The area of the larger of two similar polygons is 315 squared centimeters. If the scale factor of their perimeters is 2:3 what is the area of the smaller polygon?
the areas of similar figures are proportional to the square of their corresponding sides
let the area of the smaller be x
x/315 = 2^2/3^2 = 4/9
9x = 1260
x = 1260/9 = 140
To find the area of the smaller polygon, we need to use the concept of scale factors.
Let's assume that the area of the smaller polygon is A square centimeters.
The scale factor of the perimeters is given as 2:3. This means that the ratio of the lengths of corresponding sides of the two polygons is 2:3.
Since the polygons are similar, the ratio of their areas is equal to the square of the scale factor. Therefore, we have:
(Area of larger polygon) / (Area of smaller polygon) = (Scale factor)^2
Plugging in the given values, we have:
315 cm^2 / A cm^2 = (2/3)^2
Simplifying the equation:
315 cm^2 / A cm^2 = 4/9
Next, we can cross-multiply:
4A = 315 * 9
Dividing both sides by 4:
A = (315 * 9) / 4
Calculating this expression:
A = 708.75 square centimeters
So, the area of the smaller polygon is approximately 708.75 square centimeters.