2D shapes and their properties

problem

Circumspection

Age

14 to 16

Challenge level

3 out of 3

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.
problem

Square Areas

Age

11 to 14

Challenge level

3 out of 3

Can you work out the area of the inner square and give an explanation of how you did it?
problem

2001 Spatial Oddity

Age

11 to 14

Challenge level

3 out of 3

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
problem

Semi-Square

Age

14 to 16

Challenge level

3 out of 3

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?
problem

Circumnavigation

Age

14 to 16

Challenge level

3 out of 3

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

Gold yet Again

Age

16 to 18

Challenge level

3 out of 3

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."
problem

Shapes in the Alphabet

Age

5 to 7

Challenge level

1 out of 3

In this problem, we're going to find sets of letter shapes that go together.
problem

Logic Block Collections

Age

5 to 7

Challenge level

1 out of 3

What do you think is the same about these two Logic Blocks? What others do you think go with them in the set?
problem

Opening Patterns

Age

5 to 7

Challenge level

1 out of 3

Look at some of the patterns in the Olympic Opening ceremonies and see what shapes you can spot.
problem

Rectangles With Dominoes

Age

5 to 7

Challenge level

2 out of 3

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?