# Thread: Fragment of a circle (What am I looking for?)

1. ## Fragment of a circle (What am I looking for?)

I am trying to reason out what a question in my textbook is trying to ask me.

An archaeologist found a fragment of a circular plate on a dig. How could you determine the original size of the plate?

Is this asking for the area formula for the sector of a circle which is

A= πr^2 x θ/360 ?

2. Originally Posted by (?)G
I am trying to reason out what a question in my textbook is trying to ask me.

An archaeologist found a fragment of a circular plate on a dig. How could you determine the original size of the plate?

Is this asking for the area formula for the sector of a circle which is

A= πr^2 x θ/360 ?

Not quite,
You cannot tell what the angle is until you have the centre.

To find the radius, draw two chords across the arc of the plate.

Their perpendicular bisectors cross at the centre.

Having located the centre, you have the radius,
from which area and circumference can be measured if required.

3. Originally Posted by (?)G
I am trying to reason out what a question in my textbook is trying to ask me.

An archaeologist found a fragment of a circular plate on a dig. How could you determine the original size of the plate?

Is this asking for the area formula for the sector of a circle which is

A= πr^2 x θ/360 ?

Hi (?)G,

You don't know if the fragment is the size of a sector or not.
If it's just a fragment with a curve and some interior,
we could make a sketch.

Then, draw two chords in the fragment, construct the perpendicular bisector of these chords and the intersection of these bisectors would be the center of the plate.

I don't know why you might need the area to determine the size of the plate. Circumference would seem to work fine.

See attachment.

4. Originally Posted by Archie Meade
Not quite,
You cannot tell what the angle is until you have the centre.

To find the radius, draw two chords across the arc of the plate.

Their perpendicular bisectors cross at the centre.

Having located the centre, you have the radius,
from which area and circumference can be measured if required.
But I have the correct formula right?

5. Originally Posted by (?)G
But I have the correct formula right?
It's not really in keeping with the book question though.

I mean, the fragment would have to an exact sector of a circle,
whose point is the circle centre and the two sides absolutely straight.
It's like a slice of pizza or cheese.

You're not likely to find one of those at an archaeological dig.

masters tried to show that the plate would have an arc and a jagged edge
and the centre of the plate may even be in mid-air.

So we won't have an actual sector at all.

A sector is a fraction of a circle with straight edges that touch at the centre.

That's why we use the chords to locate the centre.

6. Draw on chord of length 2x. Find its midpoint. From that point, find the distance of the circumference. Let this by y.
Then x^2 = y(2R - y).
This relation is true for a diameter bisecting a chord.
Now solve for R.

7. $\displaystyle c^2+z^2=(2R)^2$

$\displaystyle x^2+y^2=z^2$

$\displaystyle x^2+(2R-y)^2=c^2$

$\displaystyle x^2+(2R-y)^2+x^2+y^2=(2R)^2$

$\displaystyle 2x^2=(2R)^2-y^2-(2R-y)^2=(2R)^2-y^2-\left[(2R)^2-4Ry+y^2\right]$

$\displaystyle =4Ry-2y^2=y(4R-2y)=2y(2R-y)$

$\displaystyle x^2=y(2R-y)$

$\displaystyle \frac{x^2}{y}=2R-y\ \Rightarrow\ \frac{x^2+y^2}{2y}=R$