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Broad Topics > 2D Geometry, Shape and Space > Squares

Eight Hidden Squares

Stage: 2 and 3 Challenge Level:

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?

Squares in Rectangles

Stage: 3 Challenge Level:

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 Square

Stage: 1 Challenge Level:

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

Square Coordinates

Stage: 3 Challenge Level:

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?

Ten Hidden Squares

Stage: 2 Challenge Level:

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

Square It

Stage: 1, 2, 3 and 4 Challenge Level:

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.

On the Edge

Stage: 3 Challenge Level:

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 to. . . .

Tilted Squares

Stage: 3 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Semi-detached

Stage: 4 Challenge Level:

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.

Fitted

Stage: 2 Challenge Level:

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?

Halving

Stage: 1 Challenge Level:

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

Square Corners

Stage: 2 Challenge Level:

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?

Zig Zag

Stage: 4 Challenge Level:

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?

Compare Areas

Stage: 4 Challenge Level:

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

Chain of Changes

Stage: 1 Challenge Level:

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?

Square it for Two

Stage: 1, 2, 3 and 4 Challenge Level:

Square It game for an adult and child. Can you come up with a way of always winning this game?

Peg and Pin Boards

Stage: 1 and 2

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

Data Shapes

Stage: 1 Challenge Level:

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?

Vector Journeys

Stage: 4 Challenge Level:

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?

Opposite Vertices

Stage: 3 Challenge Level:

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

Sorting Logic Blocks

Stage: 1 and 2 Challenge Level:

This interactivity allows you to sort logic blocks by dragging their images.

Jig Shapes

Stage: 1 Challenge Level:

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

Baravelle

Stage: 2, 3 and 4 Challenge Level:

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

Folding Fractions

Stage: 4 Challenge Level:

What fractions can you divide the diagonal of a square into by simple folding?

Logic Block Collections

Stage: 1 Challenge Level:

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

Sorting Shapes

Stage: 1 Challenge Level:

This activity challenges you to make collections of shapes. Can you give your collection a name?

Shaping Up

Stage: 2 Challenge Level:

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Squares

Stage: 1 and 2 Challenge Level:

This collection of resources is designed to enable children to explore concepts and ideas associated with squares.

Overlapping Squares

Stage: 2 Challenge Level:

Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.

Fraction Fascination

Stage: 2 Challenge Level:

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

LOGO Challenge 4 - Squares to Procedures

Stage: 3 and 4 Challenge Level:

This LOGO Challenge emphasises the idea of breaking down a problem into smaller manageable parts. Working on squares and angles.

Logo Challenge 3 - Star Square

Stage: 2, 3 and 4 Challenge Level:

Creating designs with squares - using the REPEAT command in LOGO. This requires some careful thought on angles

Gold Yet Again

Stage: 5 Challenge Level:

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

Trig Rules OK

Stage: 5 Challenge Level:

Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...

Something in Common

Stage: 4 Challenge Level:

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.

Circle Box

Stage: 4 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?

A Square in a Circle

Stage: 2 Challenge Level:

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

The Pi Are Square

Stage: 3 Challenge Level:

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?

Inside Seven Squares

Stage: 2 Challenge Level:

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

Squares, Squares and More Squares

Stage: 3 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?

Triangles in a Square

Stage: 4 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.

LOGO Challenge - the Logic of LOGO

Stage: 3 and 4 Challenge Level:

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

LOGO Challenge - the Humble Square

Stage: 2, 3 and 4 Challenge Level:

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.

Hidden Squares

Stage: 3 Challenge Level:

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Diagrams

Stage: 2 Challenge Level:

A group activity using visualisation of squares and triangles.

LOGO Challenge 7 - More Stars and Squares

Stage: 3 and 4 Challenge Level:

Can you use LOGO to create a systematic reproduction of a basic design? An introduction to variables in a familiar setting.

LOGO Challenge 5 - Patch

Stage: 3 and 4 Challenge Level:

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

LOGO Challenge 1 - Star Square

Stage: 2, 3 and 4 Challenge Level:

Can you use LOGO to create this star pattern made from squares. Only basic LOGO knowledge needed.

First Forward Into Logo 1: Square Five

Stage: 2, 3 and 4 Challenge Level:

A Short introduction to using Logo. This is the first in a twelve part series.

Zooming in on the Squares

Stage: 2 and 3

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?