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The area and perimeter of a circular sector
What fraction of a circle is a sector with an angle of 90?
If you'll be snacking on this, you'll probably call it a slice of pizza. But this is math and here it's called a sector of a circle. You get a sector of a circle when you divide a circle with two cuts along the radii so that they meet at the exact center of the circle. So how much food is there in the sector? What is its area?
We start by calculating the area of the whole pizza, and for this, we use pi times the radius squared. We measure the radius - 15 centimeters - and pi is approximately 3,14. 3,14 X 15 to the power of 2 is 706,5. The whole pizza is 706 and 1/2 square centimeters. That was the whole pizza's surface, but we have just a slice of it.
To measure the sector's part of the circle, we need the angle between the two radii along which the pizza is cut. Use the protractor and measure. 45 degrees exactly. This sector is 45 degrees out of 360. Since the whole circle is 360 degrees, this sector's part of the whole circle is 45 divided by 360.
Now, we can summarize. The area of a sector of a circle equals the whole circle's area times the sector's part of the whole circle. 706,5 X 45 divided by 360 is approximately 88 square centimeters. You won't get full on that. Perhaps you want to know how long the edge of a slice is. You calculate this in the same way.
First, how long is the edge of the whole pizza? Its length is the circumference. The circumference equals pi X 2 X the radius. Plug in the radius, 15, and calculate. It's a little more than 94 centimeters, and the sector's part of the whole circle is calculated the same way - 45 degrees divided by 360.
Take the circumference of the whole circle X this sector's fraction of the whole circle, and you'll get the length of this arc. An arc is the edge of the pizza slice. The arc length is just under 12 centimeters. Let's put everything together now. A sector of a circle is a part of a circle you get if you cut it along the two radii. The area of a sector equals the area of a circle X the sector's angle divided by 360.
The arc of a sector is the curved outer edge of the sector. To get the length of an arc, you take the circumference of the whole circle X the sector's angle in degrees divided by 360.