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There are **70** NRICH Mathematical resources connected to **Area - squares and rectangles**, you may find related items under Measuring and calculating with units.

Problem
Primary curriculum
Secondary curriculum
### Alison's Quilt

Nine squares are fitted together to form a rectangle. Can you find its dimensions?

Age 11 to 14

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
### The Square Under the Hypotenuse

Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Completing Quadrilaterals

We started drawing some quadrilaterals - can you complete them?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Perimeter Challenge

Can you deduce the perimeters of the shapes from the information given?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shape Draw

Use the information on these cards to draw the shape that is being described.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Through the Window

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

Age 7 to 11

Challenge Level

Game
Primary curriculum
Secondary curriculum
### Dicey Perimeter, Dicey Area

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ribbon Squares

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Perimeter Possibilities

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Changing Areas, Changing Perimeters

How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Place Your Orders

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shaping It

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Area and Perimeter

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Rectangles

A task which depends on members of the group noticing the needs of others and responding.

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Growing Rectangles

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Can They Be Equal?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Wallpaper

These pieces of wallpaper need to be ordered from smallest to largest. Can you find a way to do it?

Age 5 to 7

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
### Sizing Them Up

Can you put these shapes in order of size? Start with the smallest.

Age 5 to 7

Challenge Level

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

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Warmsnug Double Glazing

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fence It

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tilted Squares

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras for a Tetrahedron

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.

Age 16 to 18

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
### Numerically Equal

Can you draw a square in which the perimeter is numerically equal to the area?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Compare Areas

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Boxes

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Trig Rules OK

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

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### From One Shape to Another

Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.

Age 7 to 14

Problem
Primary curriculum
Secondary curriculum
### A Square in a Circle

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Inside Seven Squares

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

Age 7 to 11

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
### 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
### Framed

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of the pictures.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Wrapping Presents

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Lawn Border

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Squared

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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
### Fencing

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fencing Lambs

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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
### 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