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Puzzles of the Month + Solutions



Back to PuzzleoftheMonth
page  Home 
Puzzle
# 127


Square
vs Annulus
In
mathematics, an annulus is a ringshaped 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 cm^{2}.
Difficulty
level: ,
general math knowledge.
Category:
Geometry.
Keywords:
Pythagorean theorem, annulus, concentric circles, area.
Related
puzzles:
 Soccer
ball,
 Achtung
minen.
 Anello
al quadrato
Con i seguenti dati, trova un modo molto, ma molto
semplice per calcolare l'area dell’anello in
giallo qui sopra:
a) il punto K è il baricentro del
quadrato ABCD,
b) i vertici A e C del quadrato
appartengono alla circonferenza esterna dell’anello,
c) l'area del quadrato ABCD è di
80 cm^{2}.
Parole
chiave: corona, anello, quadrato, circonferenze
concentriche.
Suggerisci un'altra
soluzione Chiudi
 Une
couronne surprenante
Avec
les donnée cidessous, trouvez un moyen vachement
simple pour calculer la surface de la couronne jaune:
a) le point K est le centre du carré ABCD,
b) les sommets A e C du carré appartiennent à la
circonférence extérieure de l’anneau,
c) l'aire du carré ABCD est de 80
cm^{2}.
Mots
clés: couronne, carré, cercles
concentriques.
Propose une
autre solution Fermer

Source
of the puzzle:
© G. Sarcone. You
cannot reproduce any part of this page without prior written
permission. 



The
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(R^{2 } r^{2})
= πd^{2}
Knowing
that d is half the diagonal AC of
the blue square ABCD:
d^{2} = 80/2 = 40 [cm^{2}]
Thus,
the area of the annulus is: 40 x π = 125,66... [cm^{2}]

The
5 Winners of the Puzzle of the Month are:
Ian Glynn, Canada  Walter
Jacobs, Belgium  Rutvik
Oza, India  Jakub
Nogly, Poland  Matteo
Andreatta, Italy
Congratulations!



Math
fact behind the puzzle

In
mathematics, an annulus (the
Latin word for "little ring", with plural annuli)
is a ringshaped geometric figure, or more generally,
a term used to name any ringshaped 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:
S^{1} 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.
A solid
annulus is a region of a Euclidean space
of the dimension n >=
3 comprised between two concentric spheres

© 2011 G.
Sarcone, www.archimedeslab.org
You can reuse 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! 




