Circles – The Perfect Shape
The circle is one of the most perfect shapes in mathematics and nature. From wheels to planets, circles appear everywhere. Knowing the parts of a circle and how to calculate its measurements is essential geometry.
The Greek mathematician Archimedes (c. 287–212 BCE) was the first to calculate an accurate bound on the value of π by inscribing and circumscribing regular polygons with up to 96 sides around a circle, squeezing the true value between 3 10/71 and 3⅓. Today, computers have calculated π to tens of trillions of decimal places, even though only around 40 digits are needed to compute the circumference of the observable universe to the precision of a single atom. The circle's usefulness is just as old as its mathematics: the wheel, one of humanity's most important inventions, first appears in the archaeological record in Mesopotamia around 3500 BCE, and the same principle — that every point on the rim stays exactly the same distance from the centre — is why gears, pipes, and ball bearings are still built as circles today.
Parts of a Circle
| Part | Definition |
|---|---|
| Centre | The fixed point at the middle, equidistant from every point on the circle |
| Radius (r) | Distance from the centre to any point on the circle |
| Diameter (d) | A chord through the centre; d = 2r |
| Chord | A line segment joining any two points on the circle |
| Arc | A portion of the circumference (boundary curve) |
| Sector | A pie-slice region bounded by two radii and an arc |
| Segment | The region between a chord and the arc it cuts off |
| Tangent | A line that touches the circle at exactly one point |
Circle Area Through Rearranging Sectors
Where does A = πr² actually come from? Cut a circle into equal pie-slice sectors, then lay those sectors out in a row, alternating which way each one points.
With only a few sectors the rearranged row looks jagged, but the more (thinner) sectors you cut, the straighter its top and bottom edges become — approaching a genuine rectangle. That rectangle has height r (the radius) and width equal to half the circumference, πr. Its area is therefore r × πr = πr² — exactly the circle area formula, because rearranging the sectors never changes the total area, only their arrangement.
Circumference and Area
Circumference (perimeter) = 2πr = πd. Area = πr². Use π ≈ 3.14159 or leave answers in terms of π. See our dedicated Pi page for the history of π, why it never ends, and where it shows up beyond circles.
Circumference = 2 × π × 7 = 14π ≈ 43.98 cm. Area = π × 7² = 49π ≈ 153.94 cm².
r = 6 cm. Circumference = 2 × π × 6 = 12π ≈ 37.70 cm.
Arc Length and Sector Area
| Measurement | Formula |
|---|---|
| Arc length | (θ / 360) × 2πr, where θ is the angle in degrees |
| Sector area | (θ / 360) × πr² |
Arc length = (90/360) × 2π(10) = ¼ × 20π = 5π ≈ 15.71 cm. Sector area = (90/360) × π(100) = 25π ≈ 78.54 cm².
Circle Theorems (Key Rules)
- A tangent meets a radius at exactly 90°.
- The angle at the centre is twice the angle at the circumference (for the same arc).
- Angles in a semicircle are always 90°.
- Opposite angles in a cyclic quadrilateral sum to 180°.
Key Takeaways
- Circumference = 2πr; Area = πr².
- A sector is a slice of a circle; its area and arc length use the angle fraction (θ/360).
- A tangent to a circle is always perpendicular to the radius at the point of contact.
- Angles in a semicircle = 90° (a fundamental circle theorem).
Practice: Circumference, Area & Sectors
Related Topics
Continue exploring related topics:
- Pi (π) – The Number That Never Ends
- Angles – Measuring Turns and Corners
- Area – How Much Space a Shape Covers
- Congruence – Identical Shapes in Every Way
- Coordinate Geometry – Algebra Meets Geometry
- Lines – Paths That Define Shape and Direction
- Geometry – The Mathematics of Shape and Space
- Spheres
- Cylinders