Pythagoras' theorem

problem
Babylon numbers

Age

11 to 18

Challenge level

Can you make a hypothesis to explain these ancient numbers?
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
Under the ribbon

Age

14 to 16

Challenge level

A ribbon is nailed down with a small amount of slack. What is the largest cube that can pass under the ribbon ?
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
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
Nicely similar

Age

14 to 16

Challenge level

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
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
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?
problem
Right angled possibilities

Age

14 to 16

Challenge level

If two of the sides of a right-angled triangle are 5cm and 6cm long, how many possibilities are there for the length of the third side?
problem
Walk the plank

Age

14 to 16

Challenge level

A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?