Pythagoras' theorem

problem

Squaring the Circle and Circling the Square

Age

14 to 16

Challenge level

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
problem

Circumnavigation

Age

14 to 16

Challenge level

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
problem

The Dodecahedron

Age

16 to 18

Challenge level

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?
problem

Squ-Areas

Age

14 to 16

Challenge level

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more triangular areas are enclosed. What is the area of this convex hexagon?
problem

Circle Scaling

Age

14 to 16

Challenge level

Describe how to construct three circles which have areas in the ratio 1:2:3.
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?