How to find the area of a circle - Math

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Question

Circle B has a radius $\frac{2}{3}$ as long as that of Circle A.

Which is the greater quantity?

(a) The area of Circle A

(b) Twice the area of Circle B

Answer

If we call the radius of Circle A , then the radius of Circle B is $$\frac{2}{3}$ r$ .

The areas of the circles are:

(a) $\pi $r^{2}$$

(b) $\pi \left( $\frac{2}{3}$ r \right ) ^{2} = $\frac{4}{9}$ \pi $r^{2}$$

Twice the area of Circle B is

$2 \cdot \pi \left( $\frac{2}{3}$ r \right ) ^{2} =2 \cdot $\frac{4}{9}$ \pi $r^{2}$ = $\frac{8}{9}$ \pi $r^{2}$ < \pi $r^{2}$$ ,

making (a) the greater number.

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Question

Circle 1 is inscribed inside a square. The square is inscribed inside Circle 2.

Which is the greater quantity?

(a) Twice the area of Circle 1

(b) The area of Circle 2

Answer

If the radius of Circle 1 is , then the square will have sidelength equal to the diameter of the circle, or . Circle 2 will have as its diameter the length of a diagonal of the square, which by the $45 ^{\circ }-45 ^{\circ $}-90^{\circ }$$ Theorem is $\sqrt{2}$ times that, or $2r $\sqrt{2}$$ . The radius of Circle 2 will therefore be half that, or $r $\sqrt{2}$$ .

The area of Circle 1 will be $A = \pi $r^{2}$$ . The area of Circle 2 will be $A = \pi \left ( r$\sqrt{2}$ \right ) ^{2} = \pi \left ( $r^{2}$ \cdot 2 \right ) = 2 \pi $r^{2}$$ , twice that of Circle 1.

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Question

Compare the two quantities:

Quantity A: The area of a circle with radius

Quantity B: The circumference of a circle with radius

Answer

Recall for this question that the formulae for the area and circumference of a circle are, respectively:

$A=\pi * $r^2$$

$C=2\pir$

For our two quantities, we have:

Quantity A: $\pi * $13^2$ = 169\pi$

Quantity B: $2 * \pi * 63 = 126\pi$

Therefore, quantity A is greater.

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Question

The radius of a circle is . Give the area of the circle in terms of .

Answer

The area of a circle with radius can be found using the formula

$A = \pi $r^{2}$$

Since , the area is

$A = \pi (x- $5)^{2}$$

$A = \pi $(x^{2}$ - 2 \cdot x \cdot 5 + $5^{2}$)$

$A = \pi $(x^{2}$ - 10 x + 25)$

$A = \pi $x^{2}$ - 10 \pi x +25 \pi$

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Question

The radius of a circle is . Give the circumference of the circle in terms of .

Answer

The circumference of a circle is $2 \pi$ times its radius. Therefore, since the radius is , the circumference is

$2 \pi \left (x - 5 \right ) = 2 \pi x - 2 \pi \cdot 5 = 2 \pi x - 10 \pi$

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Question

The radii of six circles form an arithmetic sequence. The radius of the second-smallest circle is twice that of the smallest circle. Which of the following, if either, is the greater quantity?

(a) The area of the largest circle

(b) Twice the area of the third-smallest circle

Answer

Call the radius of the smallest circle . The radius of the second-smallest circle is then , and the common difference of the radii is .

The radii of the six circles are, from least to greatest:

The largest circle has area

The third-smallest circle has area:

Twice this is

$2 $A_{3}$ = 2 \cdot 9 \pi $r^{2}$ = 18 \pi $r^{2}$$

The area of the sixth circle is greater than twice that of the third-smallest circle, so the correct choice is that (a) is greater.

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Question

The areas of six circles form an arithmetic sequence. The second-smallest circle has a radius twice that of the smallest circle.

Which is the greater quantity?

(a) The area of the largest circle.

(b) Twice the area of the third-largest circle.

Answer

Let be the radius of the smallest circle. Then the second-smallest circle has radius . Their areas, respectively, are

and

The areas form an arithmetic sequence, so their common difference is

$4 \pi $r^{2}$ - \pi $r^{2}$ = 3 \pi $r^{2}$$ .

The six areas are

$\pi r ^{2}, 4 \pi r ^{2}, 7 \pi r ^{2}, 10 \pi r ^{2}, 13 \pi r ^{2}, 17 \pi r ^{2}$

The third-largest circle has area $10 \pi r ^{2}$ ; twice this is $20 \pi r ^{2}$ . This is greater than the area of the largest circle, which is $17\pi r ^{2}$ . (b) is the greater quantity.

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Question

In the above figure, .

Which is the greater quantity?

(a) Twice the area of inner gray ring

(b) The area of the white ring

Answer

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi $r^{2}$$ :

$A _{1}= \pi $r^{2}$ = \pi \cdot 1 ^{2} = \pi$

$A _{2}= \pi $r^{2}$ = \pi \cdot $2^{2}$ =4 \pi$

$A _{3}= \pi $r^{2}$ = \pi \cdot $3^{2}$ =9 \pi$

$A _{4}= \pi $r^{2}$ = \pi \cdot $4^{2}$ =16 \pi$

The white ring has as its area the difference of the areas of the second-largest and third-largest circles:

The inner gray ring has as its area the difference of the areas of the third-largest and smallest circles:

.

Twice this is $2 \cdot 3 \pi = 6 \pi$ , which is greater than the area of the white ring.

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Question

In the above figure, .

Which is the greater quantity?

(a) Six times the area of the white circle

(b) The area of the outer ring

Answer

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi $r^{2}$$ :

$A _{1}= \pi $r^{2}$ = \pi \cdot 1 ^{2} = \pi$

$A _{2}= \pi $r^{2}$ = \pi \cdot $2^{2}$ =4 \pi$

$A _{3}= \pi $r^{2}$ = \pi \cdot $3^{2}$ =9 \pi$

$A _{4}= \pi $r^{2}$ = \pi \cdot $4^{2}$ =16 \pi$

The outer gray ring is the region between the largest and second-largest circles, and has area

$A _{4} - A _{3} = 16 \pi - 9 \pi = 7 \pi$

Six times the area of the white (inner) circle is $6 $A_{1}$ = 6 \pi$ , which is less than the area of the outer ring, $7 \pi$ .

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Question

A star is inscribed in a circle with a diameter of 30, given the area of the star is 345, find the area of the shaded region, rounded to one decimal.

Answer

The area of the circle is (30/2)2*3.14 (π) = 706.5, since the shaded region is simply the area difference between the circle and the star, it’s 706.5-345 = 361.5

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Question

If a circle has circumference $8\pi$ , what is its area?

Answer

If the circumference is $8\pi$ , then since $C=2\pi r$ we know . We further know that $A=\pi r2$ , so $A=\pi $(4)^2$=16\pi.$

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Question

The diameter of a circle increases by 100 percent. If the original area is 16π, what is the new area of the circle?

Answer

The original radius would be 4, making the new radius 8 and by the area of a circle (A=π(r)2) the new area would be 64π.

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Question

Two equal circles are cut out of a rectangular sheet of paper with the dimensions 10 by 20. The circles were made to have the greatest possible diameter. What is the approximate area of the paper after the two circles have been cut out?

Answer

The length of 20 represents the diameters of both circles. Each circle has a diameter of 10 and since radius is half of the diameter, each circle has a radius of 5. The area of a circle is A = πr2 . The area of one circle is 25π. The area of both circles is 50π. The area of the rectangle is (10)(20) = 200. 200 - 50π gives you the area of the paper after the two circles have been cut out. π is about 3.14, so 200 – 50(3.14) = 43.

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Question

Find the area of a circle that has a radius of .

Answer

Use the following formula to find the area of a circle:

$\text{Area}$=\pi \times $$\text{radius}$^2$

For the circle in question, plug in the given radius to find the area.

$$\text{Area}$=\pi \times $(96)^2$=9216\pi$

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Question

Find the area of a circle given radius 4.

Answer

To solve, simply use the formula for the area of a circle. Thus,

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Question

A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?

Answer

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

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Question

If a circle has a circumference of 16π, what would its area be if its radius were halved?

Answer

The circumference of a circle = πd where d = diameter. Therefore, this circle’s diameter must equal 16. Knowing that diameter = 2 times the radius, we can determine that the radius of this circle = 8. Halving the radius would give us a new radius of 4. To find the area of this new circle, use the formula A=πr² where r = radius. Plug in 4 for r. Area will equal 16π.

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Question

Kate has a ring-shaped lawn which has an inner radius of 10 feet and an outer radius 25 feet. What is the area of her lawn?

Answer

The area of an annulus is

$\pi $(R^{2}$$-r^{2}$)$

where is the radius of the larger circle, and is the radius of the smaller circle.

$\pi $(25^{2}$$-10^{2}$)$

$\pi (625-100)$

$525\pi$

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Question

A 6 by 8 rectangle is inscribed in a circle. What is the area of the circle?

Answer

The image below shows the rectangle inscribed in the circle. Dividing the rectangle into two triangles allows us to find the diameter of the circle, which is equal to the length of the line we drew. Using a2+b2= c2 we get 62 + 82 = c2. c2 = 100, so c = 10. The area of a circle is $A=\pi $r^2$$ . Radius is half of the diameter of the circle (which we know is 10), so r = 5.

$A=\pi $*5^2$=25\pi$

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Question

A circle with a diameter of 6” sits inside a circle with a radius of 8”. What is the area of the interstitial space between the two circles?

Answer

The area of a circle is πr2.

The diameter of the first circle = 6” so radius of the first circle = 3” so the area = π * 32 = 9π in2

The radius of the second circle = 8” so the area = π * 82 = 64π in2

The area of the interstitial space = area of the first circle – area of the second circle.

Area = 64π in2 - 9π in2 = 55π in2

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