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Pearson Geometry Common Core, 2011

PG

Pearson Geometry Common Core, 2011 View details

7. Areas of Circles and Sectors

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Exercise 23 Page 664

To find the area of a segment for a minor arc, draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed.

22.1cm^2

Practice makes perfect

A part of a circle bounded by an arc and the segment joining its endpoints is a segment of a circle.

To find the area of a segment for a minor arc, we need to draw radii to form a sector. The area of the segment equals the area of the sector minus the area of the triangle formed.

For the given diagram, we will find the area of the sector and then the area of the triangle. Finally, we will find their difference to find the area of the segment.

Area of the Sector

The area of a sector of a circle is the product of the measure of the arc divided by 360 and the area of the circle.

With this in mind, let's consider the given diagram.

We can see that the radius of the circle is 6cm, and the measure of the arc is 120^(∘). We have all the information we need to calculate the area of the sector. Let's do it!

Measure of the arc/360* π r^2

SubstituteValues

Substitute values

120/360 * π ( 6^2)

▼

Simplify

a/b=.a /120./.b /120.

1/3 * π (6^2)

Calculate power

1/3 * π (36)

CommutativePropMult

Commutative Property of Multiplication

1/3 * 36π

MoveRightFacToNumOne

1/b* a = a/b

36π/3

MovePartNumRight

a* b/c=a/c* b

36/3 * π

Calculate quotient

12 π

The area of the sector is 12π cm^2.

Area of the Triangle

The area of a triangle can be found by taking half the product of two side lengths and the sine of the included angle. With this in mind, let's consider the triangle in our diagram.

The length of both of the known sides is 6cm and the included angle measures 120^(∘). This is enough information to find the area of the triangle.

1/2( 6)( 6)sin 120^(∘)

▼

Simplify

Multiply

1/2(36)sin 120^(∘)

MoveRightFacToNumOne

1/b* a = a/b

36/2sin 120^(∘)

Calculate quotient

18 sin 120^(∘)

The area of the triangle is 18sin120^(∘)cm^2.

Area of the Segment

Finally, to obtain the area of the segment we need to subtract the area of the triangle from the area of the sector.

12π-18sin120^(∘)

Use a calculator

22.11065...

Round to 1 decimal place(s)

22.1

The area of the segment, to the nearest tenth, is 22.1cm^2.

Subchapter links

Lesson Check p.663

Practice and Problem-Solving Exercises p.663-666

Standardized Test Prep p.666

Mixed Review p.666