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#873397 - 03/12/13 02:18 PM Maths Fun(?)
Urban Worrier Offline

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Registered: 11/16/04
Posts: 13864
Loc: Bucks, England
-1. Draw a regular hexagon then draw the six "alternate" vertices. (Skip one vertex - 1 to 3, 2 to 4 etc). If the area of the original hexagon is 6, what is the area of the internal hexagon that is formed?

-2. Inscribe a square in a circle of radius R. Draw an arc from one corner of the square with radius equal to the side of the square. The area formed between this arc and the circle is a crescent. Find the area of this crescent in terms of R.

-3. In the equations below, replace each letter with a number from 1 to 9, using each number only once:

a/b + c/12 = 1 1/12
d/e + 5/f = 7/8
g/h + 3/i = 1 1/2
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#873457 - 03/12/13 04:38 PM Re: Maths Fun(?) [Re: Urban Worrier]
manxman Offline
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Registered: 07/23/02
Posts: 13934
Loc: Unionville
2. 1\2*R^2 wave
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#873480 - 03/12/13 06:28 PM Re: Maths Fun(?) [Re: Urban Worrier]
CCbomber Offline
Addicted Boomer

Registered: 01/16/03
Posts: 2911
Loc: Mojave desert, California
3.

2/3 + 5/12 = 1 1/12
1/4 + 5/8 = 7/8
7/6 + 3/9 = 1 1/2

smile

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#873586 - 03/13/13 11:21 AM Re: Maths Fun(?) [Re: Urban Worrier]
CanukDenis Offline
Adept Boomer

Registered: 11/26/00
Posts: 14921
Loc: Ottawa Ontario Canada
1. Draw a regular hexagon then draw the six "alternate" vertices.
(Skip one vertex - 1 to 3, 2 to 4 etc). If the area of the original hexagon is 6,
what is the area of the internal hexagon that is formed?

So I don't have to type "sqrt(3)"!):
u = sqrt(3)
v = 1/u (ratio radii outer:inner)

R = outer radius
Pentagon area: 3u(R^2) / 2

Given: area = 6 ; soooooo:
3u(R^2) / 2 = 6
R = sqrt(4 / u) : ~1.5196....

r = inner radius
r = R * v : ~.8773....
area = 3u(r^2)/2
area = (drum rolls!) 2 exactement.






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#873654 - 03/13/13 06:30 PM Re: Maths Fun(?) [Re: Urban Worrier]
Urban Worrier Offline

Adept Boomer

Registered: 11/16/04
Posts: 13864
Loc: Bucks, England
CCb - very nice!

Denis - good solve. It can be done by similar triangles. The inner hexagon contains 6 identically-sized equilateral triangles and the outer hexagon contains these 6 plus 6 more plus 12 halves for a total of 18 triangles. Thus, the area of the outer hexagon is 3 times the area of the inner one. smile

manxman - if I read your answer correcly (one over 2 R-squared) then we beg to differ. What's your working? think
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I married Miss Right. I just didn't know her first name was Always.

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#873701 - 03/13/13 11:45 PM Re: Maths Fun(?) [Re: Urban Worrier]
manxman Offline
Adept Boomer

Registered: 07/23/02
Posts: 13934
Loc: Unionville
Sorry, checked my work over and got R^2 smile


Edited by manxman (03/14/13 07:32 AM)
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#873814 - 03/14/13 01:26 PM Re: Maths Fun(?) [Re: Urban Worrier]
Urban Worrier Offline

Adept Boomer

Registered: 11/16/04
Posts: 13864
Loc: Bucks, England
Great solve! grin
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