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| 8 Theory slides |
| 11 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here is some recommended reading before getting started with this lesson.
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
Like the pizza problem, numerous real-life problems can be modeled by sectors of a circle.
Tiffaniqua has a trapezoid-shaped yard whose side lengths are shown on the diagram. To water the lawn, she sets up a water sprinkler that can water the grass within a 4-meter radius. as shown.
Tiffaniqua knows that ∠ABC measures 135∘.
Substitute values
Add terms
Multiply
b1⋅a=ba
Calculate quotient
When the area of a sector is given, the measure of the corresponding central angle can be calculated
Consider a two-dimensional image of Pac-Man. The area covered by Pac-Man is about 66.5 square millimeters.
If the radius of the circle used to draw Pac-Man is 5 millimeters, find the measure of the central angle formed in the colored region. If necessary, round the answer to the nearest degree.Pac-Man is essentially a sector of a circle. Use the formula for the area of a sector of a circle to find the measure of the angle.
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
The diagram shows three circles that are congruent and have a radius of 1 centimeter.
What percentage of the square does the shaded area with the diagonal lines occupy? Round the answer to the nearest whole percent.To determine the percentage that the striped area occupies of the rectangle, we ought to find values for the rectangle's area A_R and the striped area A_(Strpd). The rectangle's area is the product of its width and length. A_R=(4)(2+sqrt(3)) cm^2 ⇕ A_R= 14.92820... cm^2 The striped area is the rectangle's area minus the sum of the area of the three circles and the area of the region between the circles. Therefore, we will also need to find the areas of the circles and the area between them. We know that each circle has a radius of 1 centimeter. We can determine the circles' combined area by calculating the area of one circle A_C and multiplying it by 3. 3A_C=3(π(1)^2) ⇔ 3A_C= 3π cm^2
To determine the area of the region between the circles, we will draw radii to the points of tangency between the circles. Since these radii meet at right angles, they form straight line segments. Additionally, the radii are all congruent which means these segments form an equilateral triangle. In such a triangle all angles are congruent with a measure of 60^(∘).
Further examining the diagram, we notice that the area of the region between the circles is the difference between the triangle's area and the area of three sectors, each with a central angle of 60^(∘).
As stated, we need to find the area of a sector A_S in order to find the area of the region between the circles. To do so, we can use the following formula.
A = θ/360^(∘)* π r^2
Let's substitute θ for 60^(∘) and r for the value 1. Notice that we have three identical sectors. To account for the value of all three sectors, we will multiply both sides of the formula by 3.
Continuing our quest to obtain the area of the region between the circles, we need to find the area of the triangle. By drawing the altitude of such an equilateral triangle, we see that this is a 30-60-90 triangle. Therefore, the leg opposite the 60^(∘)-angle is sqrt(3) times greater than the leg opposite the 30^(∘)-angle. Additionally, the smaller leg is half the length of the hypotenuse.
We have now collected enough measurements to calculate the area of the triangle A_T.
To find the area of the region between the circles, we will subtract the area of the three sectors 3A_s from the area of the triangle A_T.
Now we have everything we need to determine the striped area A_(strpd). It equals the rectangle's area subtracted by the sum of the circles and the area of the region between the circles. We have found each of those variables. Let's calculate the striped area.
Alas, by dividing the area of the striped area by the area of the rectangle, we can find the percentage that the striped area makes up of the rectangle's area. 5.34217.../14.92820...≈ 0.36 = 36 %