Area of a lune (and a segment)

By Tracey Mcdonald (M1079) on Monday, April 9, 2001 - 07:15 am :

Please please please can someone explain waht the formula for the area of a lune is before I explode!

Please can it be a detailed explanation as I have absolutely no idea what my book is talking about.

Cheers
Tracey

By Emma McCaughan (Emma) on Monday, April 9, 2001 - 09:21 am :

Can I just check whether we're talking about the shape shaded here: Diagram
Also, are you happy with finding areas of sectors and segments of circles?

By Tracey Mcdonald (M1079) on Monday, April 9, 2001 - 10:56 pm :

Yes it is the area shaded.

Tracey

No to your question.

By Emma McCaughan (Emma) on Tuesday, April 10, 2001 - 10:03 am :

I'm sure you know that the area of a circle is

π r^{2}

. The orange and green areas together are a sector, which is just a fraction of a circle. What fraction? Well,

θ / 360

if you're in degree, or

θ / (2 π)

in radians. So the formula for the area of a sector is

π r^{2} θ / 360

in degrees, or

\frac{1}{2} r^{2} θ

in radians.

We can also find the area of the green triangle easily, if you know the version of the formula for the area of a triangle which is $\frac{1}{2} a b \sin C$ . On this triangle, taking the angle at the centre as $C$ , we get $\frac{1}{2} r r \sin θ = \frac{1}{2} r^{2} \sin θ$ .

Now you can work out the area of the orange segment by subtraction.

The lune is made up of two segments: if you can find the area of each, just add them together.

By Anonymous on Friday, April 13, 2001 - 06:50 pm :

Now I'm confused... I thought lunes were formed by two arcs both concave, rather than an arc and a straight line... more like an orange segment or a mouth?

[In this case you will be subtracting the areas of two segments. - The Editor]

If anyone cares, lunes are quite historically important - Hippocrates of Chios did far too much on them in the days when the Greeks were arguing about whether circles could be said to have area at all...