Geometry - How to find circumference

Find the circumference of a circle inscribed in a square that has a diagonal of $64\sqrt2$ .

$64\pi$

$32\pi$

$4096\pi$

$1024\pi$

Explanation

When you draw out the circle that is inscribed in a square, you should notice two things. The first thing you should notice is that the diagonal of the square is also the hypotenuse of a right isosceles triangle that has the side lengths of the square as its legs. The second thing you should notice is that the diameter of the circle has the same length as the length of one side of the square.

First, use the Pythagorean theorem to find the length of a side of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Substitute in the length of the diagonal to find the length of the square.

$\text{side}=\frac{64\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=64$

Now, recall the relationship between the diameter of the circle and the side of the square.

$\text{diameter}=\text{side}=64$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\pi\times\text{diameter}$

Substitute in the diameter you just found to find the circumference.

$\text{Circumference}=64\pi$

A circle with an area of 13_π_ in2 is centered at point C. What is the circumference of this circle?

2√13_π_

√13_π_

26_π_

√26_π_

13_π_

Explanation

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_.

Find the circumference of a circle that is inscribed in a square that has side lengths of .

$3\pi$

$\frac{3\pi}{2}$

$\frac{9\pi}{4}$

$6\pi$

Explanation

Notice that when a circle is inscribed in a square, the side length of the square is also the diameter of the circle.

Recall how to find the circumference of a circle:

$\text{Circumference}=\text{Diameter}\times\pi$

Plug in the given diameter to find the circumference.

$\text{Circumference}=3\pi$

The rectangle and the circle share a center, . Find the circumference of the circle.

$3\sqrt{17}\pi$

The circumference of the circle cannot be determined.

$36\pi$

$4\sqrt{10}\pi$

Explanation

Notice that the diagonal of the rectangle is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the diagonal.

$\text{length}^2+\text{width}^2=\text{diagonal}^2$

$\text{diagonal}=\sqrt{\text{length}^2+\text{width}^2}$

Substitute in the values of the length and the width to find the length of the diagonal.

$\text{diagonal}=\sqrt{3^2+12^2}$

Simplify.

$\text{diagonal}=\sqrt{153}$

Reduce.

$\text{diagonal}=3\sqrt{17}$

Now, recall the relationship between the diagonal and the diameter.

$\text{diagonal}=\text{diameter}=3\sqrt{17}$

Recall how to find the circumference of a circle:

$\text{Circumference}=\text{diameter}\times\pi$

Substitute in the value of the diameter to find the circumference.

$\text{Circumference}=3\sqrt{17}\pi$

Find the circumference of a circle given the radius is 3.

$6\pi$

$9\pi$

$12\pi$

$18\pi$

Explanation

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

Given that the radius is 3, substitute 3 in for the r in the circumference formula below.

Thus,

$\C=2\pi{r}\C=2*\pi*3\C=6\pi$ .

Find the circumference of a circle that is inscribed in a square that has side lengths of .

$4\pi$

$8\pi$

$16\pi$

$2\pi$

Explanation

Notice that when a circle is inscribed in a square, the side length of the square is also the diameter of the circle.

Recall how to find the circumference of a circle:

$\text{Circumference}=\text{Diameter}\times\pi$

Plug in the given diameter to find the circumference.

$\text{Circumference}=4\pi$

If a circle has an area of $16\pi ; in^{2}$ , what is the circumference?

$8\pi ; in$

$12\pi ; in$

$4\pi ; in$

$6\pi ; in$

$10\pi ; in$

Explanation

For a circle, the formula for area is $A = \pi r^{2}$ and the formula for circumference is $C=\pi d$ , where is the radius and is the diameter.

Plug the known quantities into the area formula and solve for the radius:

$16\pi =\pi r^{2}$

Now plug this value into the circumference formula to solve:

$C = \pi d = \pi 2r = \pi 2(4) = 8\pi; in$

If a rectangle with a diagonal of $4\sqrt{10}$ is inscribed in a circle, what is the circumference of the circle?

$4\sqrt{10}\pi$

$8\sqrt{10}\pi$

$160\pi$

$80\pi$

Explanation

Notice that the diagonal of the rectangle is the same as the diameter of the circle.

$\text{Diameter}=\text{Diagonal}=4\sqrt{10}$

Now, recall how to find the circumference of a circle.

$\text{Circumference}=\text{diameter}\times\pi$

Substitute in the given diameter to find the circumference.

$\text{Circumference}=4\sqrt{10}\pi$

Find the circumference of the circle if the lengths of the legs of the inscribed isosceles triangle are $5\sqrt2$ .

$10\pi$

$20\pi$

$25\pi$

$30\pi$

Explanation

Notice that the hypotenuse of the triangle in the figure is also the diameter of the circle.

Use the Pythagorean theorem to find the length of the hypotenuse.

$\text{Hypotenuse}^2=\text{leg}^2+\text{leg}^2$

$\text{Hypotenuse}^2=2(\text{leg}^2)$

$\text{Hypotenuse}=\sqrt{2(\text{leg}^2)}=\text{leg}\sqrt2$

Substitute in the length of triangle’s legs to find the missing length of the hypotenuse.

$\text{Hypotenuse}=5\sqrt2(\sqrt2)$

Simplify.

$\text{Hypotenuse}=10$

Now, recall that the hypotenuse of the triangle and the diameter of the circle are the same:

$\text{Hypotenuse}=\text{Diameter}=10$

Now, recall how to find the circumference of a circle:

$\text{Circumference}=\text{Diameter}\times\pi$

Substitute in the value for the diameter to find the circumference of the circle.

$\text{Circumference}=10\pi$

If a circle has an area of $81\pi$ , what is the circumference of the circle?

$9\pi$

$81\pi$

$18\pi$

$27\pi$

$90\pi$

Explanation

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.

How to find circumference

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