Find the length and width of a rectangle if its perimeter is 16cm and area is 15cm2.

What are the dimensions of a rectangle whose perimeter is 68cm and area is 225cm2.

Find the dimensions of a rectangle whose perimeter is 39cm and area is 87.5cm2.

Find the dimensions of all the rectangles whose perimeters in cm are twice the value of their areas in cm2.

## A Mathematics Lesson Starter Of The Day

• Transum,
•
• The first three problems are in increasing order of difficulty. The first can probably be guessed, the second needs more of a strategy as the numbers are larger and the third involves a fraction so probably requires the application of a more generalised method. Refreshing the page generates three different rectangles to further consolidate any techniques learned in the first round. The fourth problem is only for Nobel prize candidates!
• Simon Fletcher, Bolton UK
•
• 52 cm rectangle perimeter 105 area. Is this correct? I've checked and can't find a solution.
• Transum,
•
• Hello Simon, so you are looking for two numbers (the height and the width) that add up to half the perimeter, 26cm, and multiply together to give an area of 105cm2.
We can write two simultaneous equations showing this information:
h + w = 26
hw = 105
We can combine the information in these two equations and eliminate one of the unknowns:
From the first equation we see that h = 26 – w
Substituting this into the second equation gives (26 – w)w = 105
Expanding the brackets gives 26w - w2=105
Rearranging gives w2 - 26w + 105 = 0
This is a quadratic equation which can be solved by factorising
(w – 5)(w – 21) = 0
So w could be 5cm in which case h would be 21cm or, w could be 21cm in which case h would be 5cm.
• Audrey Macmillan, @audproctor77
•
• Clarifying area/perimeter confusion using a Transum Starter. Great #mathschat on squares and rows.

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