Pythagoras' theorem

problem
Spherical triangles on very big spheres

Age

16 to 18

Challenge level

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.
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.
problem
Centre square

Age

14 to 16

Challenge level

What does Pythagoras' Theorem tell you about the radius of these circles?
problem
The old goats

Age

11 to 14

Challenge level

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?
problem
Tilted squares

Age

11 to 14

Challenge level

It's easy to work out the areas of most squares that we meet, but what if they were tilted?
problem
Garden shed

Age

11 to 14

Challenge level

Can you minimise the amount of wood needed to build the roof of my garden shed?
problem
Pythagoras proofs

Age

11 to 16

Challenge level

Can you make sense of these three proofs of Pythagoras' Theorem?
problem
Babylon numbers

Age

11 to 18

Challenge level

Can you make a hypothesis to explain these ancient numbers?
problem
Where is the dot?

Age

14 to 16

Challenge level

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?