Area of a Circle - Pre-Algebra

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Question

What is the area of a circle with a diameter of one unit?

Answer

Write the area of a circle.

$A=\pi r^2$

Since we are given the diameter of the circle and the area equation calls for the radius, we will need to calculate the radius first.

Divide the diameter by two to get the radius.

$r=\frac{D}{2}= \frac{1}{2}$

$A=\pi \left(\frac{1}{2}\right)^2 = \frac{1}{4}\pi$

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Question

Find the area of a circle with a diameter of 15.

Answer

Write the area of a circle.

$A=\pi r^2$

To obtain the radius, divide the diameter by two.

$r=\frac{d}{2} = \frac{15}{2}$

Substitute the radius into the area equation and solve.

$A=\pi( \frac{15}{2})^2 = \frac{225}{4}\pi$

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Question

What is the area of a circle with a radius of ?

Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius and solve for area.

$A=\pi (15)^2 = 225\pi$

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Question

Find the area of a circle with a circumference of .

Answer

Write the circumference formula for the circle.

$C=2\pi r$

Substitute the circumference and solve for the radius.

$1=2\pi r$

$r=\frac{1}{2\pi}$

Write the formula for the area of the circle.

$A=\pi r^2$

Substitute the radius.

$A=\pi (\frac{1}{2\pi})^2 = \pi(\frac{1}{4\pi^2}) = \frac{1}{4\pi}$

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Question

Find the area of a circle with a diameter of .

Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Given the diameter, we will need to divide this by two to obtain the radius.

$r=\frac{d}{2}= \frac{1.8}{2} = 0.9$

Substitute the radius and solve.

$A=\pi (0.9)^2 = 0.81 \pi$

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Question

Find the area of a circle with a circumference of $18\pi$ .

Answer

Write the circumference formula and substitute the circumference.

$C=2\pi r$

$18\pi =2\pi r$

Now solve for r by dividing $2\pi$ on each side.

Write the formula to find the area of a circle.

$A=\pi r^2$

Since we solved for the radius in the above calculation, we will substitute that found value into the area formula and solve.

$A=\pi(9)^2 =81\pi$

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Question

Find the area of a circle with a radius of $\frac{2}{7}$ .

Answer

Write the formula for area of a circle.

$A=\pi r^2$

Substitute the radius.

$A=\pi (\frac{2}{7})^2 = \frac{4}{49}\pi$

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Question

Find the area of a circle with a diameter of 12.

Answer

First identify the relationship between diameter and radius of a circle. Then, divide the diameter by two to get the radius.

$r=\frac{d}{2}= \frac{12}{2}=6$

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius.

$A=\pi (6)^2$

$A=36\pi$

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Question

Find the area of a circle in $\textup{feet}$ if the radius is $2 \textup{ inches}$ .

Answer

Convert $\textup{2 inches}$ to $\textup{feet}$ . There are $\textup{12 inches}$ in a foot.

$r= 2 \textup{ inches }(\frac{1 \textup{ foot}}{12 \textup{ inches }}) = \frac{1}{6} \textup{ foot}$

Write the formula for area of a circle and substitute the new radius.

$A=\pi r^2 = \pi (\frac{1}{6})^2 =\frac{\pi}{36}\textup{ feet}$

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Question

Find the area of a circle with the diameter of $\frac{1}{10}$ .

Answer

Write the formula to find the area of a circle.

$A=\pi r^2$

The radius is half the diameter.

$r=\frac{1}{2}d = \frac{1}{2} \times \frac{1}{10} = \frac{1}{20}$

Substitute the radius into the formula.

$A=\pi( \frac{1}{20})^2 = \frac{\pi}{400}$

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Question

Find the area of a circle with a diameter of $\pi$ .

Answer

Write the formula for the area of a circle.

$A=\pi r^2$

To obtain the radius, divide the diameter by

$r=\frac{d}{2} = \frac{\pi}{2}$

Substitute the radius into the area.

$A=\pi \left (\frac{\pi}{2} \right )^2 = \frac{\pi^3}{4}$

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Question

The diameter of a circle is 21. Which of the following is the closest area?

Answer

Divide the diameter by two to obtain the radius.

$r=\frac{d}{2} = \frac{21}{2} = 10.5$

Write the area of a circle.

$A=\pi r^2$

Substitute the radius and solve. Assume $\pi = 3.14$ .

The rounded answer is .

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Question

Find the area of a circle if the radius is 24.

Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius.

When multiplying two digit numbers together remember to do it in steps.

$24\ \underline{\times\ 24}$

First multiply $4\times 4=16$ , the six will stay in the ones place and the one will be carried to the tens place above the two.

${\color{Red} 2}4\ \underline{\times\ 2{\color{Red} 4}}$

Now multiply four and the two (highlighted in red above) then add the one that was carried over.

$2\times 4=8+1=9$

This number goes next to the six.

$2{\color{Red}4}\ \underline{\times\ {\color{Red} 2}4}$

Placing a zero in the ones places signifies that we are in our second step of multiplication. We will multiply the two and four (highlighted in red).

$2{\color{Red}4}\ \underline{\times\ {\color{Red} 2}4}$

Now we will multiply two and two,

${\color{Red}2}4\ \underline{\times\ {\color{Red} 2}4}$

From here add down to get the final product.

Therefore the final area is,

$A=\pi (24)^2 =576 \pi$ .

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Question

Find the area of the circle if the radius is .

Answer

Substitute the radius into the formula for the area of a circle.

$A=\pi r^2 = \pi (50 )^2 = 2500\pi$

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Question

Find the area of a circle if the radius is $2\sqrt2$ .

Answer

Write the formula for the area of a circle.

$A=\pi r^2$

Substitute the radius into the formula.

Remember when squaring the radius, square both terms. Also recall that when two square roots that have the same base are multiplied together, the square root sign disappears and you are left with the number underneath the square root sign.

$(2\sqrt{2})^2=2^2\cdot \sqrt2\cdot \sqrt2=4\cdot 2$

Therefore the area becomes,

$A=\pi (2\sqrt2)^2 = \pi (4\times 2) = 8\pi$ .

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Question

Determine the area of a circle with a radius of $4\pi$ .

Answer

Write the equation to find the area of a circle.

$A=\pi r^2$

Substitute the radius.

$A=\pi (4\pi)^2 = \pi (16\pi^2) = 16\pi^3$

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Question

Suppose the circle is inscribed inside a square touching all four sides of the square. The side length of the square is 2. What is the area of the circle?

Answer

Since the side of the square is 2, the diameter of the circle must also be 2. The radius of the circle is half the diameter, which would be 1.

Write the formula for the area of the circle.

$A=\pi r^2$

Substitute the radius.

$A=\pi r^2 = \pi (1)^2 = \pi$

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Question

Find the area of a circle if the circle has a perimeter of .

Answer

The circle's perimeter is the same as the circle's circumference.

Write the circumference formula for the circle.

$C=2\pi r$

Substitute the circumference and solve for the radius.

$6=2\pi r$

$r=\frac{3}{\pi}$

Write the formula for the area of a circle.

$A=\pi r^2$

$A=\pi (\frac{3}{\pi})^2 = \pi (\frac{9}{\pi^2}) =\frac{9}{\pi}$

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Question

Solve for the area of a circle if the radius is .

Answer

Write the formula to find the area of a circle.

$A=\pi r^2$

Substitute the radius.

$A=\pi (2x)^2 = 4\pi x^2$

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Question

Find the area of a circle with a diameter of .

Answer

Write the area of the circle.

$A=\pi r^2$

The radius is half the diameter.

$r= \frac{d}{2} = \frac{x}{2}$

Substitute the radius into the area formula.

Remember that when a fraction is squared, both the numerator and denominator are squared.

$A=\pi \left(\frac{x}{2}\right)^2 = \pi (\frac{x\cdot x}{2\cdot 2})=\frac{\pi x^2}{4}$

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