Geometry brainteasers 2
1. Find an expression, in terms of π and
quarter-circular arcs each of radius
2. Let
and
, for the area of the figure below, made up of
units.
be points on opposite sides of a rectangle such that
is parallel to the
other two sides of the rectangle. A triangle is formed with vertices
and the two corners
of the rectangle which are on the side of the rectangle containing the point . A second
triangle is formed with vertices
and the two corners of the rectangle which are on the
side of the rectangle containing the point
a fraction of the area of the rectangle.
. Find the area common to these triangles as
3. Find the area of the cross inside a square of side 2, made by joining its vertices to the
mid-points of the sides as shown below.
1
1
4. A polygon, not necessarily regular, is drawn so that its sides touch a given circle of radius
. The perimeter of the polygon has length
and
. Find the area of the polygon in terms of
.
5. In triangle
the sides
and
are each 6 units in length and angle
is a
right angle. The diagrams below show the triangle with a circle and two squares
and
inscribed. Determine which of the three inscribed figures has the greatest
area.
A
A
A
Y
R
Q
X
Z
B
C
B
P
C
Extension Through Depth/Spatial Reasoning/Geometry brainteasers 2
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B
W
C
6.
is any triangle,
=
Now
=
=
and
triangle
is extended to
,
to
and
. Show that triangle
to
=
so that
,
has seven times the area of
.
is extended to
,
to
and
to
. What multiple of the area of triangle
so that
=
,
=
and
is the area of the six-sided figure
?
7. A closed hollow glass cone with its axis vertical is partially filled with sand so that this
reaches exactly half way up the cone as illustrated below:
fig i
fig ii
If this cone is inverted so that its axis is again vertical, as in fig.(ii), how far up the cone will
the sand now reach? (The volume
is given by
=
π
of a cone with a base radius
.)
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and vertical height
8.
are drawn as
is a square of side 1 metre. Arcs of circles with centres at
shown
is
+
π
in
−
the
diagram
below.
Show
that
the
area
of
the
shaded
region
square metres.
A
B
D
C
9. The figure below shows a circle inscribed in a square with four circular arcs symmetrically
placed, intersecting at the centre of the square and the mid-points of the sides. Show that
the shaded area is equal to the unshaded area.
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10. A plot of land has four straight sides, all of different lengths but each measuring a whole
number of metres, and two right angles at opposite corners, as illustrated in the diagram.
Determine the least possible area of the plot.
Find also the dimensions of a plot of the same type (that is, having four straight sides of
different lengths but each measuring a whole number of metres and with two right angles
at opposite corners) and having area equal to twice the least possible area.
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