There are 74 NRICH Mathematical resources connected to Squares, you may find related items under Angles, polygons, and geometrical proof.
Broad Topics > Angles, polygons, and geometrical proof > Squares
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 what shape is made when this piece of paper is folded up using the crease pattern shown?
Can you find the squares hidden on these coordinate grids?
Sara and Will were sorting some pictures of shapes on cards. "I'll collect the circles," said Sara. "I'll take the red ones," answered Will. Can you see any cards they would both want?
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 recreate squares and rhombuses if you are only given a side or a diagonal?
This activity focuses on similarities and differences between shapes.
Can you each work out what shape you have part of on your card? What will the rest of it look like?
What can you see? What do you notice? What questions can you ask?
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
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?
Complete the squares - but be warned some are trickier than they look!
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?
These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?
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.
If you move the tiles around, can you make squares with different coloured edges?
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.
Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?
How would you move the bands on the pegboard to alter these shapes?
These pictures show squares split into halves. Can you find other ways?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
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?
Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?
What is the greatest number of squares you can make by overlapping three squares?
A Short introduction to using Logo. This is the first in a twelve part series.
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?
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?
How many centimetres of rope will I need to make another mat just like the one I have here?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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 opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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.
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.
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.
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?
What is the total number of squares that can be made on a 5 by 5 geoboard?
What is the minimum number of squares a 13 by 13 square can be dissected into?
Can you work out the area of the inner square and give an explanation of how you did it?
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
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?
Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
This activity investigates how you might make squares and pentominoes from Polydron.
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
Can you make five differently sized squares from the interactive tangram pieces?