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Some math help?

M

MournSanity

Diamond Member
I have this math project that my teacher gave us. He didn't teach us what it was about but I was able to get most of with my past geometry knowledge. There is just one problem I don't understand.

Amos has a garden enclosed by a trianglar fence 40' on each side, and surrounded by a flat field of grass. Billy, the goat, is tethered outside of the garden by a 30' chain attached to the midpoint of a side of the fence. Draw a figure that represents the grazing region and label the appropriate angles and lengths. FIND THE AREA OF THIS GRAZING REGION.

Ok, I drew a triangular garden and attached Billy to the midpoint of the fence and drew the grazing area. The picture looks like a pizza with an equilateral triangle's base sticking through one side. How would I go about finding the area? I know that a circle's area formula thing is pi R squared, where R is the radius(30 in this problem). But how do I calculate the area minus the part of the triangle that is sticking into the circle?

Any help at all would be apprectiated 🙂
 
This problem is too simplistic to be dealt with by the superior minds at ATOT. Please head to a kindergarden teacher to get this question resolved. =P
 
You are thinking about it wrong if you think the circle extends beyond the triangles edge. There is a half circle with a radius of 30' formed along the egde of the triange, but when the teather of the goat gets to the edge of the triangle 20' of it is gone. There is only 10' of rope that can be used around the corners of the triangle. so you have 180 degrees of a 30' circle and 120 degrees of a 10' circle on either side of the triangle.
 
Originally posted by: Kyteland
You are thinking about it wrong if you think the circle extends beyond the triangles edge. There is a half circle with a radius of 30' formed along the egde of the triange, but when the teather of the goat gets to the edge of the triangle 20' of it is gone. There is only 10' of rope that can be used around the corners of the triangle. so you have 180 degrees of a 30' circle and 120 degrees of a 10' circle on either side of the triangle.
Click to expand...


I made a crude drawing of what I think it looks like. Here. The question marks are the areas I dont understand.
 
What you fail to realize is that the rope cannot pass through the fence. It must wrap around at the corners. Look at this diagram.

math_diagram.jpg

Edit: So the final math is that there is a 180 degree section of a 30' radius circle and two 120 degree sections of 10 ' circles.

PI*30^2*(180/360) + PI*10^2*(120/360) + PI*10^2*(120/360)
 
Originally posted by: Kyteland
What you fail to realize is that the rope cannot pass through the fence. It must wrap around at the corners. Look at this diagram.

math_diagram.jpg
Click to expand...


Ohh, I had a feeling it was something like that. So how would I go about calculatingt he area? I know I have to calculate the half circle seperately and add the smaller peices to it, but how would you coalculate on of the smaller sides? Are the smaller sides considered 1/3 circle with a 10' radius?
 
Originally posted by: hypersonic5
Originally posted by: Kyteland
What you fail to realize is that the rope cannot pass through the fence. It must wrap around at the corners. Look at this diagram.

math_diagram.jpg
Click to expand...


Ohh, I had a feeling it was something like that. So how would I go about calculatingt he area? I know I have to calculate the half circle seperately and add the smaller peices to it, but how would you coalculate on of the smaller sides? Are the smaller sides considered 1/3 circle with a 10' radius?
Click to expand...

Yes. See the edit above.
 
One more hint, the formula for the area of a circle is

2Pi r^2, if you use radian measure for the angle, there are 2Pi radians in a circle so the area formula is just
A= Angle *r^2 where the angle is in radians.

Since you know that the angle of the triangle is 60deg = Pi/3 radians you can easily compute the area of the small portions of the circle at each side.
 
Originally posted by: RossGr
One more hint, the formula for the area of a circle is

2Pi r^2, if you use radian measure for the angle, there are 2Pi radians in a circle so the area formula is just
A= Angle *r^2 where the angle is in radians.

Since you know that the angle of the triangle is 60deg = Pi/3 radians you can easily compute the area of the small portions of the circle at each side.
Click to expand...

Ok, that's not right. The area of a circle is PI*r^2 no matter what measure of the angle you use. It is true that there are 2PI radians in a circle, so if you were using them then the formula for the partial area of a circle would be
A= (1/2) * radians *r^2
 
Since we already determined that the two smaller arcs are each 120 degrees, why don't we just call them 1/3 of a circle each? So we have half of a circle with a radius of R1 = 30 feet, and 2 * (1/3 of a circle) with a radius of R2 = 10 feet.

The answer would be 1/2 * pi * R1^2 + 2/3 * pi * R2^2, where you plug in R1 and R2. No need to be confusing about radians.
 
Originally posted by: oog
Since we already determined that the two smaller arcs are each 120 degrees, why don't we just call them 1/3 of a circle each? So we have half of a circle with a radius of R1 = 30 feet, and 2 * (1/3 of a circle) with a radius of R2 = 10 feet.

The answer would be 1/2 * pi * R1^2 + 2/3 * pi * R2^2, where you plug in R1 and R2. No need to be confusing about radians.
Click to expand...

Indeed 🙂
 
Originally posted by: Kyteland
Originally posted by: RossGr
One more hint, the formula for the area of a circle is

2Pi r^2, if you use radian measure for the angle, there are 2Pi radians in a circle so the area formula is just
A= Angle *r^2 where the angle is in radians.

Since you know that the angle of the triangle is 60deg = Pi/3 radians you can easily compute the area of the small portions of the circle at each side.
Click to expand...

Ok, that's not right. The area of a circle is PI*r^2 no matter what measure of the angle you use. It is true that there are 2PI radians in a circle, so if you were using them then the formula for the partial area of a circle would be
A= (1/2) * radians *r^2
Click to expand...


Well, duh! Since when is the area of a circle 2Pi r^2 ! Didn't it used to be simply Pi r^2! Jeess sorry about that I stand corrected......

using radian measure the area of a segment of a circle is given by

A = .5 Theta R^2

Where Theta is the angle of the arc in radians

Yes Pi r^2 is the area in any unit it just that radians are a direct plug in while degrees are not.
 
Originally posted by: oog
Since we already determined that the two smaller arcs are each 120 degrees, why don't we just call them 1/3 of a circle each? So we have half of a circle with a radius of R1 = 30 feet, and 2 * (1/3 of a circle) with a radius of R2 = 10 feet.

The answer would be 1/2 * pi * R1^2 + 2/3 * pi * R2^2, where you plug in R1 and R2. No need to be confusing about radians.
Click to expand...

You are essentially doing exactly what I said. 1/3 of a circle is 2Pi/3 radians. the area is of one sector is then .5* 2/3 Pi R1^2, What is confusing about that? The real confusing was my wild claim that the area of a circle is 2pi r^2!

If you plan on going into the sciences to any degree get used to radian, it is a much better angle measure simply because it is dimensionless.
 
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