizza as a mental workout. Feel free to tell yourself that the next time you grab a slice.
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Back To CourseGeometry: High School
15 chapters | 160 lessons