Explore Area Calculations in Circular Objects Using Trigonometric Ratios

Based on the features and theorems of the circles, this lesson will relate the circles with the real-life examples.

Catch-Up and Review

Here are some recommended readings before getting started with this lesson.

Area of a Circle
Trigonometric Ratios
Cylinder
Volume of a Cylinder
Tangent Line to a Circle

Explore

Investigating Circular Shaped Objects

Tennis balls are sold in cylindrical containers that contain three balls each. The tennis balls and container can be modeled as follows.

There are three tennis balls inside a cylindrical container.

The container, lying on its side, passes through an X-ray scanner in Tallahassee International Airport, where Jordan works as a security guard. Depending on the opacity of the container and the balls, what should Jordan expect to see when she reads the screen of the X-ray scanner? Explore the applet by adjusting the opacity of the container and the tennis balls.

Example

Modeling the Area of Circular Objects

Izabella and her friends are ordering pizza from their favorite spot. They can afford to buy either two $9 -$ inch pizzas or one $13 -$ inch pizza. The thickness of the pizza is the same for each.

On the left is a 9-inch pizza and on the right is a 13-inch pizza.

They cannot quite decide which option means more food. By modeling each pizza as a circle, help Izabella and her friends determine the best option.

Hint

Since both pizzas have the same thickness, the area of the pizzas will help to make a decision.

Solution

It is given that the pizzas have the same thickness. Therefore, areas of the pizzas will determine the best option. Begin by modeling the pizzas as circles with diameters of $9$ and $13$ inches.

Because the radius of a circle is half of its diameter, the radii of the pizzas can be found by dividing their diameter by

2 .

\underline{9 - inch Pizza} \frac{9}{2} = 4.5 in \underline{13 - inch Pizza} \frac{1 3}{2} = 6.5 in

Now that the radii of the pizzas were found, the formula for the area of a circle can be used to find their areas. Begin by finding the area of

9 -

inch pizza.

A = π r^{2}

Substitute

$r = 4.5$

A = π (4.5)^{2}

▼

Evaluate right-hand side

CalcPow

Calculate power

A = π (20.25)

UseCalc

Use a calculator

A = 63.617251 \dots

RoundDec

Round to $2$ decimal place(s)

A \approx 63.62

The area of the pizza is about

63.62

square inches. Following the same procedure, the area of

13 -

inch pizza can also be found.

	Radius	Substitution	Area
$9 -$ inch pizza	$4.5$	$A = π (4.5)^{2}$	$\approx 63.62$
$13 -$ inch pizza	$6.5$	$A = π (6.5)^{2}$	$\approx 132.73$

Recall that Izabella and her friends can afford two

9 -

inch pizzas.

Area of two 9 - inch Pizzas : \approx 127.24 Area of one 13 - inch Pizza : \approx 132.73

As a result, the best option is to buy a

13 -

inch pizza. One of the friends was so close to ordering the two

9 -

inch pizzas. Thanks for helping them make the right choice.

Example

Modeling the Radius of Circular Objects

The Knights of the Round Table are the highly-esteemed knights of the mythical fellowship of King Arthur. In the beginning, there were only twelve knights. They met at the Round Table that is a symbol of equality among all of its legendary members.

Assume that each knight has the same circular chair around the Round Table whose radius is $10$ feet. Given that all chairs and the Round Table are externally tangent to each other, what is the radius of each chair? Round the answer to the nearest tenth.

Hint

Begin by modeling the Round Table and the chairs using circles. Draw a triangle whose vertices are the center of the inner circle and two adjacent outer circles. Then use the trigonometric ratios.

Solution

The Round Table and the chairs can be modeled by using circles.

By labeling the centers of the inner circle and two adjacent outer circles, a triangle passing through these centers can be drawn.

Segments that connect the centers of the tangent circles pass through the points of tangency. Assume that the radius of each chair is $r$ feet.

Notice that

△ A B C

is an isosceles triangle where

A B = A C .

From here, to find the value of

r,

the vertex angle of

△ A B C

needs to be found. Since there are

12

chairs, the measure of

∠ B A C

can be found by dividing

36 0^{\circ}

12 .

\frac{3 6 0 ^{\circ}}{1 2} = 3 0^{\circ}

Now that the measure of the vertex angle was found, the altitude of the base can be drawn to use the trigonometric ratios. Remember that the altitude of the base of an isosceles triangle bisects the vertex angle and the base.

Finally, using the sine ratio in

△ A B D,

the value of

r

can be found.

sin θ = \frac{B D}{A B}

SubstituteValues

Substitute values

sin 1 5^{\circ} = \frac{r}{1 0 + r}

▼

Solve for

r

MultEqn

$LHS \cdot (10 + r) = RHS \cdot (10 + r)$

10 sin 1 5^{\circ} + r sin 1 5^{\circ} = r

SubEqn

$LHS - r sin 1 5^{\circ} = RHS - r sin 1 5^{\circ}$

10 sin 1 5^{\circ} = r - r sin 1 5^{\circ}

FactorOut

Factor out $r$

10 sin 1 5^{\circ} = r (1 - sin 1 5^{\circ})

DivEqn

$LHS / (1 - sin 1 5^{\circ}) = RHS / (1 - sin 1 5^{\circ})$

\frac{1 0 sin 1 5 ^{\circ}}{1 - sin 1 5 ^{\circ}} = r

RearrangeEqn

Rearrange equation

r = \frac{1 0 sin 1 5 ^{\circ}}{1 - sin 1 5 ^{\circ}}

UseCalc

Use a calculator

r = 3.491981 \dots

RoundDec

Round to $1$ decimal place(s)

r \approx 3.5

Therefore, the radius of each chair is about

3.5

feet. For reference, most common chairs these days are about

16

inches in diameter.

Example

Modeling Bodies Using Cylinders

Jordan and Ali are comparing their weights. Jordan is $5$ feet tall, and Ali is $6$ feet tall. Their shoulder widths are $1$ foot and $1.4$ feet, respectively.

Consider their bodies to be roughly cylindrical, and their shoulder widths could represent the diameter. What is the ratio of Jordan's weight to Ali's weight?

Answer

$4 : 9$ or $\frac{4}{9}$

Hint

Volume of a cylinder is the product of the base area and height of the cylinder.

Solution

When a body is modeled using a cylinder, one's height and shoulder width will represent the cylinder's height and diameter, respectively.

By comparing the volumes of these cylinders, a relation between the weights of Jordan and Ali can be set up. To do this, the radii of the cylinders should be found. Recall that the radius of a cylinder is half of its diameter.

\underline{Shorter Cylinder} \frac{1}{2} = 0.5 ft \underline{Taller Cylinder} \frac{1 . 4}{2} = 0.7 ft

Now that the radii have been found, the formula for the volume of a cylinder can be used to find their volumes.

	$h$	$r$	$π r^{2} h$	$V$
$Shorter Cylinder$	$5$	$0.5$	$π (0.5)^{2} (5)$	$\approx 4$
$Taller Cylinder$	$6$	$0.7$	$π (0.7)^{2} (6)$	$\approx 9$

The volumes of the cylinders imply that if Jordan weighs $80$ pounds, Ali weighs $180$ pounds. Therefore, the ratio of Jordan's weight to Ali's weight is $4 : 9$ or $\frac{4}{9} .$

Example

Modeling Arc Measures

Arc measure is also an important concept to interpret the situations modeled by circles. For example, consider a time zone wheel.

A time zone wheel is a tool used to find the time in different locations across the world. For example, to find the time in Tokyo when it is $1 P . M .$ in Amsterdam, rotate the small wheel until the labels of $1 P . M .$ and Amsterdam align.

As can be seen, it is $9 P . M .$ in Tokyo.

a What is the arc measure between each time zone on the wheel?

b What is the measure of the minor arc from the Istanbul zone to the Los Angeles zone?

Hint

a How many time zones are there on the wheel?

b How many time zones are there between the Istanbul zone and Los Angeles zone?

Solution

a It can be seen that the wheel is divided into

24

time zones. Since the measure of the whole wheel is

36 0^{\circ},

the arc measure between each time zone on the wheel can be found by dividing

36 0^{\circ}

24 .

\frac{3 6 0 ^{\circ}}{2 4} = 1 5^{\circ}

b Notice that the minor arc between the Istanbul zone and Los Angeles zone contains

11

time zones. Since the arc measure between each time zone is

1 5^{\circ},

the measure of the minor arc can be found by multiplying

1 5^{\circ}

11 .

1 5^{\circ} \times 11 = 16 5^{\circ}

Closure

Modeling Real Life Situations Using Circles

In real life, there are plenty of situations where circles can be appreciated. In this lesson, a few common situations have been covered. How about some other fun ones? Take, for example, one of the most complex applications of circles — the global positioning system.

When the three circular regions scanned by each satellite intersect, the position of the device is determined. Much more detail can be learned about GPS, this is just a brief introduction. How else are circles used in everyday life?

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Catch-Up and Review

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