Intersecting squares
Weekly Problem 32 - 2014
Three overlapping squares are shown. If you know the areas of the overlapping and non-overlapping parts, can you work out the side lengths of the squares?
Three overlapping squares are shown. If you know the areas of the overlapping and non-overlapping parts, can you work out the side lengths of the squares?
Age
11 to 14
Challenge level
Problem
Image
Three congruent squares overlap as shown.
The areas of the three overlapping sections are $2\;\mathrm{cm}^2$, $5\;\mathrm{cm}^2$ and $8\;\mathrm{cm}^2$ respectively.
The total area of the non-overlapping parts of the squares is $117\;\mathrm{cm}^2$.
What is the side length of each square?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Each of the overlapping areas contributes to the area of exactly two squares. So the total area of the three squares is equal to the area of the non-overlapping parts of the squares plus twice the total of the three overlapping areas, i.e. $(117 + 2(2 + 5 + 8))\;\mathrm{cm}^2 = (117 + 30)\;\mathrm{cm}^2 = 147\;\mathrm{cm}^2$.
So the area of each square is $(147 \div 3)\;\mathrm{cm}^2 = 49\;\mathrm{cm}^2$. Therefore the length of the side of each square is $7\;\mathrm{cm}$.