Find the distance of the shortest air route at an altitude of 6000
metres between London and Cape Town given the latitudes and
longitudes. A simple application of scalar products of vectors.
Use simple trigonometry to calculate the distance along the flight
path from London to Sydney.
Where should runners start the 200m race so that they have all run the same distance by the finish?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
A spherical balloon lies inside a wire frame. How much do you need
to deflate it to remove it from the frame if it remains a sphere?
An equilateral triangle rotates around regular polygons and
produces an outline like a flower. What are the perimeters of the
Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".
Find the perimeter and area of a holly leaf that will not lie flat
(it has negative curvature with 'circles' having circumference
greater than 2πr).
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
The ten arcs forming the edges of the "holly leaf" are all arcs of
circles of radius 1 cm. Find the length of the perimeter of the
holly leaf and the area of its surface.
If I print this page which shape will require the more yellow ink?
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
A belt of thin wire, length L, binds together two cylindrical
welding rods, whose radii are R and r, by passing all the way
around them both. Find L in terms of R and r.
A and B are two points on a circle centre O. Tangents at A and B
cut at C. CO cuts the circle at D. What is the relationship between
areas of ADBO, ABO and ACBO?
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
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. . . .
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
P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?