Circle Properties

Diagram: Parts of a Circle (A labeled diagram identifying all key parts)

• Centre: Marked 'O' in the exact middle.
• Circumference: The outer boundary line.
• Radius: A straight line from the centre to the circumference.
• Diameter: A straight line passing through the centre, touching the circumference at both ends.
• Chord: A straight line connecting two points on the circumference.
• Tangent: A straight line outside the circle touching the circumference at exactly one point.
• Arc: A highlighted section of the circumference.
• Sector: A shaded region bounded by two radii and an arc.
• Segment: A shaded region bounded by a chord and an arc.

The distance from the centre to any point on the circumference is the radius (plural: radii). A straight line segment passing entirely through the centre with both endpoints on the circumference is the diameter.

The diameter is exactly twice the length of the radius ($d = 2r$), and it acts as the circle's line of symmetry. The circumference is the total perimeter or boundary line running around the outside edge of the circle.

Lines and Edges

A straight line can slice right through a shape, balance on its outer edge, or stop completely inside it. A chord is a straight line segment joining any two points on the circumference.

The diameter is actually the longest possible chord you can draw in a circle. A tangent, however, is a straight line outside the circle that touches the circumference at exactly one point and does not cross it.

An arc is simply a portion or section of the circumference itself. Arc length is measured in linear units (such as $\text{cm}$ or $\text{m}$).

Slices and Pieces

Cutting a cake from the middle creates a very different shape compared to chopping off a straight edge. A sector is the region bounded by two radii and an arc, looking much like a classic slice of pizza.

A segment, on the other hand, is the region bounded by a chord and an arc. Any triangle formed inside a circle by two radii and a chord will always be an isosceles triangle, because both radii are equal in length.

When a chord or two radii divide a circle, they create two unequal parts. The smaller portion (with a central angle $< 180^\circ$) is called the minor part, while the larger portion (central angle $> 180^\circ$) is called the major part.

Key Circle Properties

Understanding geometric properties explains why circles are perfectly balanced and predictable. A tangent is always perpendicular ($90^\circ$) to the radius at the exact point of contact.

The perpendicular bisector of a chord will always pass straight through the centre of the circle. Furthermore, if you draw two tangents to a circle from the same external point, those two tangent lines will be exactly equal in length.

Another fundamental property is $\pi$ (Pi). $\pi$ is the constant ratio of every circle's circumference to its diameter, which is why the formula for circumference is $C = \pi d$.

Calculating Fractional Parts

How do you mathematically measure just a fraction of a whole circle? Lengths and areas of arcs and sectors are calculated as a fraction of the whole circle using the central angle $\theta$ (out of $360^\circ$).

The formulas for finding the size of a sector use this fraction multiplier:

$$\text{Arc Length} = \frac{\theta}{360} \times \pi d$$ $$\text{Sector Area} = \frac{\theta}{360} \times \pi r^2$$

To find the area of a segment, you must calculate the area of the entire sector first, and then subtract the area of the internal triangle ($\frac{1}{2} r^2 \sin(\theta)$).

Worked Example

Calculate the total perimeter of a semicircle with a diameter of $10\text{ cm}$.

Step 1: Calculate the arc length of the curved edge.

• A semicircle is half of a full circle.
• $\text{Arc Length} = \frac{\pi d}{2}$
• $\text{Arc Length} = \frac{10\pi}{2} = 5\pi \approx 15.71\text{ cm}$

Step 2: Add the straight edge to find the total perimeter.

• The straight edge is the diameter ($10\text{ cm}$).
• $\text{Total perimeter} = 15.71 + 10$
• $\text{Total perimeter} = 25.7\text{ cm}$