Radius
The radius of a circle or sphere is the distance from the centre to any point on its boundary. It is one of the most fundamental measurements in circular geometry, directly determining the size of the circle. The radius relates to the diameter by the equation r = d/2 and appears in formulas for circumference, area, and arc length.
Key Formula
r = d / 2
LaTeX: r = \dfrac{d}{2}
| Symbol | Meaning | Unit |
|---|---|---|
| r | Radius | m (or any length unit) |
| d | Diameter | m (or any length unit) |
Worked Example
Problem
A circular park has a diameter of 84 m. What is its radius, circumference, and area?
Solution
Step 1 — Radius: r = d/2 = 84/2 = 42 m. Step 2 — Circumference: C = 2πr = 2 × 3.14159 × 42 ≈ 263.89 m. Step 3 — Area: A = πr² = 3.14159 × 42² = 3.14159 × 1764 ≈ 5541.77 m².
Answer
Radius = 42 m, Circumference ≈ 263.89 m, Area ≈ 5541.77 m²
Radius in Common Geometric Shapes
| Shape | Radius Meaning | Formula Using r | Example (r = 5 cm) |
|---|---|---|---|
| Circle | Centre to boundary | C = 2πr | C ≈ 31.42 cm |
| Circle | Centre to boundary | A = πr² | A ≈ 78.54 cm² |
| Sphere | Centre to surface | V = (4/3)πr³ | V ≈ 523.6 cm³ |
| Cylinder | Centre of base to edge | A_base = πr² | A_base ≈ 78.54 cm² |
| Semicircle | Centre to curved edge | Perimeter = πr + 2r | Perimeter ≈ 25.71 cm |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Diameter
The diameter of a circle is a chord that passes through the centre, connecting two points on the circumference; it is the longest possible chord and equals twice the radius. The diameter determines the scale of a circle and is used directly in calculating circumference (C = πd) and indirectly in area calculations. In three dimensions, the diameter of a sphere is analogously the longest line segment passing through the centre between two surface points.
Area (geometry)
Area is the measure of the two-dimensional region enclosed within a closed geometric figure, expressed in square units. It quantifies how much flat surface a shape covers and is fundamental in fields ranging from architecture and land surveying to physics and engineering. Different shapes have distinct area formulas derived from their geometric properties, such as A = πr² for a circle or A = ½bh for a triangle.
From Latin "radius" meaning "spoke of a wheel" or "ray". The term entered mathematical usage in the 17th century, with René Descartes and other geometers using it to describe the spoke-like line from centre to edge of a circle.