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There are **76** NRICH Mathematical resources connected to **Squares**, you may find related items under Angles, polygons, and geometrical proof.

Problem
Primary curriculum
Secondary curriculum
### Seeing 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.

Age 5 to 11

Challenge Level

Problem
Live Primary curriculum
Secondary curriculum
### Always, Sometimes or Never? Shape

Are these statements always true, sometimes true or never true?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Paper Patchwork 1

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Hidden Squares

Can you find the squares hidden on these coordinate grids?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Data Shapes

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?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Vector Journeys

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?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Opposite Vertices

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sorting Logic Blocks

This activity focuses on similarities and differences between shapes.

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Jig Shapes

Can you each work out what shape you have part of on your card? What will the rest of it look like?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Baravelle

What can you see? What do you notice? What questions can you ask?

Age 7 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Eight Hidden 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 hidden squares?

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fraction Fascination

This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Torn Shapes

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squares in Rectangles

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Complete the Square

Complete the squares - but be warned some are trickier than they look!

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Coordinates

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ten Hidden Squares

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square It

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.

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### On the Edge

If you move the tiles around, can you make squares with different coloured edges?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Let Us Reflect

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Semi-detached

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fitted

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Transformations on a Pegboard

How would you move the bands on the pegboard to alter these shapes?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Halving

These pictures show squares split into halves. Can you find other ways?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Corners

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Zig Zag

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Chain of Changes

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?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Three Squares

What is the greatest number of squares you can make by overlapping three squares?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Coloured Edges

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rope Mat

How many centimetres of rope will I need to make another mat just like the one I have here?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squaring the Circle and Circling the Square

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Take a Square

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Tilted Square

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Similarly So

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tetra Square

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.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 2001 Spatial Oddity

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Folding Squares

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pinned Squares

What is the total number of squares that can be made on a 5 by 5 geoboard?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Dissect

What is the minimum number of squares a 13 by 13 square can be dissected into?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Areas

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fitting In

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Just Opposite

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cut and Make

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Squares

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### My New Patio

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tiles on a Patio

How many ways can you find of tiling the square patio, using square tiles of different sizes?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Polydron

This activity investigates how you might make squares and pentominoes from Polydron.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Geoboards

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tangrams

Can you make five differently sized squares from the interactive tangram pieces?

Age 5 to 11

Challenge Level