3.2.4: Applying Area of Circles

Lesson

Lets's find the areas of shapes made up of circles.

Exercise \(\PageIndex{1}\): Still Irrigating the Field

The area of this field is about 500,000 m². What is the field’s area to the nearest square meter? Assume that the side lengths of the square are exactly 800 m.

Figure \(\PageIndex{1}\): A square where the vertical distance is labeled 800 meters. The largest possible circle is drawn inside the square with a line segment is that begins from the center of the circle to a point on the edge of the circle.

• \(502,400\text{m}^{2}\)
• \(502,640\text{m}^{2}\)
• \(502,655\text{m}^{2}\)
• \(502,656\text{m}^{2}\)
• \(502,857\text{m}^{2}\)

Exercise \(\PageIndex{2}\): Comparing Areas Made of Circles

1. Each square has a side length of 12 units. Compare the areas of the shaded regions in the 3 figures. Which figure has the largest shaded region? Explain or show your reasoning.

Figure C is a square with nine identical circles inside of it that do not overlap. The circles are arranged in three rows of three circles, are tangent to each other and to the sides of the square. The area outside of the circles is shaded blue.

2. Each square in Figures D and E has a side length of 1 unit. Compare the area of the two figures. Which figure has more area? How much more? Explain or show your reasoning.

Figure D is composed of one square and two semi circles on top, and two bottom halves of circles and one square on the bottom. The top and bottom are divided by a horizontal dashed line. Fiqure E is composed of 4 squares and 6 quarter circles. The 4 square are arranged diagonally, where the top right vertex of the first square is the same point as the bottom left vertex of the second square, the top right vertex of the second square is the same point as the bottom left vertex of the third square, and the top right vertex of the third square is the same point as the bottom left vertex of the fourth square. The corner circles align in between the sides of each square.

Are you ready for more?

Which figure has a longer perimeter, Figure D or Figure E? How much longer?

Exercise \(\PageIndex{3}\): The Running Track Revisited

The field inside a running track is made up of a rectangle 84.39 m long and 73 m wide, together with a half-circle at each end. The running lanes are 9.76 m wide all the way around.

Figure \(\PageIndex{4}\): A picture of a field inside a running track. The field inside the track is composed of a rectangle, indicated by two dashed vertical lines labeled 73 meters and a horizontal length labeled 84 point 3 9 meters. There is a semi circle on each vertical side of the rectangle. The running track goes completely around the field and has a width of 9 point 7 6 meters.

What is the area of the running track that goes around the field? Explain or show your reasoning.

Summary

The relationship between \(A\), the area of a circle, and \(r\), its radius, is \(A=\pi r^{2}\). We can use this to find the area of a circle if we know the radius. For example, if a circle has a radius of 10 cm, then the area is \(\pi\cdot 10^{2}\) or \(100\pi\) cm². We can also use the formula to find the radius of a circle if we know the area. For example, if a circle has an area of \(49\pi\) m² then its radius is 7 m and its diameter is 14 m.

Sometimes instead of leaving \(\pi\) in expressions for the area, a numerical approximation can be helpful. For the examples above, a circle of radius 10 cm has area about 314 cm². In a similar way, a circle with area 154 m² has radius about 7 m.

We can also figure out the area of a fraction of a circle. For example, the figure shows a circle divided into 3 pieces of equal area. The shaded part has an area of \(\frac{1}{3}\pi r^{2}\).

Figure \(\PageIndex{5}\): A circle divided into three equal sections. From the center of the circle three line segments extend to a point on the circle. The line segment extending downward and to the right is labeled r. The upper left region of the circle is shaded.

Glossary Entries

Definition: Area of a Circle

If the radius of a circle is \(r\) units, then the area of the circle is \(\pi r^{2}\) square units.

For example, a circle has radius 3 inches. Its area is \(\pi 3^{2}\) square inches, or \(9\pi\) square inches, which is approximately 28.3 square inches.

Definition: Squared

We use the word squared to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s\cdot s\), or \(s^{2}\).

Practice

Exercise \(\PageIndex{4}\)

A circle with a 12-inch diameter is folded in half and then folded in half again. What is the area of the resulting shape?

Exercise \(\PageIndex{5}\)

Find the area of the shaded region. Express your answer in terms of \(\pi\).

Figure \(\PageIndex{6}\): A shaded rectangle with three unshaded circles stacked one above the other inside the rectangle. The horizontal length of the rectangle is 18 inches. On the right side of the rectangle, the length of the vertical side is subdivided into 4 different lengths using arrows. The first arrow indicates that the distance from the top of the rectangle to the top of the first circle is 3 inches. The second arrow indicates that the distance from the top to the bottom of the first circle is 6 inches. The third arrow indicates that the distance from the top to the bottom of the second circle is 9 inches. The fourth arrow indicates that the distance between the top to the bottom of the third circle is 12 inches.

Exercise \(\PageIndex{6}\)

The face of a clock has a circumference of 63 in. What is the area of the face of the clock?

(From Unit 3.2.3)

Exercise \(\PageIndex{7}\)

Which of these pairs of quantities are proportional to each other? For the quantities that are proportional, what is the constant of proportionality?

1. Radius and diameter of a circle
2. Radius and circumference of a circle
3. Radius and area of a circle
4. Diameter and circumference of a circle
5. Diameter and area of a circle

(From Unit 3.2.2)

Exercise \(\PageIndex{8}\)

Find the area of this shape in two different ways.

Figure \(\PageIndex{7}\)

(From Unit 3.2.1)

Exercise \(\PageIndex{9}\)

Elena and Jada both read at a constant rate, but Elena reads more slowly. For every 4 pages that Elena can read, Jada can read 5.

1. Complete the table.

   Table \(\PageIndex{1}\)

   pages read by Elena	pages read by Jada
   \(4\)	\(5\)
   \(1\)
   \(9\)
   \(e\)
   	\(15\)
   	\(j\)

2. Here is an equation for the table: \(j=1.25e\). What does the \(1.25\) mean?
3. Write an equation for this relationship that starts \(e=\ldots\)

(From Unit 2.2.2)