To find the area of a circle, we need to use the correct formula. We have several options to consider:
A. [tex]\( A = 2 \pi r \)[/tex]
B. [tex]\( A = \pi r d^2 \)[/tex]
C. [tex]\( A = \pi^2 r \)[/tex]
D. [tex]\( A = \pi r^2 \)[/tex]
Let's analyze each option:
Option A: [tex]\( A = 2 \pi r \)[/tex]
- This formula actually resembles the formula for the circumference of a circle, which is given by [tex]\(C = 2 \pi r \)[/tex]. Hence, this is incorrect for the area.
Option B: [tex]\( A = \pi r d^2 \)[/tex]
- The diameter [tex]\(d\)[/tex] of the circle is twice the radius [tex]\(r\)[/tex], so [tex]\(d = 2r\)[/tex]. If we substitute [tex]\(d = 2r\)[/tex] into the expression [tex]\( \pi r d^2\)[/tex], it would become [tex]\( \pi r (2r)^2 = \pi r \cdot 4r^2 = 4 \pi r^3\)[/tex], which is far from the correct formula for the area. Thus, this is incorrect.
Option C: [tex]\( A = \pi^2 r \)[/tex]
- This formula includes [tex]\(\pi^2\)[/tex], which is not involved in the formula for the area of a circle. Hence, this is incorrect too.
Option D: [tex]\( A = \pi r^2 \)[/tex]
- This is the standard and correct formula for the area of a circle, where [tex]\(A\)[/tex] is the area, [tex]\(\pi\)[/tex] (Pi) is a constant approximately equal to 3.14159, and [tex]\(r\)[/tex] is the radius of the circle.
Therefore, the correct formula for the area of a circle is indeed:
[tex]\( \boxed{A = \pi r^2} \)[/tex]
So the correct answer is option D, which corresponds to the number 4 in your list.
