Radius - ACT Math

The formula for the area of a circle is A = _πr_2.

We are given the area, and by substitution we know that 13_π_ = _πr_2.

We divide out the π and are left with 13 = _r_2.

We take the square root of r to find that r = √13.

We find the circumference of the circle with the formula C = 2_πr_.

We then plug in our values to find C = 2√13_π_.

First you must draw the diagram. The diagonal of the rectangle is also the diameter of the circle. The diagonal is the hypotenuse of a multiple of 2 of a 3,4,5 triangle, and therefore is 10.
Circumference = π * diameter = 10_π_.

In order to solve this question, first calculate the length of each side of the room.

The length of each side of the room is also equal to the length of the diameter of the largest circular rug that can fit in the room. Since , the circumference is simply

The shape of the garden consists of a rectangle and two semi-circles. Since they are building a fence we need to find the perimeter. The perimeter of the length of the rectangle is 24. The perimeter or circumference of the circle can be found using the equation C=2π(r), where r= the radius of the circle. Since we have two semi-circles we can find the circumference of one whole circle with a radius of 4, which would be 8π.

The formula for the area of a circle is πr2. For this particular circle, the area is 81π, so 81π = πr2. Divide both sides by π and we are left with r2=81. Take the square root of both sides to find r=9. The formula for the circumference of the circle is 2πr = 2π(9) = 18π. The correct answer is 18π.

If the radius is 18 inches, the diameter is 3 feet. The circumference of the tire is therefore 3π by C=d(π). After 200 revolutions, the tire and car have gone 3π x 200 = 600π feet.

The radius is 3. Yielding a circumference of .

We first must calculate the distance between these two points. Recall that the distance formula is:√((x2 - x1)2 + (y2 - y1)2)

For us, it is therefore: √((4 - 2)2 + (6 - 5)2) = √((2)2 + (1)2) = √(4 + 1) = √5

If d = √5, the circumference of our circle is πd, or π√5.

To find the circumference given the radius, simply double the radius and mulitply by .

Thus:

To find the circumference of a circle we use the formula

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In order to solve we must use the given area to find the radius. Area of a circle has a formula of

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So we manipulate that formula to solve for the radius.

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Then we plug in our given area.

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Now we plug our radius into the circumference equation to get the final answer.

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To find the circumference of a circle we use the formula

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In order to solve we must use the given area to find the radius. Area of a circle has a formula of

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So we manipulate that formula to solve for the radius.

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Then we plug in our given area.

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Now we plug our radius into the circumference equation to get the final answer.

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The questions really asks us to find the perimeter of the track.This would be the two long sides of the rectangle plus the circumferences of each of the two semicircles.

Each long side of the triangle is .

Each semicircle has a circumference that is because the whole circumference is

thus the semicircle circumference would be half of that or,

and .

Since the diameter of the semicircle is equal to the width of the rectangle, . That means each semicircle has a circumference of . Since we have two semicircles on the track, we can sum the two semicircle circumferences and the two long sides of the rectangle that comprise the track to get .

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

In this particular case the radius of 6 should be substituted into the following equation to solve for the circumference.

Thus,

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.

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.

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

Explanation: Visualizing the rectangle inside the circle (corners touching the circumference of the circle and the center of the rectangle is the center of the circle) you will see that the rectangle can be divided into 8 congruent right triangles, with the hypotenuse as the radius of the circle. Calculating the radius you divide each side of the rectangle by two for the sides of each right triangle (giving 6 and 8). The hypotenuse (by pythagorean theorem or just knowing right triangle sets) the hypotenuse is give as 10. Area of a circle is given by πr2. 102 is 100, so 100π is the area.

The area of a circle is A=πr2. Since the radius was tripled 144π =π(3r)2. Divide by π and then take the square root of both sides of the equal sign to get 12=3r, and then r=4. The diameter (d) is equal to twice the radius so d= 2(4) = 8.

The equation is already in a circle equation, and the right side of the equation stands for r2 → r2 = 81 and r = 9

The area of a circle is πr2, so the area of this circle is 81π.