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A 12x16 rectangle is inscribed in a circle. What is the area of the circle?
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.
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If the radius of a circle is tripled, and the new area is 144π what was the diameter of the original circle?
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.
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If the equation of a circle is (x – 7)2 + (y + 1)2 = 81, what is the area of the circle?
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.
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If the radius of Circle A is three times the radius of Circle B, what is the ratio of the area of Circle A to the area of Circle B?
We know that the equation for the area of a circle is π r2. To solve this problem, we pick radii for Circles A and B, making sure that Circle A’s radius is three times Circle B’s radius, as the problem specifies. Then we will divide the resulting areas of the two circles. For example, if we say that Circle A has radius 6 and Circle B has radius 2, then the ratio of the area of Circle A to B is: (π 62)/(π 22) = 36π/4π. From here, the π's cancel out, leaving 36/4 = 9.
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Area of a circle = A = πr2
R = 1/2d = ½(10) = 5
A = 52π = 25π
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A circle with center (8, **–**5) is tangent to the y-axis in the standard (x,y) coordinate plane. What is the radius of this circle?
For the circle to be tangent to the y-axis, it must have its outer edge on the axis. The center is 8 units from the edge.
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A circle has an area of . What is the radius of the circle, in inches?
We know that the formula for the area of a circle is πr_2. Therefore, we must set 49_π equal to this formula to solve for the radius of the circle.
49_π_ = _πr_2
49 = _r_2
7 = r
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Find the circumference of a circle with radius 6.
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,
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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?
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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If a circle has a circumference of 16π, what would its area be if its radius were halved?
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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A circle with an area of 13_π_ in2 is centered at point C. What is the circumference of this circle?
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_.
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A 6 by 8 rectangle is inscribed in a circle. What is the circumference of the circle?
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_.
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Ashley has a square room in her apartment that measures 81 square feet. What is the circumference of the largest circular area rug that she can fit in the space?
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
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A gardener wants to build a fence around their garden shown below. How many feet of fencing will they need, if the length of the rectangular side is 12 and the width is 8?
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.
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If a circle has an area of , what is the circumference of the circle?
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.
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A car tire has a radius of 18 inches. When the tire has made 200 revolutions, how far has the car gone in feet?
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.
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A circle has the equation below. What is the circumference of the circle?
(x – 2)2 + (y + 3)2 = 9
The radius is 3. Yielding a circumference of .
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The diameter of a circle is defined by the two points (2,5) and (4,6). What is the circumference of this circle?
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.
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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?
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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What is the circumference of a cirle with a radius of seven?
Leave your answer in terms of .
Plug the radius into the circumference formula:
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