# Calculating the Area of a sector to mm^2

#### Biggsy100

Joined Apr 7, 2014
88
I need to calculate a sector of an area. As understand it, it can be calculated as Area x 2 Pi (- Sector).

In the case of the PDF file I have attached would read; 45mm X 2*Pi (-15mm). If I a correct, it just seem to straight forward?

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#### djsfantasi

Joined Apr 11, 2010
7,867
Where did you get that formula from? I would approach it as follows.

First calculate the area of the ring (as if it was an entire circle, not the segment shown).

Area = 2*Pi*45mm - 2*Pi*30mm
Edit: This is not the equation for area. To see the correct equation, look at studiot's post following.

Then, since the sector only takes up Pi/2, the final area is 1/4 of the ring { (Pi/2)/(2*Pi) or (0.5*Pi)/(2*Pi) = 0.5/2 = 1/4.}

You may be able to combine the two steps, but I think in steps. First, I k now how to do that, then I need to do the other thing... (old mathematicians joke).

Last edited:

#### studiot

Joined Nov 9, 2007
4,998
Not sure what is going on here.

Area is not measured in mm but in millimetres squared.

So any formula based on the length of the circumference of a circle will fail.

If you want the shaded area between the two part circles then yes

calculate the area of the large circle,
subtract the area of the smaller circle
divided the result by four since you only want a quarter of it.

$$Area = \frac{{\pi r_1^2 - \pi r_2^2}}{4} = \frac{\pi }{4}\left( {r_1^2 - 1*r_2^2} \right)$$

$$= \frac{\pi }{4}\left( {{{45}^2} - {{30}^2}} \right) = \frac{\pi }{4}\left( {2025 - 900} \right) = 883.1m{m^2}$$

#### djsfantasi

Joined Apr 11, 2010
7,867
Not sure what is going on here.

Area is not measured in mm but in millimetres squared.

So any formula based on the length of the circumference of a circle will fail.

If you want the shaded area between the two part circles then yes

calculate the area of the large circle,
subtract the area of the smaller circle
divided the result by four since you only want a quarter of it.

$$Area = \frac{{\pi r_1^2 - \pi r_2^2}}{4} = \frac{\pi }{4}\left( {r_1^2 - 1*r_2^2} \right)$$

$$= \frac{\pi }{4}\left( {{{45}^2} - {{30}^2}} \right) = \frac{\pi }{4}\left( {2025 - 900} \right) = 883.1m{m^2}$$
D'oh... I knew the point I was trying to make, but was suckered in by the OP's original equation without thinking...

Thanks for the clarification!

#### Biggsy100

Joined Apr 7, 2014
88
Ok great, however I am a little confused with the area so I put in pi/4 (2/1*-1*x2/2) = 0.7853 ?

#### djsfantasi

Joined Apr 11, 2010
7,867
Ok great, however I am a little confused with the area so I put in pi/4 (2/1*-1*x2/2) = 0.7853 ?
I don't understand your equation!? Let's start with do you know the equation for the area of a circle?

$$Area_{circle} = \pi * radius^{2}$$

In your problem, we have the following.

$$Area_{bigcircle} = \pi * 45^{2}$$
$$Area_{smallcircle} = \pi * 30^{2}$$

The difference between the two gives you the area of the ring.

$$Area_{ring} = \pi * 45^{2} - \pi * 30^{2}$$

But you only want a segment of the ring, equal to $$\pi/2$$ radians

$$Ratio_{segment} = (\pi / 2) / 2\pi = 0.5\pi/\2pi = 0.5/2 = 1/4$$

So applying this ratio to the ring area, we get (what studiot posted):

$$Area_{segment} = \frac{\pi * 45^{2} - \pi * 30^{2}}{4}$$