The Area of a Circle

The Area of a Circle exercise

• Demonstrate your understanding of the area of a circle.

  Hints

  The radius or $r$ of this circle is $7$ cm.

  The diameter or $d$ of this circle is $14$ cm.

  The radius is half of the diameter.

  As long as the $r$ (radius) is given, it is possible to find the area of a circle. The radius can be substituted in the formula for $r$.

  $r=7$ cm

  $A=\pi r^2$

  $A=\pi (7^2)$

  Solution

  The circle has a radius of 10 feet which is half of 20 feet. The green line is the diameter of the circle. To find the area of this circle, the formula used is $\bf{A= \pi r^2}$.

  ${ }$

  The distance from the center of the circle to the outside is known as the radius and in this circle is 5 feet. Therefore the diameter is double, or 10 feet. Using the formula $A= \pi r^2$, the $5$ can replace the $r$ in the formula like this, $\bf{A= \pi (5^2)}$. This can then be calculated to find the approximate area of the circle.

• Determine the steps used to find the area of a circle.

  Hints

  The radius of a circle, $r$ is the distance from the center of a circle to the outside.

  When a number is squared ($g^2$), it means that the number is multiplied by itself ($g \times g$).

  For example, $8^2=8\times 8$.

  After squaring the radius ($r=4$) in the formula,

  $A=4^2 \pi$

  $A=16 \pi$

  multiply the value by $\pi$ using the button for $\pi$ on a calculator.

  $16 \times \pi = 50.2654825...$

  When giving a value of a 2D shape's area, square units are always used.

  Some examples include: $\text{cm}^2$, $\text{in}^2$ or $\text{ft}^2$.

  Solution

  Step 1

  Identify the formula needed to find the area of a circle.

  $A=\pi r^2$

  ${ }$

  Step 2

  Determine the measurement of the circle's radius and substitute that value into the formula.

  $r=5$

  $A=\pi (5^2)$

  ${ }$

  Step 3

  Evaluate the radius squared.

  $5^2=25$

  $A=25 \pi$

  ${ }$

  Step 4

  Calculate the radius squared multiplied by $\pi$.

  $25 \times \pi = 78.539...$

  ${ }$

  Step 5

  Round your answer as directed, and include the appropriate units for the area of the circle.

  $A \approx 78.5 \: \text{ft}^2$

• Use a formula to find the area of a circle.

  Hints

  If you know the radius, this value can be substituted in the formula for $r$.

  Here is a an example of how to find the area if we had a radius of 11 inches.

  $r = 11\:\text{in}$

  $A = \pi r^2$

  $A= \pi (11^2)$

  $A \approx 380.132711...$

  $A \approx 380 \:\text{in}^2$

  To round to the nearest square inch, it is the same as rounding to the nearest whole number, or ones digit.

  Remember to be sure to include your squared units at the end of the measurement.

  Instead of $\text{in}$, use $\text{in}^2$.

  Solution

  $r = 12\:\text{in}$

  $A = \pi r^2$

  $A= \pi (12^2)$

  $A \approx 452.389342...$

  $A \approx 452 \:\text{in}^2$

• Apply the formula for the area of a circle.

  Hints

  The $A$ is the area, and the $r$ is the radius of the circle.

  This example shows the steps taken to find the area of a circle.

  Radius: The radius of a circle is like a straight line going from the very center of the circle to its edge. It's like the distance from the middle of a pizza to its crust.

  Diameter: The diameter of a circle is a straight line that goes from one side of the circle to the other side, passing through the center. It's like measuring a pizza straight across from one edge to the opposite edge. The diameter is always two times longer than the radius.

  Always be sure to check if you are given the radius or the diameter.

  If given the diameter, you need to halve the value to find the radius.

  For example, if we were given a diameter of 22 inches, half of 22 is 11, so the radius is 11 inches.

  Solution

  $r=8\:\text{cm}$ ; $A \approx 201.1\:\text{cm}^2$

  $r=4\:\text{cm}$ ; $A \approx 50.3\:\text{cm}^2$

  $d=8.5\:\text{cm}$ ; $A \approx 56.7\:\text{cm}^2$

  $d=9\:\text{cm}$ ; $A \approx 63.6\:\text{cm}^2$

• Understand how to apply the formula for the area of a circle.

  Hints

  This is the formula used to find the area of a circle.

  $A$ = Area

  $r$ = radius

  $r^2$ means $r \times r$

  For example if $r=3$, $3^2 = 9$.

  The radius in this circle is $15\:m$.

  The formula used to find the area would be $A=\pi \times 15^2$.

  Solution

  $r=9$ meters

  $A= \pi (r^2)$

  Substitute the value for $r$ into the formula.

  $A= \pi (9^2)$

• Demonstrate your knowledge for calculating the area of a circle.

  Hints

  The formula for the area of a circle is $A=\pi r^2$.

  $r$ = radius

  $A$ = Area

  For this example, we only need the area of the donut shape. Subtraction would be helpful to remove the area not needed from the hole of the donut.

  Steps to find the area of the donut:

  • Find the area of the large circle.

  • Find the area of the smaller inner circle.

  • Subtract them to find just the area of what is asked.

  Solution

  Area of the entire circle:

  $A=\pi (7^2)$

  $A \approx 154\:\text{cm}^2$

  Area of the inner circle:

  $A=\pi (2^2)$

  $A \approx 13\:\text{cm}^2$

  Subtract the values to just find what is being asked.

  $154 - 13 = 141$

  $A \approx 141 \:\text{cm}^2$