Area and Perimeter

3 x 3 Areas

Which of the following shaded regions has an area different from the other shaded regions?

Maximising Area

Drag point C in the window on the left. Click on the “Capture Data” button to populate the spreadsheet with the length, width, and area data.

Variety of Areas with Fixed Perimeter

Move the point about to create as many shapes as you can with the same perimeter

Perimeter and Area


  • for a constant area, when is the perimeter greatest/least?
  • why is the perimeter usually even and can it be odd?
  • how come the perimeter stays the same when bites are taken out of a shape?
  • how is the perimeter of a square related to the area?

Nested Squares

The diagram shows three nested squares.

What proportion of the diagram is shaded?


The three regular hexagons are all the same size. X, Y and Z denote the values of the shaded areas in the hexagons as shown.

Which of these areas are equal to each other?

Areas Investigation Transum

Area of a polygon – Part 1

Question prompts:

How do you find the area of a rectangle?

How do you find the area of a triangle?

Area of a polygon – Part 2

Question prompts:

How do you find the area of a parallelogram?

How do you find the area of a triangle?

Perimeter Expressions

Charlie took a sheet of paper and cut it in half.
Then he cut one of those pieces in half, and repeated
until he had five pieces altogether.

He labelled the sides of the smallest rectangle a to represent the the shorter side and b for the longer side.

Here is a shape that Charlie made by combining the largest and smallest rectangles.

Check you agree that the perimeter is 10a+4b.
Alison combined the largest and smallest rectangles in a different way.
Her shape had perimeter 8a+6b. Can you find how she might have done

Create some other shapes by combining two or more rectangles, making
sure they meet edge to edge and corner to corner. What can you say
about the areas and perimeters of the shapes you can make?


Two identical rectangular cards are glued together as shown, to form an ‘L’ shape.
The perimeter of this ‘L’ shape is 40 cm.
What can you say about the length and width of the original rectangles?

Sheep Pens

Farmer Jones has to create a new grazing pen for his sheep. He has 36 metres of fencing and all the necessary posts. However, he also has a stone wall that he can use for one side of his pen, so he only has to worry about the other three sides.

He is building a rectangular pen. How should he lay out this pen in order to give the sheep the maximum area to graze? What is the maximum area in square metres that the sheep can have and what will be the dimensions of the pen?

Experiment with different dimensions.  Record your results in a table like the one below:

  • Now try to find the biggest area you can for 144 metres of fencing.
  • This time try out other shaped pens, not just rectangular.
  • If every sheep needs a minimum of 1m² to live in, what is the maximum number of sheep the farmer can have?  What if each sheep needs 2m²?
Nested Squares
Perimeter Expressions

Alison’s shape with a perimeter of 8a + 6b combined the largest and smallest rectangles, with the smaller rectangle’s side of length a against one of the larger rectangle’s sides.

I have made the largest possible perimeter, 12a + 12b, using all the pieces because I have put the shorter side of each shape against the one before it.

However you arrange the two rectangles, the perimeter always equals:
(the longest length + the longest width) x 2


The perimeter of the shape is 4 x length, so the original rectangles are 10 cm long.
There is insufficient information to be able to say anything about the width – except that it is less than 10 cm.