Find the ratios of the perimeters and areas of the pairs of two regular pentagons with corresponding sides that measure 48cm and 54 cm.
Since perimeter is a linear relationship, the ratio of perimeters of two similar figures is proportional to the ratio of their sides, thus
48 : 54 or 8 : 9
Since area is a quadratic relationship , .....
the square of their sides, thus
48^2 : 54^2 = 2304 : 2916
= 64 : 81
What do you call a shape that loves to work out? A flex-agon!
Now, let's get down to business. The ratio of the perimeters of two similar shapes is the same as the ratio of their corresponding sides. So, the ratio of the perimeters of the two regular pentagons with sides measuring 48cm and 54cm would be 48/54.
As for the ratio of their areas, it is equal to the square of the ratio of their corresponding sides. So, the ratio of the areas of the two regular pentagons with sides measuring 48cm and 54cm would be (48/54)^2.
Simplify the fractions and you'll get your answer. Just remember, math can be complex, but there's always room for a little humor!
To find the ratio of the perimeters, divide the larger perimeter by the smaller perimeter:
Perimeter ratio = (Perimeter of larger pentagon) / (Perimeter of smaller pentagon)
= 54 cm / 48 cm
Simplifying the ratio:
Perimeter ratio = 9/8
To find the ratio of the areas, divide the larger area by the smaller area:
Area ratio = (Area of larger pentagon) / (Area of smaller pentagon)
The area of a regular pentagon can be calculated using the formula:
Area = (5/4) * s^2 * cot(π/5)
where s is the length of the side of the pentagon.
Area of larger pentagon = (5/4) * (54 cm)^2 * cot(π/5)
Area of smaller pentagon = (5/4) * (48 cm)^2 * cot(π/5)
Area ratio = [(5/4) * (54 cm)^2 * cot(π/5)] / [(5/4) * (48 cm)^2 * cot(π/5)]
= (54 cm)^2 / (48 cm)^2
Simplifying the ratio:
Area ratio = (54/48)^2
= (9/8)^2
= 81/64
Therefore, the ratio of the perimeters is 9/8 and the ratio of the areas is 81/64.
To find the ratio of the perimeters, divide the perimeter of one regular pentagon by the perimeter of the other. In this case, we have a regular pentagon with a side length of 48 cm and another regular pentagon with a side length of 54 cm.
The perimeter of a regular pentagon is given by the formula P = 5s, where P is the perimeter and s is the length of each side.
For the first pentagon with a side length of 48 cm, the perimeter is P1 = 5 * 48 = 240 cm.
For the second pentagon with a side length of 54 cm, the perimeter is P2 = 5 * 54 = 270 cm.
The ratio of the perimeters is P1/P2 = 240/270 = 8/9.
To find the ratio of the areas, divide the area of one regular pentagon by the area of the other. The formula for the area of a regular pentagon is A = (5/4) * s^2 * cot(π/5), where A is the area and s is the length of each side.
For the first pentagon with a side length of 48 cm, the area is A1 = (5/4) * (48)^2 * cot(π/5).
For the second pentagon with a side length of 54 cm, the area is A2 = (5/4) * (54)^2 * cot(π/5).
The ratio of the areas is A1/A2 = [(5/4) * (48)^2 * cot(π/5)] / [(5/4) * (54)^2 * cot(π/5)] = (48^2)/(54^2) = (16/18)^2 = (8/9)^2.
Therefore, the ratio of the perimeters of the two regular pentagons is 8:9, and the ratio of the areas is 64:81.