A = P
Introduction
In today's lesson you will need to be familiar with finding the area (A) and perimeter (P) of a rectangle.
If you are unsure how to do this, click here for a video and do these simple questions to practise.
If you are confident, try the more challenging questions on the right:
Investigation A = P
Work out the areas and perimeters of the rectangles on the right. Make sure you show all your working.
Which of these has an area greater than its perimeter?
What about a bigger perimeter than area?
Put them in the table on the right.
Now draw some more rectangles and work out the area and perimeter for each one and put them in the table.
Describe the types of rectangle where A < P. What do they look like?
Can you find any rectangle where A = P. How many did you find?
How can you convince your teacher that you have found them all?
What if you don’t have to stick to integers (whole numbers)? Can you find some other rectangle where A = P that have side lengths that are decimals or fractions?
Can you find a way to work out more examples where A = P?
For any more examples where A = P, plot the corresponding width and height on a set of axes (width on the x-axis and height on the y-axis, see graph on the right.
If you join the points up with a smooth curve, what does it look like?
What conjectures can you make?
Can you prove your conjectures?
Further Questions and Challenges
Look at the pictures below:
What size should the new second rectangle be so that the area and perimeter of the combined shape are the same number?
What if we put the second rectangle on the top of the first one?
Further Practice
Some of the essential skills introduced in this lesson are "area" and "perimeter" of a rectangle. Should you need to you may also practise decimal and fraction arithmetics (+, -, x, /). If you would like to study ahead, you may wish to look at the areas and perimeters of other shapes such as square, triangle, parallelogram, trapezium, kite L-shape and circle etc. You will learn about some of these in the coming lessons. Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.
Transum
Area and Perimeter of a Rectangle - try this self-marked exercise. There are altogether 2 levels plus an investigation at the end.
Area of a Triangle - try this self-marked exercise. There are altogether 3 levels.
Area of a Trapezium - try this self-marked exercise. There are altogether 2 levels.
Area and Perimeter of a Parallelogram - try this self-marked exercise. There are altogether 6 levels.
Kite Maths Exercise - try this self-marked exercise.
Circles - try this self-marked exercise. There are altogether 6 levels.
Areas of Composite Shapes - try this self-marked exercise. There are altogether 5 levels.
Area and Perimeter - try this self-marked exercise. Name the shapes then find formulas to calculate the area and perimeters from the given lengths.
For more goodies on area on transum, click on the hyperlink.
Extension
Transum
Area Wall puzzle - there are altogether 5 levels.
Area Maze puzzle - there are altogether 4 levels. More area maze puzzles by Don Steward is here.
nRich
Can They Be Equal? - today's lesson is based on this activity.
Numerically Equal - this an easier investigation based on a similar idea but for a square.
Area and Perimeter - this an even easier investigation based on a similar idea but for nor regular shapes.
Framed - here is another interesting/more challenging activity based on areas and perimeters of rectangles.
Desmos
Look at Area v Perimeter on Desmos. In this brilliant investigation, you will use a graph to consider a subtle and sophisticated relationship between the area and perimeter of a rectangle.