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5. Modeling in Two & Three Dimensions
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Integrated Math 3

5.01 Areas of circles and sectors

Lesson
Practice
Lesson

Concept summary

The full length around a circle is known as its circumference, and a part of the circumference of a circle is called an arc. Arcs can be further classified as follows:

Semicircle

An arc of a circle whose endpoints lie on a diameter.

An arc of a circle with endpoints on the diameter of the circle.
Major arc

An arc larger than a semicircle.

An arc of a circle larger than a semicircle.
Minor arc

An arc smaller than a semicircle.

An arc of a circle smaller than a semicircle.

Any arc of a circle has a corresponding central angle formed by the radii which meet the arc at its endpoints.

Central angle

An angle that has its vertex at the center of a circle with radii as its sides.

A circle with two radii drawn. The angle between them is marked

The measure of the length of an arc is called the arc length. Since an arc is a fraction of the circumference, we can calculate this as a portion of the total circumference by considering the central angle of the arc as a portion of a full rotation:

\displaystyle s = \frac{\theta}{360} \cdot 2 \pi r
\bm{s}
Arc length
\bm{\theta}
Central angle
\bm{r}
Radius

Adjacent arc lengths can be combined by the following postulate:

Arc addition postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Circle C with minor arcs A B and B D.

This theorem is helpful to connect the central angle with the minor arc.

Congruent central angles theorem

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.

Circle C with diameters A Y and X B. Central angles A C B and X C Y are congruent. Minor arcs A B and X Y are congruent.

An arc and the radii which form its corresponding central angle border a region inside a circle. We call this region a sector of a circle.

Sector

A region inside a circle bounded by an arc and the two radii which form its central angle

A circle with two radii. The smaller area of the circle between the two radii is highlighted

We can find the perimeter of a sector using the arc length formula:

\displaystyle P = s + 2r
\bm{P}
Perimeter of the sector
\bm{s}
Arc length
\bm{r}
Radius

We can calculate the area of the sector in a similar way to its arc length. Since a sector is a fraction of the region inside a circle, we can find its area by taking a portion of the total area of the circle corresponding to the central arc's portion of a full rotation:

\displaystyle A = \frac{\theta}{360} \pi r^2
\bm{A}
Area of the sector
\bm{\theta}
Central angle
\bm{r}
Radius

Since \dfrac{\theta}{360} represents the proportion of the full circle for an arc or sector, we can think of arc length or sector area as: \begin{aligned} \text{Arc length}&=\text{Proportion of the circle}\cdot\text{Circumference of the full circle} \\ \text{Sector area}&=\text{Proportion of the circle}\cdot\text{Area of the full circle} \end{aligned}

Worked examples

Example 1

For the following sector, where AB = 5 inches:

Sector B C of circle A. Central angle B A C has a measure of 68 degrees.
a

Find the arc length of \overset{\large\frown}{BC}.

Solution

\displaystyle s\displaystyle =\displaystyle \frac{\theta}{360} \cdot 2 \pi rArc length formula
\displaystyle {}\displaystyle =\displaystyle \frac{68}{360} \cdot 2\pi (5)Substitute known values
\displaystyle {}\displaystyle =\displaystyle \frac{17\pi}{9}Simplify

The arc length of the sector is \dfrac{17 \pi }{9} inches.

Reflection

Since there was no instruction to approximate the solution by rounding the answer, we should keep the answer as an exact value.

b

Find the area of the sector.

Solution

\displaystyle A\displaystyle =\displaystyle \frac{\theta}{360}\pi r^2Area formula
\displaystyle {} \displaystyle =\displaystyle \frac{68}{360} \pi (5)^2Substitute known values
\displaystyle {}\displaystyle =\displaystyle \frac{85 \pi}{18}Simplify

The area of the sector is \dfrac{85 \pi }{18}\, \text{in}^2.

Example 2

Consider the given diagram:

Circle Q with radii Q J, Q K, Q L and Q M placed clockwise on the circle. Angle J Q K has a measure of 115 degrees, K Q L has a measure of 9 degrees, and L Q M has a measure of 70 degrees.
a

Use the Arc addition postulate to write an expression that represents m\overset{\large\frown}{JL}

Solution

\displaystyle m\overset{\large\frown}{JL}\displaystyle =\displaystyle m\angle JQK + m\angle KQLArc addition postulate
\displaystyle m\overset{\large\frown}{JL}\displaystyle =\displaystyle 115 \degree + 9 \degreeSubstitute known values
b

Find m\overset{\large\frown}{JL}

Solution

Adding the expression obtained from part (a), m\overset{\large\frown}{JL} = 124 \degree

Example 3

Valentim wants to make a hand fan using a sector, where AB = 12 inches.

Sector B C of circle A. Central angle B A C has a measure of 120 degrees.
a

Find what fraction of a full circle the hand fan will be.

Approach

The central angle of a full circle is 360 \degree. To find the fraction, we can divide the central angle of the sector by the central angle of the full circle.

Solution

\displaystyle \text{Fraction}\displaystyle =\displaystyle \frac{120}{360}
\displaystyle {}\displaystyle =\displaystyle \frac{1}{3}Simplify

The sector is \dfrac{1}{3} of the full circle.

b

Find the perimeter of the sector that Valentim is using for the hand fan.

Solution

\displaystyle P\displaystyle =\displaystyle s + 2rPerimeter formula
\displaystyle {}\displaystyle =\displaystyle \frac{\theta}{360} \cdot 2 \pi r + 2 rArc length formula
\displaystyle {}\displaystyle =\displaystyle \frac{120}{360} \cdot 2\pi (12) + 2 (12)Substitute known values
\displaystyle {}\displaystyle =\displaystyle 8\pi + 24Simplify

The perimeter of the sector is \left(8\pi + 24\right) inches.

Reflection

Notice the fraction found in part (a) is used in the formula for arc length.

Outcomes

M3.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.

M3.G.C.A.1

Use proportional relationships between the area of a circle and the area of a sector within the circle to solve problems in a real-world context.*

M3.MP1

Make sense of problems and persevere in solving them.

M3.MP3

Construct viable arguments and critique the reasoning of others.

M3.MP4

Model with mathematics.

M3.MP5

Use appropriate tools strategically.

M3.MP6

Attend to precision.

M3.MP7

Look for and make use of structure.

M3.MP8

Look for and express regularity in repeated reasoning.

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