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Radius of a Circle or Sphere — Formula & Examples

Radius of a Circle or Sphere

A line segment between the center and a point on the circle or sphere. The word radius also refers to the length of this segment.

 

Circle with a line segment labeled "radius" drawn from the center point to a point on the circle's edge.

Key Formula

d=2r⟺r=d2d = 2r \quad \Longleftrightarrow \quad r = \frac{d}{2}
Where:
  • rr = Radius — the distance from the center to any point on the circle or sphere
  • dd = Diameter — the distance across the circle or sphere through its center

Worked Example

Problem: A circle has a diameter of 26 cm. Find its radius, circumference, and area.
Step 1: Find the radius by halving the diameter.
r=d2=262=13 cmr = \frac{d}{2} = \frac{26}{2} = 13 \text{ cm}
Step 2: Calculate the circumference using the radius.
C=2πr=2π(13)=26π≈81.68 cmC = 2\pi r = 2\pi(13) = 26\pi \approx 81.68 \text{ cm}
Step 3: Calculate the area using the radius.
A=πr2=π(13)2=169π≈530.93 cm2A = \pi r^2 = \pi(13)^2 = 169\pi \approx 530.93 \text{ cm}^2
Answer: The radius is 13 cm, the circumference is approximately 81.68 cm, and the area is approximately 530.93 cm².

Another Example

Problem: A sphere has a radius of 6 cm. Find its surface area and volume.
Step 1: Use the surface area formula for a sphere.
S=4πr2=4π(6)2=144π≈452.39 cm2S = 4\pi r^2 = 4\pi(6)^2 = 144\pi \approx 452.39 \text{ cm}^2
Step 2: Use the volume formula for a sphere.
V=43πr3=43π(6)3=288π≈904.78 cm3V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi(6)^3 = 288\pi \approx 904.78 \text{ cm}^3
Answer: The surface area is approximately 452.39 cm² and the volume is approximately 904.78 cm³.

Frequently Asked Questions

What is the difference between the radius and the diameter?
The radius runs from the center of a circle (or sphere) to its edge, while the diameter runs all the way across, passing through the center. The diameter is always exactly twice the radius: d=2rd = 2r.
How do you find the radius from the area of a circle?
Start with the area formula A=πr2A = \pi r^2. Divide both sides by π\pi, then take the square root: r=A/πr = \sqrt{A / \pi}. For example, if the area is 100π100\pi cm², then r=100=10r = \sqrt{100} = 10 cm.

Radius vs. Diameter

The radius is the distance from the center to the boundary; the diameter is the distance across the full shape through the center. Every diameter equals two radii (d=2rd = 2r). When you know one, you immediately know the other. Most circle and sphere formulas — area, circumference, volume, surface area — are written in terms of the radius, so it is the more fundamental measurement in calculations.

Why It Matters

The radius appears in nearly every formula involving circles and spheres: circumference (C=2πrC = 2\pi r), circle area (A=πr2A = \pi r^2), sphere surface area (S=4πr2S = 4\pi r^2), and sphere volume (V=43πr3V = \frac{4}{3}\pi r^3). Engineers, architects, and scientists use the radius constantly — from sizing wheels and pipes to modeling planets and orbits. Understanding it is the key to unlocking all of circular and spherical geometry.

Common Mistakes

Mistake: Confusing radius with diameter when plugging into formulas.
Correction: Always check whether a problem gives you the radius or the diameter. If you are given the diameter, divide by 2 before using formulas like A=πr2A = \pi r^2. Using the diameter in place of rr will make your answer four times too large for area and eight times too large for sphere volume.
Mistake: Forgetting to square the radius in the area formula.
Correction: The area of a circle is πr2\pi r^2, not πr\pi r. The expression πr\pi r does not even have the correct units — it would give you a length, not an area.

Related Terms

  • Circle — The 2-D shape defined by a fixed radius
  • Sphere — The 3-D shape defined by a fixed radius
  • Diameter — Twice the radius; spans the full shape
  • Circumference — Perimeter of a circle, calculated from radius
  • Area of a Circle — Equals π times the radius squared
  • Line Segment — The radius is a specific line segment
  • Pi (π) — Constant that pairs with radius in formulas