9.5 Solve geometry applications: circles and irregular figures

In this section, we’ll continue working with geometry applications. We will add several new formulas to our collection of formulas. To help you as you do the examples and exercises in this section, we will show the Problem Solving Strategy for Geometry Applications here.

Problem Solving Strategy for Geometry Applications

1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

Use the properties of circles

Do you remember the properties of circles from Decimals and Fractions Together ? We’ll show them here again to refer to as we use them to solve applications.

Properties of circles

• $r$ is the length of the radius
• $d$ is the length of the diameter
• $d=2r$
• Circumference is the perimeter of a circle. The formula for circumference is
  
  $C=2\pi r$
• The formula for area of a circle is
  
  $A=\pi r^2$

Remember, that we approximate $\pi$ with $3.14$ or $\frac{22}{7}$ depending on whether the radius of the circle is given as a decimal or a fraction. If you use the $\pi$ key on your calculator to do the calculations in this section, your answers will be slightly different from the answers shown. That is because the $\pi$ key uses more than two decimal places.

A circular sandbox has a radius of $2.5$ feet. Find the ⓐ circumference and ⓑ area of the sandbox.

Solution

ⓐ Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the circumference of the circle
Step 3. Name. Choose a variable to represent it.	Let c = circumference of the circle
Step 4. Translate. Write the appropriate formula Substitute	$C=2\pi r$ $C=2\pi \left(2.5\right)$
Step 5. Solve the equation.	$C\approx 2\left(3.14\right)\left(2.5\right)$ $C\approx 15\text{ft}$
Step 6. Check. Does this answer make sense? Yes. If we draw a square around the circle, its sides would be 5 ft (twice the radius), so its perimeter would be 20 ft. This is slightly more than the circle's circumference, 15.7 ft.
Step 7. Answer the question.	The circumference of the sandbox is 15.7 feet.

ⓑ Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the circle
Step 3. Name. Choose a variable to represent it.	Let A = the area of the circle
Step 4. Translate. Write the appropriate formula Substitute	$A=\text{π}{r}^2$ $A=\text{π}{(2.5)}^2$
Step 5. Solve the equation.	$A\approx (3.14){\left(2.5\right)}^2$ $A\approx 19.625\text{sq. ft}$
Step 6. Check. Yes. If we draw a square around the circle, its sides would be 5 ft, as shown in part ⓐ . So the area of the square would be 25 sq. ft. This is slightly more than the circle's area, 19.625 sq. ft.
Step 7. Answer the question.	The area of the circle is 19.625 square feet.

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