Q.5 In a mathematical activity, a teacher asks students to divide a circular pizza of radius 13 cm into
5 equal parts. A student states that each part of pizza will subtend central angle of 72 degree. Is
This answer true or false? Justify your answer. Which moral value is depicted here?
Answers
Answer:
As you may remember from geometry, the area A of a circle having a radius of length r is given:
\small{ A = \pi r^2 }A=πr
2
The circumference C (that is, the length around the outside) of that same circle is given by:
\small{ C = 2\pi r }C=2πr
These are the formulas give us the area and arc-length (that is, the length of the "arc", or curved line) for the entire circle. But sometimes we need to work with just a portion of a circle's revolution, or with many revolutions of the circle. What formulas do we use then?
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If we start with a circle with a marked radius line, and turn the circle a bit, the area marked off looks something like a wedge of pie or a slice of pizza; this is called a "sector" of the circle, and the sector looks like the green portion of this picture:
circle with central angle "θ" and radius "r" marked, delineating the sector, which is shaded a light green
The angle marked off by the original and final locations of the radius line (that is, the angle at the center of the pie / pizza) is the "subtended" angle of the sector. This angle can also be referred to as the "central" angle of the sector. In the picture above, the central angle is labelled as "θ" (which is pronounced as "THAY-tuh").
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What is the area A of the sector subtended by the marked central angle θ? What is the length s of the arc, being the portion of the circumference subtended by this angle?
To determine these values, let's first take a closer look at the area and circumference formulas. The area and circumference are for the entire circle, one full revolution of the radius line. The subtended angle for "one full revolution" is 2π. So the formulas for the area and circumference of the whole circle can be restated as:
\small{ A = \left(\dfrac{\mathbf{\color{green}{2\pi}}}{2}\right) r^2 }A=(
2
2π
)r
2
\small{ C = (\mathbf{\color{green}{2\pi}}) r }C=(2π)r
What is the point of splitting the angle value of "once around" the circle? I did this in order to highlight how the angle for the whole circle (being 2π) fits into the formulas for the whole circle. This then allows us to see exactly how and where the subtended angle θ of a sector will fit into the sector formulas. Now we can replace the "once around" angle (that is, the 2π) for an entire circle with the measure of a sector's subtended angle θ, and this will give us the formulas for the area and arc length of that sector:
\small{ A = \left(\dfrac{\mathbf{\color{purple}{\theta}}}{2}\right) r^2 }A=(
2
θ
)r
2
\small{ s = (\mathbf{\color{purple}{\theta}}) r }s=(θ)r
Note: If you are working with angles measured in degrees, instead of in radians, then you'll need to include an extra conversion factor:
\small{ \begin{aligned} A &= \left(\dfrac{\theta\deg}{2}\right) \left(\dfrac{\pi}{180\deg}\right) r^2 \\[3ex] &= \left(\dfrac{\theta}{360}\right) \pi r^2 \end{aligned} }
A
=(
2
θdeg
)(
180deg
π
)r
2
=(
360
θ
)πr
2
\small{ \begin{aligned} s &= \left(\dfrac{\pi}{180\deg}\right)\left(\theta\deg\right) r \\[3ex] &= \left(\dfrac{\theta}{180}\right) \pi r \end{aligned} }
s
=(
180deg
π
)(θdeg)r
=(
180
θ
)πr
Confession: A big part of the reason that I've explained the relationship between the circle formulas and the sector formulas is that I could never keep track of the sector-area and arc-length formulas; I was always forgetting them or messing them up. But I could always remember the formulas for the area and circumference of an entire circle. So I learned (the hard way) that, by keeping the above relationship in mind, noting where the angles go in the whole-circle formulas, it is possible always to keep things straight.