# 7_05 Sectors and Segments?

7_05 Sectors and Segments

Find the area of the yellow sector of the circle with a given radius of 5 units. Use pi = 3.14.

3.925 units squared

19.625 units squared

3.125 units squared

39.25 units squared

POINT VALUE: 2 points

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Find the area of the red, purple, and yellow sectors of the circle with a given radius of 5 units.

Use pi = 3.14.

11.775 units squared

29.4375 units squared

58.875 units squared

78.5 units squared

POINT VALUE: 2 points

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Find the area of the green shaded portion of the circle with a given radius of 9 units. Use pi = 3.14

20.42 units squared

70.65 units squared

7.85 units squared

183.69 units squared

POINT VALUE: 2 points

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Find the area of the segment shaded in blue. The radius of the circle is 5 units and the base of the triangle is 8 units. Use pi = 3.14 and round your answers to the nearest hundredth.

POINT VALUE: 4 point

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Answer the questions below based on the target pictured. You may assume the dart hits the target every time. Hint: Leave your solutions in terms of pi; do not multiply by 3.14. Simply write “pi” for the pi symbol.

Part a) Find the geometric probability of throwing a dart and hitting the red circle.

Part b) Find the geometric probability of throwing a dart and hitting the blue or green rings.

Part c) Find the geometric probability of throwing a dart and hitting the yellow ring using your answers for parts a and b.

POINT VALUE: 5 point

Relevance

A "sector" of a circle is a piece cut out by the angle at the center between the two radii. Think of a piece of pie.

The area of a segment with an angle of x in a circle with radius r is:

(Pi*r^2)*(x/360) if x is in degrees

{The angle cuts off a fraction (x/360) of the area of the circle.}

If x is in radians, this simplifies to: (1/2)x*r^2

A "segment" of a circle is a piece cut off by a straight line (a chord).

This area formula is a little nastier: [A*d-L*Sqrt(d^2-L^2)]/4:

where L is the length of the chord, A is the arclength of the curved edge of the segment, and d is the diameter of the circle. (In theory, d can be calculated from A and L, but ,

An alternative area formula is: [x-Sin(x)](r^2)/2

where x is the central angle subtending the chord (in radians) and r is the radius.

Another formula is: (x/360)*Pi*r^2-L*h/2

where x is the central angle in degrees, r is the radius of the circle, L is the length of the chord, and h is the distance of the chord from the center. (You seem to be working from a very elementary textbook, so this might be the formula they use.)

A "ring" is just an outer circle minus an inner circle.

The area is the difference of the circle areas: Pi*(R^2 - r^2)

where R and r are the two radii.

A small comment:

The instructions say to use only 3 correct digits for Pi, but then the answers are given with up to 6 digits! If you use only 3 correct digits of Pi, then only (about) 3 digits of your answer will be correct! Giving additional (incorrect) digits will get you in trouble in any senior science class (as well as in my math class). Please think about rounding appropriately.

• You need to know two things here, the formula for the area of a circle, and how many degrees a circle is. The area of a circle is pi * r^2. So if r = 15m, then the area of the whole circle is pi * (15 m)^2 = 225 pi m^2. A circle is 360º. So an arc of 60º is 60/360 = 1/6 of that. So the area that is contained in an arc of 60º is 1/6 of the area of the whole circle. So the area is 225pi / 6 m^2 or 37.5 m^2