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6.8 – Areas and Sectors

Key Terms

  • Area – The space taken up by a two-dimensional figure or surface.
    • Area is measured in square units, such as square inches, square centimeters, or square feet.
  • Sector – A part of the interior of a circle bounded by an arc and the two radii that share the arc’s endpoints.

 


Review

Units of Area
  • Area is always listed in units “squared”
    • Units may be units (units), centimeters (cm), meters (m), feet (ft), inches (in), kilometers (km), etc.
    • Ex. 18\:m^2, 9\:cm^2, 3\:units^2, 6ft^2, 102km^2
Arc Length and Area of a Sector Comparison
GeoB 6.8 Area vs Arc Length

Notes

Area of a Circle
  • When we talk about the area of a circle, we mean the area enclosed by the circle (how much space is inside the circle).

GeoB 6.8 Area Enclosed

GeoB 6.8 Area Circle
  • Steps to find the area of a circle
    1. Find the radius of the circle.
    2. Square the radius.
    3. Multiply the result by using the calculator button \pi or 3.14.
    4. Round your answer to the nearest hundredth.
  • Example: What is the approximate area of the circle shown below?

GeoB 6.8 Q1E1

    • Formula: A=\pi\:r^2
    • Substitute: A=\pi\:23^2
    • Simplify: A=\pi\:529
    • Answer: A=1661.9\:cm^2

 

Sectors
  • A sector of a circle is a region bounded by an arc of a circle and radii to the endpoints of the arc.
  • The area of a sector is the area of the circle multiplied by the fraction of the circle covered by that sector.
GeoB 6.8 Sector GeoB 6.8 What is a SectorGeoB 6.8 Pizza
GeoB 6.8 Area SectorGeoB 6.8 Formula Area Sector

Examples

Area of a Circle
  • Ex 1. If a small 12-inch pizza costs $8 and a large 16-inch pizza costs $12, which is the better deal?
    • Steps to Find Out
      1. Find the area of each pizza.
      2. Use the area to find each pizza’s cost per square inch.
      3. Compare. The pizza that costs the least per square inch is the better deal.
  • Large: diameter = 16 in; so, radius = 8 in
    • Formula: Area\:=\pi r^2
    • Substitution: A\:=\pi 8^2
    • Simplify: A\:=\pi\bullet64
    • Solve: A=201\:in^2
    • Cost per Inch: cost\:\div size; so, 12\div 201=0.06\:cents\:per\:inch
  • Small: diameter = 12 in; so, 4adius = 6 in
    • Formula: Area\:=\pi r^2
    • Substitution: A\:=\pi 6^2
    • Simplify: A\:=\pi\bullet36
    • Solve: A=113\:in^2
    • Cost per Inch: cost\:\div size; so, 8\div 113=0.07\:cents\:per\:inch
  • Result: The large pizza is about 1 cent cheaper per square inch.  It doesn’t seem like much, but it will save you money in the long run if you shop “price per unit” for the lowest amount!
    • For the pizza problem, you save about 88 cents per pizza if you buy the large one (201 sq in minus 113 sq in = 88 sq in).
    • If each square inch is about 1 cent, multiply 1 cent by 88 sq in to get 88 cents!
    • If you buy 10 pizzas, you save $8.80!

 

Area of Sectors
  • Ex 1. If the area of the circle below is 12\:m^2, what is the area of the shaded sector?
    • Answer: 3\:m^2

GeoB 6.8 Q2E1

  • Ex 2. A circle has an area of 36\:m^2. What is the area of a 40° sector of this circle?
    • Answer: 4\:m^2
  • Ex 3. What is the approximate area of the shaded sector in the circle shown below?
    • Answer: 18.8\:cm^2

GeoB 6.8 Q2E2

  • Ex 4. What is the approximate area of the shaded sector in the circle shown below?
    • Answer: 297\:cm^2

GeoB 6.8 Q2E4

  • Ex 5. What is the approximate area of the shaded sector in the circle shown below?
    • Notice that the marker is NOT a chord. It is the measure of the diameter (5.4cm).
    • Answer: 11.45\:cm^2

GeoB 6.8 Q2E7


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