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?
Four identical right angled triangles are drawn on the sides of a
square. Two face out, two face in. Why do the four vertices marked
with dots lie on one line?
The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
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. . . .
What fractions can you divide the diagonal of a square into by
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
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?
A and C are the opposite vertices of a square ABCD, and have
coordinates (a,b) and (c,d), respectively. What are the coordinates
of the vertices B and D? What is the area of the square?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Can you recreate squares and rhombuses if you are only given a side
or a diagonal?
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
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
A man paved a square courtyard and then decided that it was too
small. He took up the tiles, bought 100 more and used them to pave
another square courtyard. How many tiles did he use altogether?
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.
The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
What can you see? What do you notice? What questions can you ask?
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Here are four tiles. They can be arranged in a 2 by 2 square so
that this large square has a green edge. If the tiles are moved
around, we can make a 2 by 2 square with a blue edge... Now try. . . .
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Can you work out the area of the inner square and give an
explanation of how you did it?
A tilted square is a square with no horizontal sides. Can you
devise a general instruction for the construction of a square when
you are given just one of its sides?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
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.
A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?
Just four procedures were used to produce a design. How was it
done? Can you be systematic and elegant so that someone can follow
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
It is possible to dissect any square into smaller squares. What is
the minimum number of squares a 13 by 13 square can be dissected
Creating designs with squares - using the REPEAT command in LOGO.
This requires some careful thought on angles
Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?
Can you use LOGO to create this star pattern made from squares.
Only basic LOGO knowledge needed.
Can you use LOGO to create a systematic reproduction of a basic
design? An introduction to variables in a familiar setting.
A Short introduction to using Logo. This is the first in a twelve part series.
Using LOGO, can you construct elegant procedures that will draw
this family of 'floor coverings'?
Look at how the pattern is built up - in that way you will know how
to break the final pattern down into more manageable pieces.
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.
This LOGO Challenge emphasises the idea of breaking down a problem
into smaller manageable parts. Working on squares and angles.