Pythagoras' theorem

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

Rectangular Pyramids

Age

14 to 18

Challenge level

Is the sum of the squares of two opposite sloping edges of a rectangular based pyramid equal to the sum of the squares of the other two sloping edges?
problem

Xtra

Age

14 to 18

Challenge level

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.
problem

Chord

Age

16 to 18

Challenge level

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.
problem

Square Pair Circles

Age

16 to 18

Challenge level

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
problem

Pythagoras Mod 5

Age

16 to 18

Challenge level

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.
article
Incircles Explained

This article is about triangles in which the lengths of the sides and the radii of the inscribed circles are all whole numbers.
article
All Is Number

Read all about Pythagoras' mathematical discoveries in this article written for students.