Circles and Circle Theorems - February 2004, Stage 4&5

Problems

Maximised Area

Stage: 4 Short Challenge Level:

Weekly Problem 39 - 2011
Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.

Annulus Area

Stage: 4 Short Challenge Level:

Weekly Problem 38 - 2011
Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?

Pencil Turning

Stage: 4 Short Challenge Level:

Weekly Problem 37 - 2011
Rotating a pencil twice about two different points gives surprising results...

Circle in a Semicircle

Stage: 4 Short Challenge Level:

Weekly Problem 36 - 2011
Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?

Circle Box

Stage: 4 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

Inscribed in a Circle

Stage: 4 Challenge Level:

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

Circle Scaling

Stage: 4 Challenge Level:

You are given a circle with centre O. Describe how to construct with a straight edge and a pair of compasses, two other circles centre O so that the three circles have areas in the ratio 1:2:3.

Circles in Circles

Stage: 5 Challenge Level:

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Lunar Angles

Stage: 5 Challenge Level:

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?

Flower

Stage: 5 Challenge Level:

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.