Pythagoras' theorem

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?

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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

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

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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