Pythagoras' theorem

problem

The Fire-Fighter's Car Keys

Age

14 to 16

Challenge level

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket?
problem

Folding in Half

Age

14 to 16

Challenge level

How does the perimeter change when we fold this isosceles triangle in half?
problem

Unusual Polygon

Age

14 to 16

Challenge level

What is the perimeter of this unusually shaped polygon...
problem

Six Discs

Age

14 to 16

Challenge level

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?
problem

Two Circles

Age

14 to 16

Challenge level

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?
problem

Ball Packing

Age

14 to 16

Challenge level

If a ball is rolled into the corner of a room how far is its centre from the corner?
problem

The Pillar of Chios

Age

14 to 16

Challenge level

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
problem

Crescents and Triangles

Age

14 to 16

Challenge level

Can you find a relationship between the area of the crescents and the area of the triangle?
problem

Common Tangent

Age

14 to 16

Challenge level

Two circles touch, what is the length of the line that is a tangent to both circles?
problem

Interior Squares

Age

14 to 16

Challenge level

Calculate the ratio of areas of these squares which are inscribed inside a semi-circle and a circle.