Pythagoras' theorem

problem

In a Spin

Age

14 to 16

Challenge level

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

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

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

Rectangle Dissection

Age

11 to 14

Challenge level

Weekly Problem 2 - 2009
The 16 by 9 rectangle is cut as shown. Rearrange the pieces to form a square. What is the perimeter of the square?
problem

Square Pegs

Age

11 to 14

Challenge level

Which is a better fit, a square peg in a round hole or a round peg in a square hole?
problem

Isosceles

Age

11 to 14

Challenge level

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
problem

Trice

Age

11 to 14

Challenge level

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?