The circumference of a circle is 20 pi cm. Find diameter, radius, and area of the disk, the length of an arc of 90 angle and area of a 90 angle sector. Answer in terms of pi. d= ___ r= _____,
disk area= _______ arc length =______, sector area =________
To find the diameter, radius, and area of a disk given the circumference, we can use the following formulas:
1. Diameter (d) = Circumference / π = 20π / π = 20 cm
2. Radius (r) = Diameter / 2 = 20 cm / 2 = 10 cm
Now, let's move on to finding the area of the disk. The formula for the area of a circle is:
3. Area of Circle = π * (radius)^2
Substituting the value of the radius, we have:
Area of Circle = π * (10 cm)^2 = 100π cm^2
Next, let's calculate the length of an arc of a 90-degree angle. The formula for the length of an arc is:
4. Arc Length = (angle / 360°) * 2π * radius
Since we have a 90-degree angle, the formula becomes:
Arc Length = (90° / 360°) * 2π * 10 cm = (1/4) * 20π cm = 5π cm
Lastly, let's find the area of a 90-degree sector. The formula for the area of a sector is:
5. Sector Area = (angle / 360°) * π * (radius)^2
Using the given angle of 90 degrees, the formula becomes:
Sector Area = (90° / 360°) * π * (10 cm)^2 = (1/4) * 100π cm^2 = 25π cm^2
To summarize the answers in terms of π:
d = 20 cm
r = 10 cm
Disk Area = 100π cm^2
Arc Length = 5π cm
Sector Area = 25π cm^2
Given that the circumference of a circle is 20π cm, we can use the formula for circumference to find the diameter.
Circumference = πd
20π = πd
Dividing both sides of the equation by π, we get:
20 = d
So, the diameter (d) of the circle is 20 cm.
Now, let's find the radius (r) of the circle. The radius is half the diameter, so:
r = d/2
r = 20/2
r = 10 cm
Next, let's find the area of the disk. The formula for the area of a circle is given by:
Area = πr^2
Area = π(10)^2
Area = π(100)
Area = 100π sq. cm
The length of an arc of a 90-degree angle is one-fourth of the circumference. Since the circumference is 20π cm, the length of the arc would be:
Arc Length = (1/4) * Circumference
Arc Length = (1/4) * 20π
Arc Length = 5π cm
Finally, let's find the area of a 90-degree sector. The formula for the area of a sector is given by:
Sector Area = (θ/360) * πr^2
Where θ is the angle of the sector.
For a 90-degree sector:
Sector Area = (90/360) * π(10)^2
Sector Area = (1/4) * 100π
Sector Area = 25π sq. cm
So, the answers are:
d = 20 cm
r = 10 cm
Disk Area = 100π sq. cm
Arc Length = 5π cm
Sector Area = 25π sq. cm
90° is 1/4 of a circle. So, what's 1/4 of the circumference or area?
c = πd, so d = c/π
c = 2πr, so r = c/2π
a = πr^2 = π*(c/2π)^2 = c^2/4π
Now just plug in c where you need it.