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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Get Cross

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

Two Circles

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

Giant Holly Leaf

Age 14 to 16
Challenge Level

holly leaf construction1
construction 2

This model holly leaf is made in sections and joined together. Like a real holly leaf, it will not lie flat. It has negative curvature.

To make the holly leaf, a circle centre $C$ of radius 5 cm and radii $CA$ and $CB$ with $\angle ACB = 125$ degrees are drawn. The tangents to the circle at $A$ and $B$ meet at the point $P$. Eight identical 3 sided shapes are made by cutting along $PA$ and $PB$ and around the arc $AB$ to make a 3 sided shape with 2 straight edges and one edge along the minor arc of the circle (the circles are thrown away).

Two identical 4-sided shapes are made by drawing a circle with radius 5 cm, a diameter $B$*$D$* and tangents $B$*$P$* and $D$*$Q$*equal in length to $PB$. These shapes have edges $B$*$P$*, $P$*$Q$*, $Q$*$D$* and the semicircular arc (inside the rectangle) from $B$* to $D$*.

The sketch shows (on a smaller scale) how the ten pieces are joined together to make the "holly leaf".

Find the length of the boundary of the yellow area around $P$ which is bounded by six arcs centred at $P$, each of radius $r$ cm. All points on the boundary of the yellow region are equidistant from the point $P$.

If the surface at $P$ were flat, the boundary of the region would be a circle and its length would be $2\pi r$. In this case the length of the boundary is greater than $2\pi r$ and the surface of the "holly leaf" has negative curvature at $P$.

Compare the perimeter and area of this "holly leaf" with the similar flat leaf for which $\angle ACB = 135$ degrees.

See the problem "Holly" for the flat version of this problem.

What happens to the holly leaves as the angle $\angle ACB$changes?

[For positive curvature the boundary is less than $2 \pi r$ in length.]

See the article Curvature of Surfaces to find out more about this subject.