This is part of our Secondary Curriculum collection of favourite rich tasks arranged by topic.

Scroll down to see the complete collection, or explore our subcollections on Perimeter and Area in two dimensions, and Surface Area and Volume in three dimensions.

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### Blue and White

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

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### Fence it

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

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### Isosceles Triangles

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?

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### Can they be equal?

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

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### 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?

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### Perimeter Possibilities

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

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### Perimeter Challenge

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

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### Colourful Cube

A colourful cube is made from little red and yellow cubes. But can you work out how many of each?

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### Isometric Areas

We usually use squares to measure area, but what if we use triangles instead?

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### Shear Magic

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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### On the Edge

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

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### Sending a Parcel

What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?

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### Cuboid challenge

What's the largest volume of box you can make from a square of paper?

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### Changing areas, changing volumes

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

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### Triangle in a Trapezium

Can you find and prove the relationship between the area of a trapezium and the area of a triangle constructed within it?

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### More Isometric Areas

Isometric Areas explored areas of parallelograms in triangular units. Here we explore areas of triangles...

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### Cuboids

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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### Efficient cutting

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

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### Cola Can

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Scroll down to see the complete collection, or explore our subcollections on Perimeter and Area in two dimensions, and Surface Area and Volume in three dimensions.

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### Perimeter and Area

This selection of problems is designed to help you teach Perimeter and Area.

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### Surface Area and Volume

This selection of problems is designed to help you teach Surface Area and Volume.

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*ou may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.*