How is the radius calculated using its circumference? - Examples, Exercises and Solutions

The formula for the circumference is equal to:

$2\pi r$

We use the formula:$P=2\pi r$

We replace the data in the formula:$P=2\times8\pi$

$P=16\pi$

Formula of the circumference:

$P=2\pi r$

We insert the given data into the formula:

$P=2\times6\times\pi$

$P=12\pi$

The radius is a straight line that extends from the center of the circle to its outer edge.

The radius is essential for calculating the circumference of the circle, which can be found using the following formula:

If we substitute in the radius we have, the formula will be:

2*π*2/3

To solve this, first we'll rearrange the formula like so:

π*2*2/3 =

We'll then multiply the fraction by the whole number:

π*(2*2)/3 =

π*4/3 =

4/3π

To calculate, we will use the formula:

$\frac{P}{2r}=\pi$

Pi is the ratio between the circumference of the circle and the diameter of the circle.

The diameter is equal to 2 radii.

Let's substitute the given data into the formula:

$\frac{8}{4}=\pi$

$2\ne\pi$

Therefore, this situation is not possible.

To solve the exercise, first we must remember the circumference formula:

$P= 2\pi R$

P is the circumference and Pi has a value of 3.14 (approximately).

We substitute in the known data:

$31.41=2\cdot3.141\cdot R$

Keep in mind that the result can be easily simplified using Pi:

$\frac{31.41}{3.141}=2R$

$10=2R$

Finally, we simplify by 2:

$5=R$

We use the formula:

$P=2\pi r$

We insert the known data into the formula:

$50.25=3.14\times2r$

$50.25=2\times r\times3.14$

$50.25=6.28r$

$\frac{50.25}{6.28}=\frac{6.28r}{6.28}$

$r=8$

We begin by using the formula:

$P=2\pi r$

We then insert the given data into the formula:

$14=2\times\pi\times r$

Lastly we divide Pi by 2:

$\frac{14}{2\pi}=\frac{2\pi r}{2\pi}$

$\frac{7}{\pi}=r$

To solve the problem of finding the area of the trapezoid with a semicircle on its top base, we follow these steps:

• Step 1: Identify the semicircle's radius from the given circumference.
• Step 2: Find the upper base length, which is the diameter of the semicircle.
• Step 3: Calculate the trapezoid's area using the area formula.

Let's work through each step:

Step 1: The given length of the highlighted segment is $7\pi$, which is the half-circumference of a circle (since it's a semicircle). The formula for the circumference of a full circle is $2\pi r$, so for a semicircle, it is $\pi r$. Setting this equal to the length given:

Canceling $\pi$ from both sides, we find:

Step 2: The diameter of the semicircle is twice the radius, hence:

This diameter also serves as the length of the upper base of the trapezoid.

Step 3: We use the formula for the area of a trapezoid:

Substitute the known values ($\text{Base}_1 = 14$, $\text{Base}_2 = 18$, $\text{Height} = 7$):

Thus, the area of the trapezoid is 112 cm$^2$.

First, we add letters as reference points:

Let's observe points A and B.

We know that two tangent lines to a circle that start from the same point are parallel to each other.

Therefore:

$AE=AF=3$
$BG=BF=6$

And from here we can calculate:

$AB=AF+FB=3+6=9$

Now we need the height of the parallelogram.

We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.

It is also known that the diameter is equal to two radii.

Since the circumference is 25.13.

Circumference formula:$2\pi R$
We replace and solve:

$2\pi R=25.13$
$\pi R=12.565$
$R\approx4$

The height of the parallelogram is equal to two radii, that is, 8.

And from here you can calculate with a parallelogram area formula:

$AlturaXLado$

$9\times8\approx72$

Let's draw the center of the circle and we can divide the diameter of the circle into two equal radii

Now let's calculate the length of the radii as follows:

$7\times2r=28x$

$14r=28x$

We'll divide both sides by 14:

$r=\frac{28}{14}x$

$r=2x$

Let's calculate the circumference of the circle:

$P=2\pi\times r=2\pi\times2x=4\pi x$

To determine which has the larger measurement, the triangle's perimeter or the circle's circumference, we need to compute both values.

Step 1: Calculate the perimeter of the Triangle
We are given two sides of the triangle: 6 and 5. Since it's implied to be a right triangle, we apply the Pythagorean theorem to find the third side, the hypotenuse $c$:

The perimeter $P$ of the triangle is:

Step 2: Calculate the circumference of the Circle
The circumference $C$ of a circle with radius $r$ is given by the formula:

Assuming the radius of the circle is equivalent to the '6' mentioned for the green line in the SVG:

Step 3: Compare the Triangle's Perimeter and the Circle's Circumference
We compare the values:

• Perimeter of the Triangle: $P \approx 18.81$
• Circumference of the Circle: $C \approx 37.7$

The circumference of the circle (37.7) is greater than the perimeter of the triangle (18.81).

Therefore, the circle has the larger measurement.

Conclusion: The circle has the larger perimeter.

To solve this problem, we'll go through the following steps:

• Calculate the circumference of the circular park.
• Determine the total distance Ivan runs.
• Find Ivan's average speed.

Step 1: Calculate the circumference of the circular park.
The formula for the circumference of a circle is $C = 2\pi r$, where $r$ is the radius.
Given $r = 300$ meters, $C = 2\pi \times 300 = 600\pi$ meters.

Step 2: Determine the total distance Ivan runs.
Ivan completes 5 laps, so the total distance $D$ is given by:
$D = 5 \times 600\pi = 3000\pi$ meters.

Step 3: Find Ivan's average speed.
The formula for average speed is $v = \frac{\text{total distance}}{\text{total time}}$.
Total time = 35 minutes.
Average speed $v = \frac{3000\pi}{35} \approx 269.14$ meters per minute.

Therefore, Ivan's average speed is $269.14$ meters per minute.

First, we add letters as reference points:

Let's observe points A and B.

We know that two tangent lines to a circle that start from the same point are parallel to each other.

Therefore:

$AE=AF=3$
$BG=BF=6$

From here we can calculate:

$AB=AF+FB=3+6=9$

Now we need the height of the parallelogram.

We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.

It is also known that the diameter is equal to two radii.

It is known that the circumference of the circle is 25.13.

Formula of the circumference:$2\pi R$
We replace and solve:

$2\pi R=25.13$
$\pi R=12.565$
$R\approx4$

The height of the parallelogram is equal to two radii, that is, 8.

And from here it is possible to calculate the area of the parallelogram:

$\text{Lado }x\text{ Altura}$$9\times8\approx72$

Now, we calculate the area of the circle according to the formula:$\pi R^2$

$\pi4^2=50.26$

Now, subtract the area of the circle from the surface of the trapezoid to get the answer:

$72-56.24\approx21.73$