Area of a Sector- Why squared?

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In summary, the area of a sector of a circle is found by multiplying half the radius squared by the central angle in radians. The resulting units will be squared, representing the standard units of area. This can be visualized by thinking of the sector as a triangle, with the base being the radius and the height being the arc length, which is equal to the central angle multiplied by the radius. This results in the units being squared, as the base and height are both in linear units.
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Homework Statement


Find the area, A, of a sector of a circle with a radius of 9 inches and a central angle of 30°.

Homework Equations


$$Area~of~a~Sector:$$
$$A=\left( \frac 1 2 \right)r^2θ$$

The Attempt at a Solution


[/B]
$$θ=30°$$
$$θ=30°\left( \frac π {180} \right)$$
$$θ=\left( \frac π 6 \right)$$

$$A=\left( \frac 1 2 \right)\left(9\right)^2\left(\frac π 6 \right)$$
$$A=\left( \frac {81π} {12} \right)$$
$$A≈21.2 in^2$$

My question:
I know that when you find the area of a space, it will be in ##units^2##. But I've always thought of it as a square- that is, one equal side multiplied by the other equal side obviously results in a squared result. However in this case, I don't see how the units for a sector of a circle are squared, as it doesn't seem like we're multiplying two things of equal value to each other.
So why is this result squared?
 
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  • #2
opus said:

Homework Statement


Find the area, A, of a sector of a circle with a radius of 9 inches and a central angle of 30°.

Homework Equations


$$Area~of~a~Sector:$$
$$A=\left( \frac 1 2 \right)r^2θ$$

The Attempt at a Solution


[/B]
$$θ=30°$$
$$θ=30°\left( \frac π {180} \right)$$
$$θ=\left( \frac π 6 \right)$$

$$A=\left( \frac 1 2 \right)\left(9\right)^2\left(\frac π 6 \right)$$
$$A=\left( \frac {81π} {12} \right)$$
$$A≈21.2 in^2$$

My question:
I know that when you find the area of a space, it will be in ##units^2##. But I've always thought of it as a square- that is, one equal side multiplied by the other equal side obviously results in a squared result. However in this case, I don't see how the units for a sector of a circle are squared, as it doesn't seem like we're multiplying two things of equal value to each other.
So why is this result squared?
Because it's an area. The shape doesn't matter.
The standard units of area are always squared, ##\text{length} \times \text{length}##, except for some special cases such as acres or hectares (which involve implicitly squared units such as ft2 for acres and m2 for hectares.
 
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So the length x length in this particular case, would be length(radius) x length(arc). However the length of the radius is in inches, and the length of the arc is in radians. So how can this results in inches squared?
 
  • #4
opus said:
So the length x length in this particular case, would be length(radius) x length(arc). However the length of the radius is in inches, and the length of the arc is in radians. So how can this results in inches squared?
The angle in radians is just an angle, with no length. Think about it this way, as a, say, peach pie. If an 8" diameter pie is cut into 6 pieces, each slice (a sector) will subtend an angle of ##\pi/3##, and the radius will be 4". The arc length of the curved edge of the slice has to take into account the radius, otherwise the arc length of an 8" pie would be the same as for a 16" pie. So in fact, the curved dimension of the pie sector is radius * angle (in radians), or ##4 \times \pi/3##. So you have one radius for the radius of the sector and another radius for the arc length, making the sector area equal to ##\frac 1 2 r^2 \theta##.
 
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  • #5
Ahhh ok. That makes complete sense. Great explanation, thank you Mark.
 
  • #6
opus said:
I know that when you find the area of a space, it will be in ##units^2##. But I've always thought of it as a square- that is, one equal side multiplied by the other equal side obviously results in a squared result.
Let me add that one way to visualize the "units square" is to think that a wedge with an area of 22.2 in2 has the same area as a square with sides of √(22.2) ≈ 4.6 in.
 
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  • #7
Interesting! Thanks DrClaude
 
  • #8
imagine the arc as a triangle , becuase area would be same even if you make the arc straight line.
now the base of this triangle is "r=radius" and the perpendicular side is the arc which is equal to "theta*r"

area of triangle = 0.5base*height

0.5(r)(r)(theta)=formula of area of sector
 

Related to Area of a Sector- Why squared?

1. What is the formula for finding the area of a sector?

The formula for finding the area of a sector is (θ/360) x πr², where θ is the central angle and r is the radius of the circle.

2. Why is the area of a sector squared?

The area of a sector is squared because it is calculated by multiplying the radius squared by the central angle divided by 360. This is because the area of a circle is calculated by multiplying πr², and a sector is a fraction of a circle determined by the central angle.

3. Can the area of a sector be greater than the area of the entire circle?

No, the area of a sector can never be greater than the area of the entire circle. The area of a sector is always a fraction of the area of the circle, determined by the central angle.

4. How does the central angle affect the area of a sector?

The central angle directly affects the area of a sector. As the central angle increases, the area of the sector also increases, and vice versa.

5. Can the area of a sector be negative?

No, the area of a sector cannot be negative. It is always a positive value representing the amount of space enclosed by the sector within the circle.

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