# Circles in Geometry — Parts, Formulas, and Examples

## TL;DR
A circle is the set of all points in a plane that sit the same distance from a fixed center, and that fixed distance is the radius. This hub walks through every part of a circle — radius, diameter, chord, secant, tangent, arc, sector, segment — and the two formulas that do most of the work: 
circumference C=2πr and area A=πr².

## What Is A Circle?
A circle is the set of all points in a plane that are an equal distance from a fixed point called the **center**. That equal distance is the **radius**. Nothing about a circle is arbitrary — the whole shape follows from this one equidistant rule.

A circle is a closed, two-dimensional curve. It has no straight sides and no corners. Because every boundary point obeys the same distance rule, a circle is also the most symmetric shape in plane geometry: it looks identical after any rotation about its center.

## What Are The Parts of a Circle?
Most circle problems are really problems about one specific part. Here is each part, defined once, in the order you meet them.

- **Center:** the fixed point all distances are measured from, usually labelled O.
- **Radius (r):** a segment from the center to any point on the circle. Every radius of a given circle has the same length. Read more on the [radius of a circle](/content/math/geometry/radius/index.html).
- **Diameter (d):** a chord that passes through the center; it is the longest chord and equals twice the radius, d=2r. See [diameter of a circle](/content/math/geometry/diameter-of-a-circle/index.html).
- **Chord:** a segment whose two endpoints both lie on the circle. A diameter is a special chord. More on [chords of a circle](/content/math/geometry/chords-of-a-circle/index.html).
- **Secant:** a straight line that cuts the circle at two points and keeps going past them — a chord extended in both directions. See [secant of a circle](/content/math/geometry/secant-of-a-circle/index.html).
- **Tangent:** a straight line that touches the circle at exactly one point and is perpendicular to the radius drawn to that point. More on the [tangent](/content/math/geometry/tangent/index.html).
- **Arc:** a connected portion of the circle's boundary. See [arc length](/content/math/geometry/arc-length/index.html).
- **Sector:** the pie-slice region enclosed by two radii and the arc between them. More on the [sector of a circle](/content/math/geometry/sector-of-a-circle/index.html).
- **Segment:** the region cut off by a chord and the arc above it.
- **Circumference:** the length of the circle's boundary — its perimeter.

## How Do You Find The Circumference And Area of a Circle?
Two formulas carry most circle calculations. Both are built from the radius and from π (pi), the constant ratio of any circle's circumference to its diameter, roughly 3.14159.

Circumference is the distance once around the circle:
C=2πr=πd

Area is the space the circle encloses:
A=πr²

| Symbol | Meaning | Units |
| --- | --- | --- |
| r | Radius (center to edge) | length (cm, m) |
| d | Diameter (d=2r) | length (cm, m) |
| C | Circumference (boundary length) | length (cm, m) |
| A | Area (enclosed region) | square units (cm², m²) |
| π | Ratio C/d, about 3.14159 | none |

## Examples of Circles
These worked examples move from a single direct substitution to a multi-step, real-world calculation. Each step sits on its own line.

### Example 1
**A circle has a radius of 7 cm. Find its diameter.**  
d=2r=2×7=14 cm  
Final answer: 14 cm.

### Example 2
**A circle has a radius of 10 cm. A student finds its area by multiplying π by 10. What went wrong, and what is the correct area?**  
The first instinct is to write A=πr and compute π×10≈31.4. But check the units: that answer is in centimetres, a length — and an area must be in square centimetres. The mistake is dropping the square on the radius.

The correct formula squares the radius:
A=π×10²=π×100≈314.16 cm²  
Final answer: about 314.16 cm².

### Example 3
**A circle has a diameter of 20 cm. Find its circumference. Use π≈3.14.**  
First find the radius from the diameter:
 r=d/2=20/2=10 cm
Then apply the circumference formula:
C=2πr=2×3.14×10=62.8 cm  
Final answer: 62.8 cm.

### Example 4
**The circumference of a circle is 44 cm. Find its radius. Use π≈22/7.**  
Start from C=2πr and solve for r:
r=C/2π=44/(2×22/7)=7 cm  
Final answer: 7 cm.

### Example 5
**Find the area of a sector with central angle 90° in a circle of radius 8 cm.**  
A sector is a fraction of the whole circle, and 90° is one-quarter of 360°:
A_\text{sector}=\frac{\theta}{360°}×πr²=\frac{90}{360}×π×8²=16π≈50.27 cm²  
Final answer: about 50.27 cm².

### Example 6
**A circular running track has a radius of 35 m. A runner completes 4 full laps. How far did the runner travel? Use π≈22/7.**  
One lap is the circumference:
C=2πr=2×22/7×35=220 m  
Four laps multiply that distance:
Total=4×220=880 m  
Final answer: 880 m.

## Where Circles Show Up — And Why The Definition Matters
The equidistant rule is not a textbook nicety; it is why circles do real jobs. A wheel rolls smoothly because the axle at the center stays a constant height above the ground. A satellite dish is a curved section so that signals reflect to one focal point.

Circles also anchor a chain of ideas you will meet later. The unit circle organizes all of trigonometry. The [equation of a circle](/content/math/geometry/equation-of-circle/index.html) is the distance rule rewritten in coordinates — the same equidistant idea, now in algebra.

## Tripping points to avoid
A few mistakes recur often enough that they are worth naming directly.

### Mistake 1: Confusing radius and diameter
**Where it slips in:** When a problem gives the diameter but the formula needs the radius (or the reverse).  
**Don't do this:** Plug the diameter straight into A=πr² as if it were the radius.  
**The correct way:** Convert first. If you are given the diameter, halve it to get the radius (r=d/2) before using any radius formula.

### Mistake 2: Forgetting to square the radius in the area formula
**Where it slips in:** Computing area quickly under time pressure.  
**Don't do this:** Write A=πr and report a length where an area belongs.  
**The correct way:** The area formula is A=πr². The exponent is exactly what separates the circumference formula from the area formula, and it is the single most common source of wrong answers on circle problems.

### Mistake 3: Mixing radians and degrees in arc and sector formulas
**Where it slips in:** Arc length and sector area, where the angle can be measured two ways.  
**Don't do this:** Use θ/360° with θ already in radians.  
**The correct way:** Match the formula to the angle's units.

## Conclusion
- A **circle** is the set of points equidistant from a center; that distance is the radius.
- The diameter is twice the radius: d=2r.
- Circumference is C=2πr and area is A=πr² — remember to square the radius for area.
- A chord stops at the circle, a secant extends beyond it, and a tangent touches at one point.
- Sectors and arcs are fractions of the whole circle, set by the central angle.