Squares

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Zooming in on the Squares

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
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Peg and Pin Boards

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
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Fitting In

Age

14 to 16

Challenge level

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ
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Squares, Squares and More Squares

Age

11 to 14

Challenge level

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
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Square Bisection

Age

11 to 14

Challenge level

Weekly Problem 8 - 2008
In how many ways can a square be cut in half using a single straight line cut?
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Geoboards

Age

7 to 11

Challenge level

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
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Tetra Square

Age

14 to 18

Challenge level

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
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Two Triangles in a Square

Age

14 to 16

Challenge level

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.
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Circle Box

Age

14 to 16

Challenge level

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
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Unusual Polygon

Age

14 to 16

Challenge level

What is the perimeter of this unusually shaped polygon...