Perimeter & area

This is a collection of different ideas and problems to solve using area and perimeter of shape. Many of the activities are based on squares and rectangles but some of the tasks involve various other 2D shapes, with circles making just a brief appearance (there is a separate post on circles).

Area

Lots of problems involving shape, particularly compound rectangular shapes, hinge on figuring out missing lengths. Work through the following Transum task to check that you can work these out:

Area Maze is a brilliant puzzle that gets another mention in the Further Ideas section at the end of this post. And Perimeters is useful revision for working with algebra in shape problems.

Work through this collection of activities to explore area.

In Crazy Shading, from NRICH, the aim is work out what proportion of the whole square is shaded:

Could you create your own design to pose a problem like this one for someone else? Send your designs in to the Student Showcase.

And Margins is a problem in context. A 30 cm by 40 cm page of a book includes a 2 cm margin on each side, as shown. What percentage of the page is occupied by the margins?

Hint: split the margin (shaded part) into four rectangles, working out the dimensions of each carefully. Then calculate the shaded area and the total area. Or you could work out the unshaded area to get the shaded area (as the solution below shows).

To practise working out areas of different shapes, try the following Area Challenge from TeachIt. All of the shapes have the same area (they are not drawn to scale). The challenge is to find all of the missing lengths, a, b, c, d, e and f.

Perimeter

Work through these slides about perimeter:

Try pairing up the following rectangles with their areas and perimeters

You might want to use an algebraic approach in this Line of Squares NRICH problem:

In the diagram, each of the squares touches adjacent squares at the corners and the line GH along one of the edges. If the length of GH is 24 cm, what is the total perimeter of all the squares?

Now try this Sideways Ratio problem:

Hint: You could try to use algebra, or try out some different lengths that satisfy the ratio to see what area they would give you and refine your answers from there.

In this problem Bobbly Perimeter, a square with perimeter 20 cm has a semicircle drawn onto each of its sides, as shown in the diagram.

Further ideas

The Japanese puzzle Shikaku (also known as Rectangles) can be played online:

Area maze is another great puzzle based on areas of rectangles. You can read about it in a Guardian article here or in the New York Times here, where a 3D version is included. Or try the online version here.

To further explore areas of different shapes and how they are linked, work through this extensive activity book:

ANSWERS – click below

Area
So the proportion shaded is 25%, or 0.25 or one quarter.

Perimeter