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Circles | Lesson

What are circles?

Circles are round. They don't have any corners. All of the points on a circle are the same distance from the center.
That distance is called the radius, and it's usually represented by start color #1fab54, r, end color #1fab54. The diameter, represented by start color #aa87ff, d, end color #aa87ff, is twice as long as the radius.
start color #aa87ff, d, end color #aa87ff, equals, 2, start color #1fab54, r, end color #1fab54
Because circles are so symmetrical, simply knowing either the radius or the diameter can tell us a lot about a circle.

What skills are tested?

  • Calculating the circumference of a circle
  • Calculating the radius or diameter of a circle when given the circumference
  • Calculating the area of a circle
  • Calculating the radius or diameter of a circle when given the area

What's a circumference?

The circumference of a circle is the distance around it. It's usually represented by start color #11accd, C, end color #11accd. We can calculate start color #11accd, C, end color #11accd using either start color #aa87ff, d, end color #aa87ff or start color #1fab54, r, end color #1fab54, but to do so we need to know about the number pi.
Various ancient civilizations found that the ratio of circumference to diameter, or start fraction, start color #11accd, C, end color #11accd, divided by, start color #aa87ff, d, end color #aa87ff, end fraction, is the same for circles of all sizes. This ratio, pi, is a little greater than 3. For our purposes, the approximation pi, approximately equals, 3, point, 14 works.
The formula for start color #11accd, C, end color #11accd, the circumference of the circle with diameter start color #aa87ff, d, end color #aa87ff, is:
start color #11accd, C, end color #11accd, equals, pi, start color #aa87ff, d, end color #aa87ff
The formula for start color #11accd, C, end color #11accd, the circumference of the circle with radius start color #1fab54, r, end color #1fab54, is:
start color #11accd, C, end color #11accd, equals, 2, pi, start color #1fab54, r, end color #1fab54
With each formula, we can also calculate start color #aa87ff, d, end color #aa87ff or start color #1fab54, r, end color #1fab54 when start color #11accd, C, end color #11accd is given. To do so, we:
  1. Write down the appropriate equation.
  2. Plug in the value for start color #11accd, C, end color #11accd.
  3. Solve for start color #aa87ff, d, end color #aa87ff or start color #1fab54, r, end color #1fab54.

What's a circle's area?

The area of a circle is the amount of flat space inside the circle's circumference.
The formula for start color #ca337c, A, end color #ca337c, the area of a circle with radius start color #1fab54, r, end color #1fab54, is:
start color #ca337c, A, end color #ca337c, equals, pi, start color #1fab54, r, end color #1fab54, squared
We can also calculate start color #1fab54, r, end color #1fab54 when start color #ca337c, A, end color #ca337c is given. To do so, we:
  1. Write down the area equation.
  2. Plug in the value for start color #ca337c, A, end color #ca337c.
  3. Solve for start color #1fab54, r, end color #1fab54.
  4. For the diameter start color #aa87ff, d, end color #aa87ff, multiply start color #1fab54, r, end color #1fab54 by 2.

Your turn!

try: calculating circumference
What is the circumference, in meters, of the circle shown above?
Choose 1 answer:
Choose 1 answer:

Try: calculating radius from circumference
What is the radius, in inches, of a circle with a circumference of 10, pi inches?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
inches

Try: calculating area
The radius of a circular mirror is 6 inches. What is the area, in square inches, of the mirror?
Choose 1 answer:
Choose 1 answer:

Try: calculating radius from area
The area of a circle is 100 square centimeters. What is the radius of the circle in centimeters?
Choose 1 answer:
Choose 1 answer:

Things to remember

The diameter (d) is twice as long as the radius (r), and 3, point, 14 is a common approximation for pi.
d=2rπ3.14\begin{aligned} d&=2r \\\\ \pi &\approx 3.14 \end{aligned}
The circumference (C) formulas are:
C=πdC=2πr\begin{aligned} C&=\pi d \\\\ C&=2\pi r \end{aligned}
The area (A) formula is:
A, equals, pi, r, squared

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