12. Circles

Lesson

Let's refresh our memories of how many diameters fit around the circumference of a circle by using the applet below.

- Change the diameter of the circle by dragging the blue dot.
- Then roll the circle out along the line (slide black dot along the line)
- Once the circle is unraveled we can see how many diameters were around the circumference.

We should find that there are $3$3 and a bit diameters around the circumference. More accurately, there are $\pi$π diameters around the circumference of a circle.

Circumference of a circle

Where C is the circumference of a circle and D is the diameter, we find that

$C=\pi D$`C`=π`D`

and because the diameter is twice the radius, we can also write the formula as

$C=2\pi r$`C`=2π`r`

If the radius of a circle equals $27$27 cm, find its circumference correct to one decimal place.

Find the circumference of the circle shown, correct to two decimal places.

To find the area of a circle, we need to know its radius. If we are given the diameter, we will need to halve it to get the radius.

We already know that area is the space inside a 2D shape. We can find the area of a circle, but we will need a special rule.

The following investigation will demonstrate what happens when we unravel segments of a circle.

Interesting isn't it that when we realign the segments we end up with a parallelogram shape. Which is great, because it means we know how to find the area based on our knowledge that the area of a parallelogram has formula $A=bh$`A`=`b``h`. In a circle, the base is half the circumference and the height is the radius.

Area of a circle

$Area=\pi r^2$`A``r``e``a`=π`r`2

If the radius of the circle is $9$9 cm, find its area, rounded to 2 decimal places.

So far, we have seen the formula below:

$C=2\pi r$`C`=2π`r`

$C=\pi D$`C`=π`D`

$A=\pi r^2$`A`=π`r`2

Instead of being given the radius or diameter and being asked to find the circumference or area, we can also have the reverse. That is we are given the circumference or area and are asked to find the radius of diameter. This is just solving literal equations, like we have seen many times.

**Diameter from circumference:**

$C$C |
$=$= | $\pi D$πD |
The original formula for circumference |

$\frac{C}{\pi}$Cπ |
$=$= | $D$D |
Divide both sides by $\pi$π |

$D$D |
$=$= | $\frac{C}{\pi}$Cπ |
Formula for diameter in terms of circumference |

**Radius from circumference:**

$C$C |
$=$= | $2\pi r$2πr |
The original formula for circumference |

$\frac{C}{2\pi}$C2π |
$=$= | $r$r |
Divide both sides by $2\pi$2π |

$r$r |
$=$= | $\frac{C}{2\pi}$C2π |
Formula for diameter in terms of circumference |

**Radius from area:**

$A$A |
$=$= | $\pi r^2$πr2 |
The original formula for area |

$\frac{A}{\pi}$Aπ |
$=$= | $r^2$r2 |
Divide both sides by $\pi$π |

$r$r |
$=$= | $\sqrt{\frac{A}{\pi}}$√Aπ |
Square root both sides |

It is sometimes more useful to know the radius or diameter of a circle as a decimal, rather than the exact value in terms of $\pi$π. If this is the case, we can use a calculator to evaluate the expression and round the result to the desired number of decimal places.

What is the radius $r$`r` of a circle with the circumference $C=14$`C`=14 cm?

Round your answer to two decimal places.

What is the diameter $D$`D` of a circle with the circumference $C$`C` of length $44$44 m?

Round your answer to two decimal places.

The area of a circle is $352$352 cm^{2}.

If its radius is $r$

`r`cm, find $r$`r`, correct to two decimal places.Be careful not to round your answer until the very end.

Using the rounded value from the previous part, find the circumference of the circle. Round your answer to 1 decimal place.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, cavalieri's principle, and informal limit arguments.