Suppose you want to install carpet in a hexagon room whose sides are all 15 ft long. Carpet costs $48 per sq yd and comes in rectangular rolls that are 12 ft by 90 ft. (Sold by the whole sq yd) How will you lay the carpet? Where will the seams be? How many pieces will you need? How much will it cost?
When I first did this problem I started by dividing into triangles but nothing seems to work.
Notice that if you sketch a reasonably accurate hexagon, you have opposite sides that are parallel.
So you can construct a rectangle down the centre.
On either side you will have an isosceles triangle with base angles of 30° and equal sides of 15
By simple trig, you can see that it will have a height of 7.5 and a length of 25.98 , let's say 26
Now, we could cut that large isosceles triangle into congruent right-angled triangles. There will be 4 of these and they form a rectangle 26 by 7.5
I would cut this rectangle into the 4 right-angled triangles and fit them around the large central rectangle of 26
Ok, so we need 2 cuts of the roll, of 26 by 12, the roll is 90 ft, so there is enough
1. lay down one of the large rectangles, leaving a gap of 3 ft
2. cut the second 26 by 12 piece along its length to get a 3 by 26. This will fill the large central rectangle
3. You will have a piece 12 by 26 left. Cut that along the length to have a width of 7.5 ft
4. This can now be cut into the 4 identical right-angled triangles and arranged to finish the hexagon.
You might try this with a piece of paper.
I will leave the rest of the question up to you