Search by Topic

Filter by: Content type:
Stage:
Challenge level: Challenge Level:1 Challenge Level:2 Challenge Level:3

There are 88 results

Broad Topics > Measures and Mensuration > Area

problem icon

Growing Rectangles

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Can They Be Equal?

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

All in a Jumble

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

problem icon

Torn Shapes

Stage: 2 Challenge Level: Challenge Level:1

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

problem icon

More Transformations on a Pegboard

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Warmsnug Double Glazing

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Isosceles Triangles

Stage: 3 Challenge Level: Challenge Level:1

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

problem icon

Fence It

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Tilted Squares

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

An Unusual Shape

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you maximise the area available to a grazing goat?

problem icon

Inscribed in a Circle

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

problem icon

Pick's Theorem

Stage: 3 Challenge Level: Challenge Level:1

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

problem icon

Fitted

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

problem icon

Transformations on a Pegboard

Stage: 2 Challenge Level: Challenge Level:1

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

problem icon

Numerically Equal

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Making Boxes

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Shape Draw

Stage: 2 Challenge Level: Challenge Level:1

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

problem icon

Through the Window

Stage: 2 Challenge Level: Challenge Level:1

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?

problem icon

Dicey Perimeter, Dicey Area

Stage: 2 Challenge Level: Challenge Level:1

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?

problem icon

Ribbon Squares

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Cover the Tray

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Perimeter Possibilities

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Different Sizes

Stage: 1 and 2 Challenge Level: Challenge Level:1

A simple visual exploration into halving and doubling.

problem icon

Tiling Into Slanted Rectangles

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A follow-up activity to Tiles in the Garden.

problem icon

Tiles in the Garden

Stage: 2 Challenge Level: Challenge Level:1

How many tiles do we need to tile these patios?

problem icon

Changing Areas, Changing Perimeters

Stage: 3 Challenge Level: Challenge Level:1

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?

problem icon

Cylinder Cutting

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Place Your Orders

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Shaping It

Stage: 1 and 2 Challenge Level: Challenge Level:1

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.

problem icon

Area and Perimeter

Stage: 2 Challenge Level: Challenge Level:1

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

problem icon

Extending Great Squares

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Explore one of these five pictures.

problem icon

Fit These Shapes

Stage: 1 and 2 Challenge Level: Challenge Level:1

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?

problem icon

Making Rectangles

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

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

problem icon

Cutting it Out

Stage: 1 and 2 Challenge Level: Challenge Level:1

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

problem icon

How Random!

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Tiling

Stage: 2 Challenge Level: Challenge Level:1

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

problem icon

A Day with Grandpa

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Triangle Relations

Stage: 2 Challenge Level: Challenge Level:1

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

problem icon

Overlapping Squares

Stage: 2 Challenge Level: Challenge Level:1

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.

problem icon

Approaches to Area

Stage: 1 and 2

This article for teachers gives some food for thought when teaching ideas about area.

problem icon

Exploration Versus Calculation

Stage: 1, 2 and 3

This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.

problem icon

Circle Panes

Stage: 2 Challenge Level: Challenge Level:1

Look at the mathematics that is all around us - this circular window is a wonderful example.

problem icon

From One Shape to Another

Stage: 2

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.

problem icon

Uncanny Triangles

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

problem icon

Shear Magic

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

problem icon

A Square in a Circle

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

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?

problem icon

The Pi Are Square

Stage: 3 Challenge Level: Challenge Level:1

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?

problem icon

Inside Seven Squares

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Triangle Island

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

You have pitched your tent (the red triangle) on an island. Can you move it to the position shown by the purple triangle making sure you obey the rules?

problem icon

Rati-o

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?