Parts and Properties of Circles

Circles are everywhere — wheels, pipes, ducts, clocks, gears. Before you can calculate circumference, area, or anything else involving circles, you need to understand the vocabulary. This page covers every major part of a circle and how those parts relate to each other.

Center, Radius, and Diameter

A circle is the set of all points that are exactly the same distance from a fixed point called the center. A circle is often named by its center point — for example, “circle $O$” means a circle whose center is the point $O$.

• Center: The fixed point equidistant from every point on the circle.
• Radius ($r$): The distance from the center to any point on the circle. Every radius of the same circle has the same length.
• Diameter ($d$): The distance across the circle through the center — a line segment that connects two points on the circle and passes through the center. The diameter is always twice the radius:

$d = 2r \quad \text{or equivalently} \quad r = \frac{d}{2}$

A circle has infinitely many radii and infinitely many diameters. They all have the same length within a given circle.

Chord

A chord is a line segment whose both endpoints lie on the circle. Unlike a radius, a chord does not need to pass through the center.

• The diameter is the longest possible chord — it passes through the center.
• Any chord that does not pass through the center is shorter than the diameter.
• The perpendicular from the center to a chord bisects (cuts in half) that chord. This property is useful for finding chord lengths.

Arc

An arc is a portion of the circle’s circumference — a curved segment between two points on the circle. Two points on a circle divide it into two arcs:

• Minor arc: The shorter arc (less than a semicircle, measuring less than $180\degree$).
• Major arc: The longer arc (more than a semicircle, measuring more than $180\degree$).
• Semicircle: Exactly half the circle. This occurs when the chord connecting the two endpoints is a diameter, and the arc measures exactly $180\degree$.

Arcs are measured in degrees. A full circle is $360\degree$, so the minor arc and the major arc between two points always add up to $360\degree$.

Central Angle

A central angle is an angle whose vertex is at the center of the circle and whose sides are two radii. The central angle and its intercepted arc (the arc that lies inside the angle) have the same measure in degrees.

$\text{central angle} = \text{intercepted arc measure}$

For example, a $90\degree$ central angle intercepts a $90\degree$ arc, which is one-quarter of the circle.

Sector

A sector is the “pie slice” shaped region enclosed by two radii and the arc between them. Think of a slice of pizza or a wedge of a pie chart.

The area of a sector depends on the central angle $\theta$ (in degrees) and the radius $r$:

$A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2$

This formula takes the fraction of the full circle that the sector covers and multiplies it by the full circle’s area. For a complete treatment of sector area with worked examples, see Area of Circles.

Segment

A segment (sometimes called a “circular segment”) is the region between a chord and the arc it cuts off. Imagine slicing straight across a circle with a chord — the region between that chord and the curved edge is a segment. A segment is not the same as a sector: a sector has two straight sides (radii) while a segment has one straight side (a chord).

Tangent

A tangent is a line that touches the circle at exactly one point, called the point of tangency. The key property of a tangent:

$\text{tangent} \perp \text{radius at the point of tangency}$

A tangent line is always perpendicular (forms a $90\degree$ angle) to the radius drawn to the point where it touches. This is critical in HVAC and plumbing work, where round pipes and ducts meet flat surfaces — the tangent property determines exactly how and where the connection is made.

Secant

A secant is a line that intersects the circle at two points. While a chord is a segment (it stops at the circle’s edge), a secant extends beyond the circle in both directions. You can think of a chord as the part of a secant that is inside the circle.

Circle Diagram: All Major Parts

In the diagram above, notice how the tangent touches the circle at exactly one point and makes a right angle with the radius at that point. The secant crosses through the circle, entering and exiting at two points. The shaded sector is the pie-slice region between two radii, and the central angle at the center measures the same number of degrees as its intercepted arc.

Key Relationships

Worked Examples

Example 1: Find diameter and circumference from a given radius

A circle has a radius of $7$ cm. Find the diameter and the circumference.

Step 1: Find the diameter.

$d = 2r = 2(7) = 14 \text{ cm}$

Step 2: Find the circumference.

$C = \pi d = \pi(14) = 14\pi \approx 43.98 \text{ cm}$

Answer: The diameter is 14 cm and the circumference is approximately 43.98 cm.

Example 2: Find the length of a chord using the Pythagorean theorem

A chord is 6 inches from the center of a circle with radius 10 inches. Find the length of the chord.

Setup: When a perpendicular is drawn from the center to a chord, it bisects the chord. This creates a right triangle where:

• The hypotenuse is the radius ($10$ in)
• One leg is the distance from the center to the chord ($6$ in)
• The other leg is half the chord length

Step 1: Apply the Pythagorean theorem to find half the chord.

$\text{half-chord}^2 + 6^2 = 10^2$

$\text{half-chord}^2 = 100 - 36 = 64$

$\text{half-chord} = \sqrt{64} = 8 \text{ in}$

Step 2: Double it to get the full chord length.

$\text{chord} = 2 \times 8 = 16 \text{ in}$

Answer: The chord is 16 inches long.

Example 3: Central angle and arc fraction

A central angle measures $120\degree$. What fraction of the circle is the minor arc?

Step 1: The arc and the central angle have the same measure, so the minor arc is $120\degree$.

Step 2: Find the fraction of the full circle.

$\frac{120}{360} = \frac{1}{3}$

Answer: The minor arc is one-third of the circle. The remaining major arc is $360\degree - 120\degree = 240\degree$, which is two-thirds.

Example 4: Find the area of a sector

Find the area of a sector with a central angle of $90\degree$ and a radius of $8$ ft.

Step 1: Apply the sector area formula.

$A = \frac{\theta}{360} \times \pi r^2 = \frac{90}{360} \times \pi (8)^2$

Step 2: Simplify.

$A = \frac{1}{4} \times 64\pi = 16\pi \approx 50.27 \text{ ft}^2$

Answer: The sector area is approximately 50.27 square feet. This makes sense — a $90\degree$ sector is one-quarter of the circle, and the full circle area would be $64\pi \approx 201.06 \text{ ft}^2$, and $201.06 \div 4 \approx 50.27$.

Example 5: HVAC — round duct meeting a flat wall

A round HVAC duct with a diameter of $12$ in connects to a flat wall. Where the duct meets the wall, the wall acts as a tangent line to the circular cross-section.

Key insight: The tangent (wall surface) is perpendicular to the radius at the point of contact. This means:

• The center of the duct is exactly $r = \frac{12}{2} = 6$ inches away from the wall surface.
• The duct contacts the wall at exactly one point (the point of tangency).
• If you need to anchor the duct, the mounting bracket must account for the $6$-inch distance from the wall to the duct center.

Answer: The duct center sits 6 inches from the wall, and the duct touches the wall at exactly one point — the tangent point. The perpendicular relationship guarantees the duct sits flush against the wall without a gap.

Real-World Application: HVAC Duct Connections

Understanding tangent lines is essential when working with round ductwork. Every time a round pipe or duct meets a flat surface — a wall, ceiling, floor, or rectangular duct — the contact follows the tangent property.

Why it matters:

• Mounting: The center of a round duct is always one radius away from the flat surface it touches. A 10-inch diameter duct sits with its center 5 inches off the wall.
• Sealing: Because the duct touches a flat surface at only one point (in cross-section), sealing requires a gasket or mastic that fills the gap on either side of the tangent point.
• Branch connections: When a round branch duct meets a larger round main duct, the connection point is where the two circles are tangent to each other — the tangent line at that point is perpendicular to the line connecting the two centers.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A radius goes from center to edge; a diameter goes across through the center ($d = 2r$)
• A chord has both endpoints on the circle — the diameter is the longest chord
• Arcs are curved portions of the circle, measured in degrees; minor + major arc $= 360\degree$
• A central angle equals the measure of its intercepted arc
• A sector is the pie-slice region between two radii; a segment is between a chord and an arc
• A tangent touches the circle at one point and is perpendicular to the radius there — critical for HVAC duct-to-wall connections
• A secant crosses the circle at two points; the chord is the interior portion of a secant

Return to Geometry for more topics in this section.