Among the following shapes of equal perimeter, whichone has the largest area ?(A) A square (B) An equilateral triangle(C) A circle (D) A regular pentagon

To determine which shape among the square, equilateral triangle, circle, and regular pentagon has the largest area, we need to compare their areas given that they all have equal perimeters.

Let's start by calculating the area of each shape:

A) Square: A square has all sides equal in length. Let's assume the length of each side is 's'. The perimeter of a square is given by 4s, so if the perimeter is equal, we can say that 4s = P, where P is the given perimeter. Therefore, each side of the square would be s = P/4.

The area of a square is given by the formula A = s^2. Substituting the value of s, we get A = (P/4)^2 = P^2/16.

B) Equilateral Triangle: An equilateral triangle has all sides equal in length. Let's assume the length of each side is 's'. The perimeter of an equilateral triangle is given by 3s, so if the perimeter is equal, we can say that 3s = P. Therefore, each side of the equilateral triangle would be s = P/3.

The area of an equilateral triangle is given by the formula A = (sqrt(3)/4) * s^2. Substituting the value of s, we get A = (sqrt(3)/4) * (P/3)^2 = (sqrt(3)/36) * P^2.

C) Circle: A circle has a circumference equal to the given perimeter. Let's assume the circumference of the circle is 'C'. The perimeter of a circle is given by 2πr, where r is the radius. If the perimeter is equal, we can say that 2πr = P. Therefore, the radius of the circle would be r = P/(2π).

The area of a circle is given by the formula A = πr^2. Substituting the value of r, we get A = π * (P/(2π))^2 = P^2/(4π).

D) Regular Pentagon: A regular pentagon has all sides equal in length. Let's assume the length of each side is 's'. The perimeter of a regular pentagon is given by 5s, so if the perimeter is equal, we can say that 5s = P. Therefore, each side of the regular pentagon would be s = P/5.

To calculate the area of a regular pentagon, we need to know the apothem (the distance from the center of the pentagon to the midpoint of any side). However, since the apothem is not given, we cannot calculate the exact area of the regular pentagon.

Now, to determine which shape has the largest area, we need to compare the formulas for the areas of each shape:

A = P^2/16 (Square) B = (sqrt(3)/36) * P^2 (Equilateral Triangle) C = P^2/(4π) (Circle) D = Unknown (Regular Pentagon)

Comparing the formulas, we can see that the area of the circle (C) is dependent on the value of π, which is a constant. Therefore, the area of the circle will always be the same for any given perimeter.

On the other hand, the areas of the square (A) and equilateral triangle (B) are both dependent on the value of P. However, the area of the square (A) has a larger coefficient (1/16) compared to the area of the equilateral triangle (sqrt(3)/36).

Since the coefficient of the area of the square is larger, the square will have the largest area among the given shapes of equal perimeter.

Therefore, the answer is (A) A square.