# Perimeter/Area of Shaded Region within a Circle

• zak100
In summary, the conversation discussed finding the area and perimeter of a shared region in a diagram. The formula for calculating the area and perimeter of the shared region was provided, but it was determined that the formula was incorrect. The correct formulas for finding the area and perimeter of the shared region were discussed, and it was suggested to look up the concept of a "circular segment". The correct formulas were then used to calculate the area and perimeter of the shared region.
zak100

## Homework Statement

Find the area and perimeter of shared region in the following diagram:

## Homework Equations

Perimeter of shared Region = x/360 * 2* PI* radius[/B]

## The Attempt at a Solution

I am finding the area of circle & then subtracting the area of triangle to find the area of shared region:
Thus
= 3.14 * 12 * 12
= 144 PI
A of Triangle (Note its an equilateral triangle) = sqr(s) * sqrt(3)/ 4 = 36 sqrt(3)
Therefore ans = 144PI - 36 sqrt(3) but this is a wrong answer.

Similarly perimeter of circle = 12 * 3 = 36
Circumference of circle = 2 * PI * radius
= 2* PI * 12
=24 PI
Therefore ans= 24PI - 36

Some body please guide me why my logic is not coreect.

Zulfi.

zak100 said:
I am finding the area of circle & then subtracting the area of triangle to find the area of shared region:
That will not give you the area of the shaded region. If you added the area of the triangle to the area of the shaded region, which area will you get?
zak100 said:
Similarly perimeter of circle = 12 * 3 = 36
Circumference of circle = 2 * PI * radius
= 2* PI * 12
=24 PI
Therefore ans= 24PI - 36
That's not how you calculate perimeter. You need to find minor arc length+CD.

What is the formula for arc length which uses the given information in the diagram ?

Hi,
<That will not give you the area of the shaded region. If you added the area of the triangle to the area of the shaded region, which area will you get?>
Is this not same as calculating area of Circle & then subtract the are of triangle from it. If not please guide me why?
<What is the formula for arc length which uses the given information in the diagram ?>
I gave all the formulae in the beginning:
Perimeter of shared Region = x/360 * 2* PI* radius

Zulfi.

zak100 said:
<That will not give you the area of the shaded region. If you added the area of the triangle to the area of the shaded region, which area will you get?>
Is this not same as calculating area of Circle & then subtract the are of triangle from it. If not please guide me why?
<What is the formula for arc length which uses the given information in the diagram ?>

This ^_^.
zak100 said:
I gave all the formulae in the beginning:
Perimeter of shared Region = x/360 * 2* PI* radius

What is x ?

zak100 said:
Hi,
<That will not give you the area of the shaded region. If you added the area of the triangle to the area of the shaded region, which area will you get?>
Is this not same as calculating area of Circle & then subtract the are of triangle from it. If not please guide me why?
<What is the formula for arc length which uses the given information in the diagram ?>
I gave all the formulae in the beginning:
Perimeter of shared Region = x/360 * 2* PI* radius

Zulfi.

There are formulas available for such areas; you just need to look them up on the internet, or look inside a textbook.

Hint: look up "circular segment".

Ray Vickson said:
There are formulas available for such areas; you just need to look them up on the internet, or look inside a textbook.

Hint: look up "circular segment".
He have enough information to find the area without formula.

Last edited:
Hi,
Thanks for your guidance. As you said perimeter is:
<You need to find minor arc length+CD>.
x/36 * 2* PI * radius + CD
Note x= 60 which is the central angle & is equal to the angle of arc?? (Is this correct?)
CD = 12 because its an equilateral triangle
60/360 * 2 * PI * 12 + 12
1/6* 2 PI * 12 + 12
4*PI + 12

Area of Sector COD - Area of triangle COD
60/360 * PI * RADIUS * RADIUS - 12 * 12 Sqrt(3)/4
1/6 * PI * 12 * 12 -36 sqrt(3)
24PI -36sqrt(3)
Since the shaded region is part of sector O so it is not needed to calculate area of circle.
Zulfi.

zak100 said:
Hi,
Thanks for your guidance. As you said perimeter is:
<You need to find minor arc length+CD>.
x/36 * 2* PI * radius + CD
Note x= 60 which is the central angle & is equal to the angle of arc?? (Is this correct?)
CD = 12 because its an equilateral triangle
60/360 * 2 * PI * 12 + 12
1/6* 2 PI * 12 + 12
4*PI + 12

Area of Sector COD - Area of triangle COD
60/360 * PI * RADIUS * RADIUS - 12 * 12 Sqrt(3)/4
1/6 * PI * 12 * 12 -36 sqrt(3)
24PI -36sqrt(3)
Since the shaded region is part of sector O so it is not needed to calculate area of circle.
Zulfi.
Looks good.

Buffu said:
He have enough information to find the area without formula.

But he HAS been using formulas---just the wrong ones.

Try finding the area of the slice of the circle and subtract the area of the triangle
You have the angle so just try to find the right formulas to use

Your calculation is far too unnecessarily complicated.
For the triangle, forget about the circle. Just take a triangle like that by itself.
Then for the <) shape, you have very probably seen processed cheeses (Philadelphia, Kraft are a couple of names to my knowledge) in a circular box. How many cheeses are there in one?
(Or at least cheesees in a box or a slice of a pie -no pun intended - looking like your diagram.)

Last edited:

## 1. What is the formula for finding the perimeter of a shaded region within a circle?

The formula for finding the perimeter of a shaded region within a circle is given by P = 2πr + 2L, where r is the radius of the circle and L is the length of the shaded region.

## 2. How do you find the area of a shaded region within a circle?

To find the area of a shaded region within a circle, you can use the formula A = πr^2 - A', where r is the radius of the circle and A' is the area of the unshaded portion within the shaded region.

## 3. Can the shaded region within a circle have a perimeter greater than the perimeter of the circle?

Yes, it is possible for the shaded region within a circle to have a perimeter greater than the perimeter of the circle. This can happen if the length of the shaded region is greater than half the circumference of the circle.

## 4. How does the radius of the circle affect the perimeter and area of the shaded region?

The radius of the circle directly affects the perimeter and area of the shaded region. As the radius increases, both the perimeter and area of the shaded region also increase. Similarly, if the radius decreases, the perimeter and area of the shaded region will also decrease.

## 5. Can the shaded region within a circle have a negative perimeter or area?

No, the shaded region within a circle cannot have a negative perimeter or area. Perimeter and area are always positive values and cannot be negative.

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