The lengths of the shortest sides of two similar hexagons are 10 cm and 8 cm. Given that the area of the larger hexagon is 200 cm^2, find the area..of the smaller hexagon.
What we have to do here is set up this equation using ratios.
We know that the ratio of the sides of the two hexagons is 10:8. We can also show that as 10/8.
We know the area of the larger hexagon, but not that of the smaller hexagon. Let us call the larger hexagon H1 and the smaller H2.
To solve this, we need to set up the following equation:
10/8 = 200/area of H2
We know this because we know that the areas of the two hexagons must have the same ratio as the similar sides.
Now it is just algebra -- we have to cross multiply. When we do, we get
10(H2) = 1600
We divide that and we get H2 = 160
So, that tells us that the area of the smaller hexagon is 160 cm^2
The area of any similar polygons are proportional to the square of the two corresponding sides.
Here, the two shotest sides are the corresponding sidesand the ratio of the sides is 10:8. Therefore, their areas are in 10^2/8^2 ratio and not10/8 as in the answer posted above.
The area of the larger hexagon = 200cm^2.Let the area of the smaller hexagon be x. Then the ratio of the areas of the two hexagon is
x = 200*8^2/10^2 or
x = 200*64/100 sq cm^2
=2*64 sq cm
So, the area of the smaller hexagon is 128 cm^2.
Let us say:
Length of the shortest line of bigger hexagon = l1 = 10 cm (given)
Length of the shortest line of smaller hexagon = l2 = 8 cm (given)
Area of the bigger hexagon = a1 = 200 cm^2 (given)
Area of the smaller hexagon = a2 (to be determined)
Then: a1/a2 = (l1^2)/(l2^2)
Or: a2 = a1*((l2^2)/(l1^2)
Substituting above values of l1, l2 and a 2 we get:
a2 = 200*((8^2)/(10^2) = 200*64/100 = 128 cm^2
Area of smaller hexagon = 128 cm^2