Re: need help with homework

- Do you know the formula for area of a rectangle? (Hint: Count rows,columns.
From: Scena <metarell...@xxxxxxxxx>
Base X Height (b x h)
Correct.

- Do you know the formula for area of a triangle? (Hint: Decompose rectangle.
1/2 b(h)
Correct.

- Do you know the formula for circumference of circle? (Hint: Definition!!)
2 pi x r

That's not the definition. PI is defined as the ratio circumference/diameter.
(It's a theorem that it's always the same value regardless of diameter.)
So if you solve for circumference you get pi x diameter,
which is what I was aiming for.
Now if you know relation between diameter and radius,
and substitute it into that formula, you get what you had.

- I hope you know the relation between radius and diameter!!
radius is half of the diameter.
Correct. Or diameter is twice the radius. Either way.
From geometry it's actually a theorem derived from premises:
- Whole equals sum of parts.
- Center of circle splits diameter into two radii.
- Each radius equals the other, so we can call either the radius.
(<---R1--->Center<---R2--->)
<--------diameter-------->
In algebra, R1 + R2 = diameter, radius=R1=R2. Substuting we get:
radius + radius = diameter
Then algebra gives it your way or mine.

- Do you know how to use the triangle-area formula together with the
circumference formula to approximate the area of a circle, first
expressed as function of diameter, then converted to function of
radius where it becomes simpler and easier to remember?
no.

Clarification: Area of circle is a misnomer. It's really area of a
circular disk, i.e. a disk whose perimiter is a circle. The circle
is just the perimeter, a curved line. The disk is the inside of
the circle, with or without the circle itself included (closed disk
or open disk respectively if you study topology or metric spaces).

Think of a circular disk as a pie. Cut it into lots of slices.
Each slice is approximately shaped like a triangle. So its area is
approximately equal to base (portion of circle) times height
(radius) divided by two. Let's say we have six slices of pie:
A1 =appx= 1/2 * R * C1
A2 =appx= 1/2 * R * C2
A3 =appx= 1/2 * R * C3
A4 =appx= 1/2 * R * C4
A5 =appx= 1/2 * R * C5
A6 =appx= 1/2 * R * C6
Using the distributive law, we have:
A1+A2+A3+A4+A5+A6 =appx= 1/2 * R * (C1+C2+C3+C4+C5+C6)
But the left side is the total area, and the (C1...C6) is the total
circumference, so we get Atotal =appx= 1/2 * R * C.
Replacing C by {pi} * D, we have :
Atotal =appx= 1/2 * R * {pi} * D.
Replacing D by 2 * R and rearranging and simplifying, we have:
Atotal =appx= 1/2 * R * {pi} * 2 * R.
Atotal =appx= R * {pi} * R.
Atotal =appx= {pi} * R^2.

(I'm assuming you don't remember the formula for area of a circle,
or maybe you already looked it up and/or memorized it but it's a
mystery why the formula is like that and you really should learn
the rationale for it if you didn't learn that already.)
Don't know this either.

The above approximation is in fact *exactly* the correct answer for
Euclidean space. area = {pi} * radius^2

- Extra credit, can you guess who first proved somewhat rigorously
that the weird/sloppy pseudo-triangle-approximation for area of
circle in fact gave *exactly* the correct answer?
No.

It depends on how rigorous you want.
Some ancient Greek mathematician originally gave an informal proof,
by means if inscribed and circumscribed regular polygons which
provide upper and lower bound but converge to the same value..
Isaac Newton gave a mostly-rigorous proof, by use of his Calculus.
More recently his proof was made totally rigorous.
At least that's how I recall the sequence. Others might know better.

Later, after you've informally derived the area of circle formula
yourself, I'll coach you toward the formula for area of trapezoid,
and then the general formula for volume of cylinder (even a
trapezoidal cylinder with non-circular weird-shaped base!).

As you saw by arranging balls in rectangular array, so area equals
height times width, likewise area of square, if you skew the
stacking into a parallelogram (nevermind trapezoid, I typed wrong
there), instead of a reactangle, you end up with the same formula,
providing you use perpendicular height rather than length of
sloping edge.

Or you can do a purely geometrical dissection to prove it:
|<---width--->| (width of either rectangle or parallelogram)
*-------------*
/| /|
/ | / |height
/ | / |
*---+---------*---+
|<---width--->| (width of rectangle)
|<---width--->| (width of parallelogram)
Note two overlaid figures:
- the center trapezoid and right triangle combining to form a rectangle
- center trapezoid and left triangle combining to form a trapezoid.
Can you see that the left and right triangles are exactly the same
size and shape, hence have the same area as each other?
Hence the parallelogram and rectangle have the same area as each
other, namely base times height.
Can you see that the width is shared at the top, and equal at the
bottom, so it doesn't matter whether you measure base at top or bottom?

The same idea works with a non-right cylinder. It doesn't matter
whether the sides of a cylinder run vertically (perpendicular to
base) or or at an oblique slope (not perpendicular to base), volume
of cylinder is still the area of the base times the perpendicular
height. It's harder to show a good proof, so I won't do that here,
OK? But you get the idea, right?

Now just as sweeping a line segment to make a rectangle (or
parallelogram) gives area = base * height, likewise sweeping a
plane disk such as circular disk to make a cylinder (right or
oblique doesn't matte) gives volume = baseArea * height.
Does it all make sense, how to compute area and volume for these
simple geometric figures? If so, now you have all the math to
understand your daughter's math problem just the same as she does.
Maybe you didn't take geometry in high school, but for this one
case of area and volume you have the basic idea now,
and even an understanding how to informally derive the formulas,
so you don't have to memorize them and worry if you got them correct.
Any time you doubt your memory, just re-derive the formula, right?

Last math class I took was college level Algebra 1 in 1997. I did
pretty good (B+), but I had to study like crazy. We did go over some
basic geometry in one chapter.

Did any of this circular-disk and cylinder-volume stuff seem
familiar now that I've gotten you thinking about it again?

(End of last part of three-part reply, whew!)
.

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