Length of an arc

An arc is a section of the circumference - it is smaller than 2πr.

To calculate it you need to know the angle, yo, made by the arc at the centre of the circle. 

y/360 = (length of arc)/2πr

You can think of this as the angle as some percentage of the full 360o.
The length of the arc is the same percentage as the full circumference 2πr

You can rearrange the formula to find the length of an arc

length of arc = (2πr mulitpled by y) divided by 360 = 2πry/360

Area of a sector

A sector is a section of the area, its shape is like a slice of pizza.

You use the same method as above

To calculate it you need to know the angle, zo, of the sector.

z/360 = (area of sector)/πr2

You can again think of this as the angle as some percentage of the full 360o.
The area of the sector is the same percentage as the area πr2

area of sector = zπr2 /360

Area of a segment

A segment is the area between a chord and edge of a circle (the smaller section when drawing a line from the edge of a circle to another part of the edge)

This is done in 2 parts.

We first draw lines from each end of the chord to the centre, we then ignore the chord, and calculate the area of the sector, as above.

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We then put the chord back in, and now calculate the 'triangle area' between the chord, and the two lines we just drew in to the centre. 

The area of the triangle = length of chord x perpendicular distance from chord to the centre.

(Notice that this triangle is two identical right angles (by properties of a circle the bisector of a chord going through the centre is a perpendicular bisector). The sum of 2 right angle will give you just base x height, which in this instance is length of chord x perpendicular distance from chord to the centre)

It may be necessary to use trigonmetry to calculate these lengths, given that we have the angle of the sector.

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Area of the segment = Area of the sector - Area of the triangle

 

 

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