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Three friends are sharing a 16-inch pizza equally. Emily is becoming a nutritionist and is curious about how many calories are in a slice. To find out, she will calculate the area of one slice.
Help Emily to find the area of one slice. How can the measure of a central angle be used to find the area?The area of a sector of a circle is calculated by multiplying the circle's area by the ratio of the measure of the central angle to 360∘.
Area of Sector=360∘θ⋅πr2
From the fact that 2π rad equals 360∘, an equivalent formula can be written if the central angle is given in radians.
Since the measure of an arc is equal to the measure of its central angle, the arc AB measures θ. Therefore, by substituting mAB for θ, another version of the formula is obtained which can also be written in degrees or radians.
Area of Sector=360∘mAB⋅πr2
or
Area of Sector=2mAB⋅r2
Consider sector ACB bounded by AC,BC, and AB.
Since a circle measures 360∘, this sector represents 360∘θ of ⊙C. Therefore, the ratio of the area of a sector to the area of the whole circle is proportional to 360∘θ.Area of Sector=360∘θ⋅πr2
Substitute values
ba=b/2πa/2π
Use a calculator
LHS⋅R=RHS⋅R
LHS/0.4=RHS/0.4
Rearrange equation
In his free time, Dylan enjoys making decorative figures by hand. He has 5 identical sectors and brings these sectors together as shown.
Dylan knows that the area of each sector is 18 square millimeters.
Mark set up a lamp in his courtyard. He uses a light bulb that illuminates a circular area with a radius of 6 meters. The diagram shows a bird's eye view of Mark's house.
If the measure of arc MN is 100∘, what is the area of the region that is illuminated outside of the courtyard area? If necessary, round the answer to two decimal places.The area of a triangle is half the product of the lengths of any two sides and the sine of the included angle.
From the diagram, it can be seen that the region bounded by MP, NP, and MN is a sector of ⊙P.
Substitute values
Substitute values
Subtract term