Circle (geometry)
A circle is the set of all points in a plane that are equidistant from a fixed central point, called the center; the common distance is called the radius. It is one of the most fundamental and perfectly symmetric shapes in geometry, and its properties underpin trigonometry, calculus, physics, and engineering. Circles appear in nature and technology ranging from planetary orbits to wheel design and clock faces.
Key Formula
Circumference C = 2πr; Area A = πr²
LaTeX: C = 2\pi r, \quad A = \pi r^2
| Symbol | Meaning | Unit |
|---|---|---|
| C | circumference (perimeter of the circle) | units |
| A | area enclosed by the circle | square units |
| r | radius (distance from center to any point on circle) | units |
| π | pi, approximately 3.14159... | unitless |
Worked Example
Problem
A circular garden has a radius of 7 m. Find its circumference and area. (Use π ≈ 3.14159)
Solution
Step 1: Circumference C = 2πr = 2 × 3.14159 × 7 = 43.98 m. Step 2: Area A = πr² = 3.14159 × 7² = 3.14159 × 49 = 153.94 m².
Answer
Circumference ≈ 43.98 m; Area ≈ 153.94 m²
Key Parts of a Circle
| Part | Definition | Formula / Relation | Example (r = 5 cm) |
|---|---|---|---|
| Radius (r) | Distance from center to circle | r | 5 cm |
| Diameter (d) | Distance across circle through center | d = 2r | 10 cm |
| Circumference (C) | Perimeter of the circle | C = 2πr = πd | 31.42 cm |
| Area (A) | Region enclosed by the circle | A = πr² | 78.54 cm² |
| Chord | Line segment with both endpoints on circle | Max chord = diameter | Any chord ≤ 10 cm |
| Arc | Part of the circumference between two points | Arc = rθ (θ in radians) | Sector boundary |
Interactive Tools
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Related Terms
Angle
An angle is the measure of rotation between two rays (sides) that share a common endpoint called the vertex. Angles are measured in degrees (°) or radians (rad) and describe the amount of turn between two directions. They are fundamental to geometry, trigonometry, physics, and engineering, appearing in everything from architectural blueprints to robotic arm movements.
Pythagorean Theorem
The Pythagorean Theorem states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides (the legs). It is one of the most famous and widely applied theorems in mathematics, used in distance calculations, navigation, construction, and virtually every branch of science and engineering.
Geometric Similarity
Two geometric figures are similar if they have the same shape but not necessarily the same size, meaning one can be obtained from the other by a combination of rigid motions and a uniform scaling (dilation). Similar figures have equal corresponding angles and proportional corresponding sides. Similarity is denoted by the symbol ~ and is the basis for scale drawings, maps, and many real-world applications including shadow calculations and indirect measurement.
From Latin "circulus" meaning a small ring or circle, diminutive of "circus" meaning a ring or circular course, from Greek "kirkos" or "krikos" meaning a ring or hoop. The Greek word for circle was "kyklos" (κύκλος), which gives the English word "cycle". The mathematical study of circles dates back to ancient Egypt and Babylon (c. 1800 BC).