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- Thread starter Ragnarok
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- Mar 5, 2012

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Hi Ragnarok!The solution to this question (whose answer is pi) is eluding me:

The radius of a circle is 3 feet. Find the approximate length of an arc of this circle, if the length of the chord of the arc is 3 feet also.

Did you make a drawing?

If you draw it, that should help to find the answer...

For starters, what kind of triangle made by the chord and two radii? What does that tell you about its angles?The solution to this question (whose answer is pi) is eluding me:

The radius of a circle is 3 feet. Find the approximate length of an arc of this circle, if the length of the chord of the arc is 3 feet also.

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Just because a book hasn't shown something doesn't make it any less true or provide any reason why you can't use it.Thank you! I understand now. I was unsure how much information we were allowed to assume as I can't remember if the book proved triangles have angle sum of 180 degrees yet.

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But I am pretty sure this book presupposes a knowledge of Euclid, so I think the sum of a triangle's angles is fine to assume.

- Feb 25, 2014

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View attachment 2013

We know radius AO (3) and chord AB.

AE = 1/2 AB

From Pythagorean Theorem OE² = AO² - AE²

OE² = 3² - 1.5²

OE² = 9 - 2.25

OE = 2.5980762114

Segment Height ED = Radius AO - Apothem OE

Segment Height ED = 3 - 2.5980762114

Segment Height ED = 0.4019237886

Angle AOE = arc tangent (AE/OE)

Angle AOE = arc tan (1.5/2.5980762114)

Angle AOE = 29.9999999996 or 30° rounded

There are 2PI radians in a circle and 30° is 1/12 of a circle.

So, 30° = 2PI/12 radians or PI/6 radians.

The answer you said was supposed to be PI. Well I get PI/6.

************************************************************

EDITED TO ADD:

Angle AOE is only*half* the central angle, so it should be 60° or PI/3 radians.

We know radius AO (3) and chord AB.

AE = 1/2 AB

From Pythagorean Theorem OE² = AO² - AE²

OE² = 3² - 1.5²

OE² = 9 - 2.25

OE = 2.5980762114

Segment Height ED = Radius AO - Apothem OE

Segment Height ED = 3 - 2.5980762114

Segment Height ED = 0.4019237886

Angle AOE = arc tangent (AE/OE)

Angle AOE = arc tan (1.5/2.5980762114)

Angle AOE = 29.9999999996 or 30° rounded

There are 2PI radians in a circle and 30° is 1/12 of a circle.

So, 30° = 2PI/12 radians or PI/6 radians.

The answer you said was supposed to be PI. Well I get PI/6.

************************************************************

EDITED TO ADD:

Angle AOE is only

Last edited:

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