Diameter and Chords

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Math › Diameter and Chords

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What is the ratio of any circle's circumference to its radius?

$2\pi :1$

$\pi :1$

$\pi ^{2}:1$

Undefined.

Explanation

The circumference of any circle is

$2\pi*r$

So the ratio of its circumference to its radius r, is

$\frac{2\pi*r}{r}$

$=2\pi$

What is the ratio of the diameter and circumference of a circle?

$\pi$

$2\pi$

$\frac{\pi}{2}$

Explanation

To find the ratio we must know the equation for the circumference of a circle is

$Circumference=\pi d$

Once we know the equation we can solve for the ratio of the diameter to circumference by solving the equation for $\frac{diameter}{Circumference}$

we divide both sides by the circumference giving us

$\frac{d}{\pi d}=\frac{1}{\pi }$

We now know that the ratio of the diameter to circumference is equal to .

What is the ratio of the diameter and circumference of a circle?

$\pi$

$2\pi$

$\frac{\pi}{2}$

Explanation

To find the ratio we must know the equation for the circumference of a circle is

$Circumference=\pi d$

Once we know the equation we can solve for the ratio of the diameter to circumference by solving the equation for $\frac{diameter}{Circumference}$

we divide both sides by the circumference giving us

$\frac{d}{\pi d}=\frac{1}{\pi }$

We now know that the ratio of the diameter to circumference is equal to .

Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?

$6.25\pi$

$6\pi$

$5\pi$

$25\pi$

$12.5\pi$

Explanation

For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.

The equation for the area of a circle is A = πr2.

$A=\pi (5/2)^2=6.25\pi$

What is the ratio of any circle's circumference to its radius?

$2\pi :1$

$\pi :1$

$\pi ^{2}:1$

Undefined.

Explanation

The circumference of any circle is

$2\pi*r$

So the ratio of its circumference to its radius r, is

$\frac{2\pi*r}{r}$

$=2\pi$

Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?

$6.25\pi$

$6\pi$

$5\pi$

$25\pi$

$12.5\pi$

Explanation

For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.

The equation for the area of a circle is A = πr2.

$A=\pi (5/2)^2=6.25\pi$

The perimeter of a circle is 36 π. What is the diameter of the circle?

Explanation

The perimeter of a circle = 2 πr = πd

Therefore d = 36

The perimeter of a circle is 36 π. What is the diameter of the circle?

Explanation

The perimeter of a circle = 2 πr = πd

Therefore d = 36

What is the diameter of a circle with a circumference of $16\pi$ ?

Explanation

To find the diameter we must understand the diameter in terms of circumference. The equation for the circumference of a circle is $C=D\pi$ , where is the circumference and is the diameter. The circumference is equal to the diameter multiplied by $\pi$ .

We can rearrange $C=D\pi$ to solve for .

$D=\frac{C}{\pi}$

All we have to do is plug in the circumference and divide by $\pi$ , and it will yield the diameter.

$D=\frac{16\pi}{\pi}=16$

The $\pi$ s cancel and the diameter is .

Sat_math_picture

If the area of the circle touching the square in the picture above is $81\pi$ , what is the closest value to the area of the square?

Explanation

Obtain the radius of the circle from the area.

$A=\pi r^2=81\pi$

Split the square up into 4 triangles by connecting opposite corners. These triangles will have a right angle at the center of the square, formed by two radii of the circle, and two 45-degree angles at the square's corners. Because you have a 45-45-90 triangle, you can calculate the sides of the triangles to be , , and $x\sqrt{2}$ . The radii of the circle (from the center to the corners of the square) will be 9. The hypotenuse (side of the square) must be $9\sqrt{2}$ .