Segment of a circle (Exact form answer)

[Matt]

Homework Statement

Find the circle segment area that has the boundaries of line segment AB and the minor arc ACB.

Give the area in an exact form in terms of surds and Pi. (see attachments for annotated picture & original question).

Homework Equations

Equation 1: Area of a segment = Sector area - Triangle area

Equation 2: Sector area = Central angle/360 * pi * diameter

Equation 3: Triangle area = 0.5 * base * height

The Attempt at a Solution

Area of sector = 120/360 * pi * 4
Simplified = 1/3 * pi/1 * 4/1 = 4/3 * pi [Units squared]

Area of one of the triangle = 0.5 * 1√3 * 1
Combined triangle area = 0.5 √3 x 2 = 1√3

Segment area = (4 * pi/3 - √3) [Units squared]
Approximated answer = 2.4567

 

[NascentOxygen]

Hi Matt. I see you are new here ... http://imageshack.com/a/img515/4884/welcomesign.gif

Your answer is right. Your working is very well presented.

I would not include the real approximation, since it's the exact form that is required.

Can you see a way to check your answer yourself?

 

[Matt]

Hi Matt. I see you are new here ... http://imageshack.com/a/img515/4884/welcomesign.gif

Your answer is right. Your working is very well presented.

I would not include the real approximation, since it's the exact form that is required.

Can you see a way to check your answer yourself?

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Hello Nascent Oxygen,

Thank you for your feedback, I appreciate it.

The only methods I can think to check if the answer is correct is to use one of the following equations. I know those equation give the same approximate answers. However I'm not sure if that is a reliable method of checking if the answer is correct.

Equation 1 (Area of a circle segment using height and radius)
Radius [squared] * cos -1 (radius - height / radius) - (radius - height) √2*r*h -h2

Equation 2 (Area of circle segment given the central angle)
Radius [Squared] * (pi/180) * central angle - sin (central angle)

 

[Matt]

Also,
Thank you very much for your warm welcome. It's good to be on here.

 

[NascentOxygen]

I did have in mind just reversing the process to arrive at the area of the full sector, as a check of your arithmetic.

But, you're right, you could compare your answer with a published formula, as you quoted.

http://d2gbom735ivs5c.cloudfront.net/m/geometry/images/circle-segment-area.gif

 

[Matt]

Just to check I understand correctly. I would calculate the approximate area of the segment area and the approximate area of the triangle and add them together.

Segment area (2.4567) + Triangle area (1.7320) = Sector area (4.1887)

4 * pi / 3 + √3 = 120/360 * pi * r [squared]

 

[NascentOxygen]

Just to check I understand correctly. I would calculate the approximate area of the segment area and the approximate area of the triangle and add them together.

Segment area (2.4567) + Triangle area (1.7320) = Sector area (4.1887)

Any arithmetic check will pick up blunders, giving you a chance to rectify mistakes before they cost you marks.

Now, 3 * 4.1887 = 12.5661

compare with the area of a circle of radius 2,
and there is agreement.

 

[Matt]

Any arithmetic check will pick up blunders, giving you a chance to rectify mistakes before they cost you marks.

Now, 3 * 4.1887 = 12.5661

compare with the area of a circle of radius 2,
and there is agreement.

------------------------------------------------------------------------------------------------
Thank you Nascent Oxygen,

That has really cleared things up for me. I have two questions for you, if you don't mind answering them. The first question refers to the answer having to be exact form. If I was to use the equation (below) that uses the circle's height and radius to calculate the segment area, would that be acceptable?

Equation (Area of a circle segment using height and radius)
Radius [squared] * cos -1 (radius - height / radius) - (radius - height) √2*r*h -h2
Simplified form (Exact form) = 4 cos -1 (0.5) - √3

The second question refers to that equation itself. I understand slightly how it works, however I'm slightly confused on exactly how it calculates the segment area.

r2 cos -1 (radius - height / radius) = 4 pi / 3
I don't understand how this works.

 

[NascentOxygen]

Simplified form (Exact form) = 4 cos -1 (0.5) - √3

Can be simplified further, you probably know cos^-1 ½ = π/3.

Where you are given radius and height, draw also that radius onto which you can mark the height, then find the lengths of all 3 sides of the right-angled triangle formed at the circle's centre.

Denote the angle at the centre as βẞand express βẞin terms of r and h.

The double angle formula will around this time come in handy, because θ = 2βẞ

 

[Matt]

I understand that cos-1 = pi/3. However I'm very confused regarding the rest. I understand using the radius to work out the hypotenuse of the triangle (Formed at the circle's centre) and the segment height is the same as the triangle's base.

I'm unsure about denoting the angle at the centre (120 degrees) as βẞ expressed in terms of r and h.

 

[NascentOxygen]

βẞis 60^o here

βẞ= cos^-1 ...
where ... is some expression in r and h

 

[Matt]

βẞis 60^o here

βẞ= cos^-1 ...
where ... is some expression in r and h

Hello Nascent Oxygen, Totally forgot to thank you for all your help. So here it is, thank you very much. :)

 

[NascentOxygen]

Appreciate your reply. Also reveals that along with the β character I've been copying a strange character apparently concealed on my browser by a backspace.