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Send to a Friend Puzzle # 127
puzzle no. 126

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Square vs Annulus
In mathematics, an annulus is a ring-shaped geometric figure.
Find a very simple way to calculate the area of the yellow annulus shown below using the following datas:
a) K is the center of the square ABCD,
b) Vertices A and C are on the larger, circonference of the annulus,
c) The area of the square ABCD is 80 cm2.

Difficulty level: bulbbulb, general math knowledge.
Category: Geometry.
Keywords: Pythagorean theorem, annulus, concentric circles, area.
Related puzzles:
- Soccer ball,
- Achtung minen.

Source of the puzzle:
© G. Sarcone.
You cannot reproduce any part of this page without prior written permission.
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Solution to puzzle 127The area A of the annulus can be obtained from the chord of the outside circle, that is the length of the longest interval that lies completely inside the annulus, 2d in the diagram opposite. This can be proven by the Pythagorean theorem; the chord of the larger circle is tangent to the smaller circle and form a right angle with its radius at that point.

Therefore, d and r are the sides of a right angled triangle with hypotenuse R and the area is given by:
A = π(R2 - r2) = πd2

Knowing that d is half the diagonal AC of the blue square ABCD:
d2 = 80/2 = 40 [cm2]

Thus, the area of the annulus is: 40 x π = 125,66... [cm2]

cup winnerThe 5 Winners of the Puzzle of the Month are:
Ian Glynn, Canada USA flag - Walter Jacobs, Belgium Belgian flag - Rutvik Oza, India Indian flag - Jakub Nogly, Poland poland flag - Matteo Andreatta, Italy Italian flag


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Math fact behind the puzzle

In mathematics, an annulus (the Latin word for "little ring", with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name any ring-shaped object. Or, it is the area or region between two concentric circles. The adjectival form is annular (for example, an annular eclipse).

The open annulus is topologically equivalent to an open cylinder:
S1 x (0,1).

In the figure below, the area of any circle whose diameter is tangent to the inner circle of an annulus and has endpoints at the outer circle is equal to the area of the annulus.

example 1

A solid annulus is a region of a Euclidean space of the dimension n >= 3 comprised between two concentric spheres

example 2


© 2011 G. Sarcone,
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!
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