The circumference vs. area of a circle.

EbenezerSon · 2014-08-08 01:22:17

- What is the circumstance of a circle with 14cm as its radius?
taking pie to be 22/7.

When one calculates for the solution of the above, he will realise that, the answer is the same as calculating for the area of a circle.
My problem is why the formula for calculating the circumference is almost the same as that of a circle, and in many cases the answers seem to be the same.

Last edited by EbenezerSon (2014-08-08 01:23:08)

Bob · 2014-08-08 01:56:08

hi EbenezerSon

??? How are you getting that these are the same ???

If you want A = C, this will happen in just one case (well, two if you count r = 0 )

Cancel r and pi

This is because 2 squared = 2 doubled.

Bob

EbenezerSon · 2014-08-10 06:58:52

Thanks Bob, I had miscalculated it, but now I have grasped the difference. But what is the meaning of SUBTEND when used in geometry or mensuration? For instance if an angle subtends another angle.
Could you please give an illustration with explanation of 'an angle subtending another' In order that I could understand its meaning properly?
Many thanks!

Last edited by EbenezerSon (2014-08-10 07:03:03)

Bob · 2014-08-10 07:10:38

hi EbenezerSon,

Silly word isn't it ? I haven't found a question yet where you cannot substitute the word 'makes'.

eg. Draw a circle and mark on three points on the circumference, A, B and P.

Then the chord AB subtends an angle at P on the circumference OR the chord AB makes an angle at P on the circumference (APB)

If you move P about along the major arc of the circle the angle subtended by AB (made by AB) stays the same.

etc.

Bob

Just doing the washing up so there may be a pause between responses.

EbenezerSon · 2014-08-10 08:22:47

Brilliant Bob!
Finally, 'subtends' = makes. Thank I have now understood it meaning properly!
Again, why is it that rectangles diagonals do not bisect at right angles? To me, it seems they bisect at right angles, what do say Bob?
Again, is it always true that, all interior angles of triangles add to 180°?
Thanks

Last edited by EbenezerSon (2014-08-10 08:24:48)

Bob · 2014-08-10 19:43:54

hi EbenezerSon

You could draw a rectangle and measure.

ABCD is a rectangle. AC crosses BD at E. AB = DC. AD = BC. Let's say that AB > BC, so AB is the length and BC is the shorter width.

The diagonal are equal so AC = BD.

In the middle the angle BEC is made up from three sides, BE and EC which are equal and BC.

The angle AEB is made up of three sides, AE and EB which are equal (and also to BE and EC) and AB.

So these two angles have two sides in common, but the third side is definitely not. Because AB > BC I know, even without drawing that AEB > BEC. They do add up to 180 so I also know that AEB is obtuse and BEC is acute. Definitely not equal to 90.

If you shorten AB until AB = BC you get a square. Now the angles at the middle are 90.

[note: To be a rectangle the shape must have four straight sides, with opposite sides equal and 90 degree corners. To be a square all four sides must be equal. So

{squares} is a subset of {rectangles} ]

In Euclidean geometry the angles of all flat triangles add up to 180. Many years ago I started a lesson by getting the children to draw any triangle, measure its angles and then add them up. After sorting out a few incorrect uses of a protractor, we wrote up the results on the board. Answers ranged from 178 to 182 degrees. I asked them what conclusion they could make from this. One lad said "The angles of a triangle add up to roughly 180 degrees" Not quite what I was expecting, but I couldn't fault his logic. In the real world you'll be lucky to be able to measure angles any better than this.

Euclidean geometry is a mathematical model for the real world based on lines than have no thickness and are perfectly straight, points that have no dimension at all and planes that are perfectly flat. It's an idealised model. In the real world you cannot measure with such perfect accuracy. Euclid showed how to prove that the angles add up to 180 and I can show you that if you would like. But don't forget it's just a model. Real triangles have pencil thick lines and aren't even on perfectly flat paper.

Bob

EbenezerSon · 2014-08-12 01:23:15

Thanks Bob! I must ask few questions concernin the above, but let me put this across, in fact a book I am using now has the following question and its final answer confuses me;
(1) A rectangle measures 15 cm by 20 cm. find
(i) the angle between its diagonals

How to get the answer confuses me. Please could you draw a rectangle for me to see, and then use it to calculate for the diagonals?
Thanks

Last edited by EbenezerSon (2014-08-12 01:27:23)

EbenezerSon · 2014-08-12 01:53:56

Again,
Does it need all the four angles created by the rectangles' diagonals at the center?

Bob · 2014-08-12 04:20:34

hi

The angle with the red dot can be found using inverse trig.

The other three are then easy by 180 - answer for the next one; and the other two will be the same again by symmetry.

Bob

EbenezerSon · 2014-08-13 06:25:45

Thanks Bob.

So, after I have gotten the angle at the red dot and have multiplied it by 2, do I need to work out for the angles of the three triangles in the same way by drawing a line through each triangle and then calculate for each angle in order to fit the question's demands?

Bob · 2014-08-13 19:17:51

Opposite angles will be equal so no need to repeat the same calculation. Also the acute and obtuse angles at the centre will add to 180 so that should speed things up.

Bob

EbenezerSon · 2014-08-14 20:48:57

Why the acute and obtuse angles will add up to 180? Please clarify it.
Thank you

Last edited by EbenezerSon (2014-08-14 20:50:48)

Bob · 2014-08-15 00:36:40

hi EbenezerSon

Let's use the diagram in post 9. Starting top left and going clockwise, call the corners A, B, C, D. Call the middle point E and the midpoint of BC call it F.

So we can work out angle BEF (red dot) with atan(BF/EF) = atan(7.5/10)*

Then double this to get BEC.

The lines AC and BD cross at E. AEC is a straight line so angle AEC = 180.

This is made up of two angles: AEB and BEC. So AEB + BEC = 180

AB > BC so AEB is obtuse and BEC is acute.

If you have ruler, pencil and protractor, I suggest you draw it for yourself. There's nothing like practical maths for making a result clear.

Bob

*atan(1/1) = 45, so even without a calculator, I can see BEF will be less than 45. So when doubled it will be less than 90.

EbenezerSon · 2014-08-16 09:26:16

I am now grasping it, Bob your explanation is excellent!
I was thinking that, after one has gotten the final answer one must states that, angle AEB = 36.2, angle BEC = 2.3, angle AED = 2.3, DEC = 36.2. This is to show the angles that exist between the diagonals just to answer exactly the question's demand, or you think writing the angles wouldn't be necessary?
(those figures were just guessed by me)
Thank you very much Bob!

Last edited by EbenezerSon (2014-08-16 09:30:34)

Bob · 2014-08-16 19:19:17

(i) the angle between its diagonals

I would expect a single answer to that; the acute angle. If the question wants more it should be clearly stated.

Bob

EbenezerSon · 2014-08-17 05:12:23

Thanks for answering, but the obtuse angle is also between its diagonals? (that's how I see it).
Thanks Bob!

EbenezerSon · 2014-08-17 05:35:46

The following is a new question, and please I want you to help me understand it properly.

(1) The diagonals of a rhombus are 5 cm and 8 cm long. Find
(i) the acute angle between the sides.
Please, draw a rhombus and solve this, that I could ask a question because the book's steps of solving it puzzles me. Please help me
Thank you.

Bob · 2014-08-17 05:45:26

Diagram may take a while, but you can probably draw one yourself.

The diagonals of a rhombus cross at 90 and bisect each other.

So draw a line 8 cm long across the paper.

Mark the midpoint and draw a line straight up and down to make the second diagonal.

Go 2.5 up and 2.5 down to fix the ends of that line.

Now you can draw the rhombus itself. It will be like a diamond on its side.

Bob

EbenezerSon · 2014-08-19 06:47:37

Thank you Bob!
I have drawn it, per your directions, and named them ABCD starting from the top of the vertical line then going clockwise. I have calculated for angle BAD and had 115.9° as the answer. Suprisingly, the book has 64° as the answer. In truth, the book calculated for angle ABD. Please which one is the correct answer and why? Please, you could calculate it and see what I am talking of
Thanks Bob.

Bob · 2014-08-19 07:10:15

hi

I get BAD = 115.9 and ABC = 64.1 Notice that they add up to 180 (why?).

Your question asks for the acute angle.

Bob

EbenezerSon · 2014-08-22 03:02:03

But is there any angle like obtuse in a rhombus? I was thinking that all the angles are acute and 115.9 and 64.1 could be the answer, please is 115.9 not the required answer? Please explain why 115.9 is not the required answer if you think so.

Many thanks!

EbenezerSon · 2014-08-22 22:42:28

Please, I just dont know much about rhombus that is why I am asking that question, please help me clear my doubts.

Many thanks

Math Is Fun Forum

#1 2014-08-08 01:22:17

The circumference vs. area of a circle.

#2 2014-08-08 01:56:08

Re: The circumference vs. area of a circle.

#3 2014-08-10 06:58:52

Re: The circumference vs. area of a circle.

#4 2014-08-10 07:10:38

Re: The circumference vs. area of a circle.

#5 2014-08-10 08:22:47

Re: The circumference vs. area of a circle.

#6 2014-08-10 19:43:54

Re: The circumference vs. area of a circle.

#7 2014-08-12 01:23:15

Re: The circumference vs. area of a circle.

#8 2014-08-12 01:53:56

Re: The circumference vs. area of a circle.

#9 2014-08-12 04:20:34

Re: The circumference vs. area of a circle.

#10 2014-08-13 06:25:45

Re: The circumference vs. area of a circle.

#11 2014-08-13 19:17:51

Re: The circumference vs. area of a circle.

#12 2014-08-14 20:48:57

Re: The circumference vs. area of a circle.

#13 2014-08-15 00:36:40

Re: The circumference vs. area of a circle.

#14 2014-08-16 09:26:16

Re: The circumference vs. area of a circle.

#15 2014-08-16 19:19:17

Re: The circumference vs. area of a circle.

#16 2014-08-17 05:12:23

Re: The circumference vs. area of a circle.

#17 2014-08-17 05:35:46

Re: The circumference vs. area of a circle.

#18 2014-08-17 05:45:26

Re: The circumference vs. area of a circle.

#19 2014-08-19 06:47:37

Re: The circumference vs. area of a circle.

#20 2014-08-19 07:10:15

Re: The circumference vs. area of a circle.

#21 2014-08-22 03:02:03

Re: The circumference vs. area of a circle.

#22 2014-08-22 22:42:28

Re: The circumference vs. area of a circle.

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