Disk
Key Formula
D={(x,y)∈R2:(x−h)2+(y−k)2≤r2}
Where:
- (h,k) = Center of the disk
- r = Radius of the disk
- (x,y) = Any point in the plane belonging to the disk
Worked Example
Problem: Determine whether the point (1, 2) lies inside the disk centered at the origin with radius 3.
Step 1: Write the condition for the disk. A point lies in a disk of radius 3 centered at the origin if its distance from the origin is at most 3.
x2+y2≤32=9
Step 2: Substitute the coordinates of the point (1, 2) into the left side.
12+22=1+4=5
Step 3: Compare the result to 9. Since 5 ≤ 9, the inequality holds.
5≤9✓
Answer: Yes, the point (1, 2) lies inside the disk because its squared distance from the origin (5) is less than or equal to the squared radius (9).
Another Example
Problem: Find the area of a disk with radius 5.
Step 1: Recall that the area of a disk equals π times the square of its radius.
A=πr2
Step 2: Substitute r = 5.
A=π(5)2=25π
Answer: The area of the disk is 25π≈78.54 square units.
Frequently Asked Questions
What is the difference between a disk and a circle?
A circle is only the curved boundary — the set of points at exactly a fixed distance from the center. A disk includes everything inside that boundary as well. Think of a circle as a ring and a disk as a filled-in plate.
What is the difference between an open disk and a closed disk?
A closed disk uses ≤ in its defining inequality, so it includes the boundary circle. An open disk uses < (strict inequality), so it contains only the interior points and excludes the boundary. When people say 'disk' without qualification, they usually mean the closed disk.
Disk vs. Circle
A circle is a one-dimensional curve (it has no area), while a disk is a two-dimensional region (it has area πr2). The circle is the boundary of the disk; the disk is the circle together with its interior.
Why It Matters
The concept of a disk appears throughout mathematics and physics. In calculus, double integrals over circular regions are integrals over disks. In topology, the open disk serves as a fundamental example of an open set in the plane and is used to define neighborhoods of points.
Common Mistakes
Mistake: Using 'circle' and 'disk' interchangeably.
Correction: A circle is just the boundary curve; a disk is the entire filled-in region. When you refer to the area enclosed by a circle, you are really talking about a disk.
Mistake: Forgetting the distinction between open and closed disks.
Correction: A closed disk (≤) includes every point on the boundary circle, while an open disk (<) does not. This matters especially in analysis and topology where boundary behavior is important.
Related Terms
- Circle — The boundary curve of a disk
- Interior — The set of points strictly inside the disk
- Union — Disk is the union of circle and interior
- Radius — Distance from center defining the disk's size
- Area of a Circle — Area formula πr² applies to the disk region
- Sphere — Three-dimensional analogue of a circle
- Ball — Three-dimensional analogue of a disk