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Broad Topics > Measuring and calculating with units > Area - squares and rectangles

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Always, Sometimes or Never? Shape

Age 7 to 11 Challenge Level:

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

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Perimeter Challenge

Age 11 to 14 Challenge Level:

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

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Shape Draw

Age 7 to 11 Challenge Level:

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

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Through the Window

Age 7 to 11 Challenge Level:

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?

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Dicey Perimeter, Dicey Area

Age 7 to 11 Challenge Level:

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?

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Ribbon Squares

Age 7 to 11 Challenge Level:

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

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Perimeter Possibilities

Age 11 to 14 Challenge Level:

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

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Different Sizes

Age 5 to 11 Challenge Level:

A simple visual exploration into halving and doubling.

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Changing Areas, Changing Perimeters

Age 11 to 14 Challenge Level:

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?

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Poly Plug Rectangles

Age 5 to 11 Challenge Level:

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

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Shaping It

Age 5 to 11 Challenge Level:

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.

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Area and Perimeter

Age 7 to 11 Challenge Level:

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

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Growing Rectangles

Age 11 to 14 Challenge Level:

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

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Can They Be Equal?

Age 11 to 14 Challenge Level:

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

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Wallpaper

Age 5 to 7 Challenge Level:

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

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Torn Shapes

Age 7 to 11 Challenge Level:

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

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Sizing Them Up

Age 5 to 7 Challenge Level:

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

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More Transformations on a Pegboard

Age 7 to 11 Challenge Level:

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

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Warmsnug Double Glazing

Age 11 to 14 Challenge Level:

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

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Fence It

Age 11 to 14 Challenge Level:

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

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Tilted Squares

Age 11 to 14 Challenge Level:

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

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Pythagoras for a Tetrahedron

Age 16 to 18 Challenge Level:

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. . . .

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Semi-detached

Age 14 to 16 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.

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Fitted

Age 7 to 11 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?

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Transformations on a Pegboard

Age 7 to 11 Challenge Level:

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

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Numerically Equal

Age 7 to 11 Challenge Level:

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

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Compare Areas

Age 14 to 16 Challenge Level:

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

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Making Boxes

Age 7 to 11 Challenge Level:

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

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Pebbles

Age 7 to 11 Challenge Level:

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?

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Paper Halving

Age 5 to 11 Challenge Level:

In how many ways can you halve a piece of A4 paper? How do you know they are halves?

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Being Resilient - Primary Measures

Age 5 to 11 Challenge Level:

Measure problems at primary level that may require resilience.

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Being Resourceful - Primary Measures

Age 5 to 11 Challenge Level:

Measure problems at primary level that require careful consideration.

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Being Collaborative - Primary Measures

Age 5 to 11 Challenge Level:

Measure problems for primary learners to work on with others.

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Being Curious - Primary Measures

Age 5 to 11 Challenge Level:

Measure problems for inquiring primary learners.

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Completing Quadrilaterals

Age 11 to 14 Challenge Level:

We started drawing some quadrilaterals - can you complete them?

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Cover the Tray

Age 7 to 11 Challenge Level:

These practical challenges are all about making a 'tray' and covering it with paper.

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Tiling Into Slanted Rectangles

Age 7 to 11 Challenge Level:

A follow-up activity to Tiles in the Garden.

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Tiles in the Garden

Age 7 to 11 Challenge Level:

How many tiles do we need to tile these patios?

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Cylinder Cutting

Age 7 to 11 Challenge Level:

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

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Place Your Orders

Age 11 to 14 Challenge Level:

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

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Extending Great Squares

Age 7 to 11 Challenge Level:

Explore one of these five pictures.

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Fit These Shapes

Age 5 to 11 Challenge Level:

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

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Making Rectangles

Age 7 to 14 Challenge Level:

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

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Cutting it Out

Age 5 to 11 Challenge Level:

I cut this square into two different shapes. What can you say about the relationship between them?

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How Random!

Age 7 to 11 Challenge Level:

Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?

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Tiling

Age 7 to 11 Challenge Level:

An investigation that gives you the opportunity to make and justify predictions.

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A Day with Grandpa

Age 7 to 11 Challenge Level:

Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?

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Triangle Relations

Age 7 to 11 Challenge Level:

What do these two triangles have in common? How are they related?

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Overlapping Squares

Age 7 to 11 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.

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Proof of Pick's Theorem

Age 16 to 18 Challenge Level:

Follow the hints and prove Pick's Theorem.