9.5 Solve geometry applications: circles and irregular figures  (Page 2/5)

We usually see the formula for circumference in terms of the radius $r$ of the circle:

But since the diameter of a circle is two times the radius, we could write the formula for the circumference in terms $\text{of}d.$

We will use this form of the circumference when we’re given the length of the diameter instead of the radius.

A circular table has a diameter of four feet. What is the circumference of the table?

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 table
Step 3. Name. Choose a variable to represent it.	Let c = the circumference of the table
Step 4. Translate. Write the appropriate formula for the situation. Substitute.	$C=\pi d$ $C=\pi \left(4\right)$
Step 5. Solve the equation, using 3.14 for $\pi .$	$C\approx \left(3.14\right)\left(4\right)$ $C\approx 12.56\text{feet}$
Step 6. Check: If we put a square around the circle, its side would be 4. The perimeter would be 16. It makes sense that the circumference of the circle, 12.56, is a little less than 16.
Step 7. Answer the question.	The diameter of the table is 12.56 square feet.

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Find the diameter of a circle with a circumference of $47.1$ centimeters.

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 diameter of the circle
Step 3. Name. Choose a variable to represent it.	Let d = the diameter of the circle
Step 4. Translate.
Write the formula. Substitute, using 3.14 to approximate $\pi$ .
Step 5. Solve.
Step 6. Check: $47.1\stackrel{?}{=}(3.14)(15)$ $47.1=47.1✓$
Step 7. Answer the question.	The diameter of the circle is approximately 15 centimeters.

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Find the area of irregular figures

So far, we have found area for rectangles, triangles, trapezoids, and circles. An irregular figure    is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.

Find the area of the shaded region.

Solution

The given figure is irregular, but we can break it into two rectangles. The area of the shaded region will be the sum of the areas of both rectangles.

The blue rectangle has a width of $12$ and a length of $4.$ The red rectangle has a width of $2,$ but its length is not labeled. The right side of the figure is the length of the red rectangle plus the length of the blue rectangle. Since the right side of the blue rectangle is $4$ units long, the length of the red rectangle must be $6$ units.

The area of the figure is $60$ square units.

Is there another way to split this figure into two rectangles? Try it, and make sure you get the same area.

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