Question

multiple answer 1 point mark wants to find the area of a circle with a diameter of 8 cm. which expressions can he use to find the area of the circle in square centimeters? choose all the correct answers. $2\cdot4\cdot\pi$ $8\cdot\pi$ $4\cdot4\cdot\pi$ $8^2\cdot\pi$ $4^2\cdot\pi$

Explanation:

Step1: Recall circle area formula

The formula for the area of a circle is \( A = \pi r^2 \), where \( r \) is the radius.

Step2: Find the radius

Given the diameter \( d = 8 \) cm, the radius \( r=\frac{d}{2}=\frac{8}{2} = 4 \) cm.

Step3: Substitute radius into formula

Substitute \( r = 4 \) into \( A=\pi r^2 \), we get \( A=\pi\times4^2=4^2\cdot\pi \) or \( A = \pi\times(\frac{8}{2})^2=\pi\times4^2 = 16\pi \)? Wait, no, wait: \( 4^2\cdot\pi=16\pi \), and also, if we consider the radius is 4, so \( r = 4 \), so \( A=\pi r^2=\pi\times4^2 = 16\pi \), and also, let's check the options:

Wait the options are: \( 2\cdot4\cdot\pi \), \( 8\cdot\pi \), \( 4\cdot4\cdot\pi \), \( 8^2\cdot\pi \), \( 4^2\cdot\pi \)

Wait \( 4\times4\times\pi=16\pi \), and \( 4^2\cdot\pi = 16\pi \). Also, let's re - check:

Radius \( r = 4 \) (since diameter is 8, radius is half of that). Area \( A=\pi r^2=\pi\times4^2=16\pi \). Now, \( 4\times4\times\pi = 16\pi \) and \( 4^2\times\pi=16\pi \). Let's check other options:

• \( 2\times4\times\pi=8\pi \) (this is the circumference formula \( C = 2\pi r \) with \( r = 4 \), so it's circumference, not area)
• \( 8\times\pi=8\pi \) (not equal to \( 16\pi \))
• \( 8^2\times\pi = 64\pi \) (wrong, since radius is 4, not 8)
• \( 4\times4\times\pi=16\pi \) (correct)
• \( 4^2\times\pi = 16\pi \) (correct)

Wait, maybe I made a mistake earlier. Let's re - do:

Diameter \( d = 8 \), so radius \( r=\frac{d}{2}=4 \). Area of circle is \( A=\pi r^2=\pi\times4^2 = 16\pi \). Now, \( 4\times4\times\pi=16\pi \), and \( 4^2\times\pi = 16\pi \). So the correct options are \( 4\cdot4\cdot\pi \) (which is \( 4\times4\times\pi \)) and \( 4^2\cdot\pi \).

Wait the last option is \( 4^2\cdot\pi \) (the one with the checkmark), and also \( 4\cdot4\cdot\pi \) is the same as \( 4^2\cdot\pi \). Let's confirm:

\( 4\times4\times\pi=16\pi \), \( 4^2\times\pi = 16\pi \). So the correct expressions are \( 4\cdot4\cdot\pi \) and \( 4^2\cdot\pi \).

Wait the options:

1. \( 2\cdot4\cdot\pi=8\pi \) (circumference, not area)
2. \( 8\cdot\pi = 8\pi \) (wrong)
3. \( 4\cdot4\cdot\pi=16\pi \) (correct)
4. \( 8^2\cdot\pi = 64\pi \) (wrong)
5. \( 4^2\cdot\pi=16\pi \) (correct)

So the correct options are \( 4\cdot4\cdot\pi \) (third option) and \( 4^2\cdot\pi \) (fifth option). Wait but in the image, the fifth option (the last one) is checked. Let's re - check the calculation:

Radius \( r = 4 \), area \( A=\pi r^2=\pi\times4^2 = 16\pi \). \( 4\times4\times\pi=16\pi \), \( 4^2\times\pi = 16\pi \). So these two are correct.

Answer:

The correct expressions are \( 4\cdot4\cdot\pi \) (or \( 4\times4\times\pi \)) and \( 4^2\cdot\pi \). So the correct options are:

• \( 4\cdot4\cdot\pi \)
• \( 4^2\cdot\pi \)