Diameter to Radius: The Simple Halving That Defines Every Circle

The radius is always exactly half the diameter — r = d/2 — a definitional truth from Euclidean geometry that holds for every circle and sphere. The diameter spans a circle's full width through its center; the radius extends from center to circumference. All circular formulas — area πr², circumference 2πr, sphere volume 4/3 πr³ — require the radius, making this conversion the essential first step. A pipe with diameter 10 cm has radius 5 cm; the Earth's mean diameter of 12,742 km gives a radius of 6,371 km. [Euclid of Alexandria, circa 300 BCE, Elements Book I Definition 15] {#deck}

Table of Contents

1. The definitional relationship between diameter and radius
2. The diameter to radius conversion formula
3. Worked examples: diameter to radius and back
4. Practical applications across disciplines
5. Conversion table for diameter to radius
6. Frequently Asked Questions

The definitional relationship between diameter and radius

The diameter-to-radius relationship is one of the oldest documented geometric concepts, appearing as Definition 15 in Euclid's Elements circa 300 BCE: "A diameter of a circle is any straight line drawn through the center and terminated in both directions by the circumference." From this definition, the radius follows naturally as half of that line — the distance from the center to the circumference.

The relationship is not a physical law or an empirical observation; it is a logical consequence of how circles are defined. A circle is the set of all points equidistant from a center point. The radius is that fixed distance. The diameter is twice that distance — the longest possible chord passing through the center. Because both measurements originate from the same geometric definition, the conversion factor of 2 is exact and universal.

Key relationships:

• r = d / 2 — radius is half the diameter
• d = 2 × r — diameter is twice the radius
• The radius is used in area (A = πr²), circumference (C = 2πr), sphere volume (V = 4/3 πr³), and sphere surface area (A = 4πr²)
• The diameter is used for practical measurements (pipe sizing, wheel fitment, tool clearance) where through-center width is directly measurable

The diameter to radius conversion formula

Radius = Diameter ÷ 2

r = d / 2

The conversion is exact for any positive real number. There are no rounding considerations for the conversion itself — the factor of 2 is integer — though subsequent calculations using the radius (area, circumference) may involve π and require appropriate precision.

Reverse formula: d = 2 × r

Unit consistency: The conversion factor 2 is dimensionless, so the radius takes the same unit as the diameter. If the diameter is measured in meters, the radius is in meters. If the diameter is in inches, the radius is in inches. No unit conversion is needed.

For example:

• Diameter 10 cm → radius 5 cm
• Diameter 1.5 m → radius 0.75 m
• Diameter 24 inches → radius 12 inches
• Diameter 0.5 mm → radius 0.25 mm
• Diameter 100 km → radius 50 km

Formulas that follow from the radius:

• Circumference: C = 2πr = πd
• Area: A = πr² = π(d/2)² = πd²/4
• Sphere volume: V = 4/3 πr³ = 4/3 π(d/2)³ = πd³/6
• Sphere surface area: A = 4πr² = 4π(d/2)² = πd²

Worked examples: diameter to radius and back

Worked example 1: Pipe radius from diameter

A steel pipe has an outer diameter of 168.3 mm (standard DN150 pipe). What is its outer radius?

• r = d / 2
• r = 168.3 / 2 = 84.15 mm
• Result: 84.15 mm

Worked example 2: Basketball rim radius

An NBA basketball hoop has a diameter of 18 inches. What is the rim radius?

• r = 18 / 2 = 9 inches
• Result: 9 inches (228.6 mm)

Worked example 3: Reverse — radius to diameter

A circular garden bed has a radius of 2.5 meters. What diameter fencing is needed?

• d = 2 × r = 2 × 2.5 = 5.0 meters
• Result: 5.0 meters diameter

Practical applications across disciplines

The diameter-to-radius conversion appears in virtually every field that involves circles, spheres, or curved surfaces. In engineering and manufacturing, drilled holes are specified by diameter, but the stress concentration factor around the hole depends on the radius. A 10 mm hole in a steel plate has a radius of 5 mm, and the stress at the edge of the hole depends on this radius value according to the formula σ_max = 3σ_avg for a circular hole in an infinite plate under tension.

In civil engineering, column and pile diameters are specified by their width, but the structural capacity depends on the cross-sectional area computed from the radius. A concrete column with 0.4 m diameter has radius 0.2 m and cross-sectional area π × 0.2² = 0.126 m². The axial load capacity is this area multiplied by the concrete's compressive strength.

In astronomy and planetary science, celestial bodies are described by their mean diameters, but gravitational calculations require radii. The Earth's mean diameter of 12,742 km yields a radius of 6,371 km, which is used in the formula for surface gravity g = GM/r², orbital velocity v = √(GM/r), and escape velocity v_esc = √(2GM/r).

In medicine, coronary artery stents are sized by diameter (typically 2.0–5.0 mm), but the cross-sectional area (from the radius) determines blood flow capacity. A 3.0 mm stent has a radius of 1.5 mm and a cross-sectional area of 7.07 mm², compared to 12.57 mm² for a 4.0 mm stent — the 33% increase in diameter produces a 78% increase in area.

Conversion table for diameter to radius

Diameter to Radius Conversion for Common Sizes in Metric and Imperial Units