The area of the larger of two similar polygons is 729 square centimeters. If the scale factor of their perimeters is 2:3 what is the area of the smaller polygon?
A/729 = (2/3)^2 = 4/9
A/729 = 4/9
A = 4/9 * 729 = 324 sq. cm.
To find the area of the smaller polygon, we need to determine the scale factor of their areas.
The scale factor of the perimeters is given as 2:3. This means that the ratio of the perimeters of the two polygons is 2/3.
Now, let's assume that the perimeters of the smaller and larger polygons are P₁ and P₂ respectively. According to the given information, we have:
P₁/P₂ = 2/3
Now, let's consider the formula for the perimeter of a polygon, which is the sum of the lengths of all its sides. If we assume that the corresponding sides of the two polygons have lengths a₁ and a₂ respectively, then we can write:
P₁ = n₁ * a₁
P₂ = n₂ * a₂
Where n₁ and n₂ represent the number of sides of the smaller and larger polygons respectively.
Since the two polygons are similar, it means that the corresponding sides are proportional. Therefore, we can write:
a₁/a₂ = 2/3
Now, let's consider the formula for the area of a polygon, which is given by the product of its perimeter and the apothem (the perpendicular distance from the center to any side). We can write:
Area₁ = 1/2 * P₁ * apothem₁
Area₂ = 1/2 * P₂ * apothem₂
Since the two polygons are similar, it means that the ratio of their areas is equal to the square of the scale factor of their corresponding sides. Therefore, we can write:
Area₁/Area₂ = (a₁/a₂)²
Now, let's substitute the values we have:
Area₁/729 = (2/3)²
Simplifying the equation:
Area₁/729 = 4/9
Cross-multiplying:
Area₁ = (4/9) * 729
Calculating:
Area₁ = 4 * 81
Area₁ = 324
Therefore, the area of the smaller polygon is 324 square centimeters.