Gerbert (ca. 950-1003), who became Pope Sylvester 2nd, claimed that the area of an equilateral triangle of side a was (a/2)(a-a/7). Show he was wrong, but close.

How can I prove that he is wrong, but close?

Results 1 to 7 of 7

- Sep 12th 2010, 07:28 PM #1

- Joined
- Sep 2010
- Posts
- 98
- Thanks
- 1

- Sep 12th 2010, 08:10 PM #2

- Joined
- Nov 2005
- From
- someplace
- Posts
- 14,972
- Thanks
- 5

- Sep 13th 2010, 01:09 PM #3

- Joined
- Oct 2009
- Posts
- 5,577
- Thanks
- 790

- Sep 13th 2010, 01:49 PM #4

- Joined
- May 2006
- From
- Lexington, MA (USA)
- Posts
- 12,028
- Thanks
- 848

Hello, matgrl!

I'll do this one "from scratch" . . .

Code:A * /|\ / | \ / | \ a / | \ a / |h \ / | \ / | \ B * - - - + - - - * C D a/2

We have equilateral triangle with side

The altitude bisects the base, so

Pythagorus says: .

. . . . . . . . . . . . .

The area of the triangle is: .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Gerbert's formula has: .

. . which is slightly smaller than the true area.

The relative error is: .

. . . . . about 1% as emakarov already stated.

- Sep 13th 2010, 06:10 PM #5

- Joined
- Sep 2010
- Posts
- 98
- Thanks
- 1

- Sep 13th 2010, 06:15 PM #6

- Joined
- Sep 2010
- Posts
- 98
- Thanks
- 1

- Sep 13th 2010, 08:22 PM #7

- Joined
- Nov 2005
- From
- someplace
- Posts
- 14,972
- Thanks
- 5