Area of Circular Path (Ring) with Inner and Outer Radii
Area of Circular Path (Ring) is the region between two concentric circles. It is found by subtracting the area of the inner circle from the area of the outer circle. This page covers the formula, radius and diameter cases, circular path problems, shortcuts, and solved numerical examples.
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What Is the Area of a Circular Path (Ring)?
The outer circle includes the inner circle and the circular path. Therefore, subtract the inner-circle area, πr², from the outer-circle area, πR². If the path has width w and the inner radius is r, then R = r + w, so the area is π[(r + w)² − r²] = πw(2r + w).
Area of Circular Path (Ring) Video Lessons
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Area of Paths, Rings and Borders
Learn how to calculate the area of paths, rings, frames and bordered rectangles using outer and inner dimensions in 2D mensuration problems.
Area of Circular Path (Ring) Formula & Tricks
Important Formulas
R is the outer radius and r is the inner radius. The result is expressed in square units.
If w is the width of the path and r is the inner radius, then the outer radius is r + w.
D and d are the outer and inner diameters respectively, since R = D/2 and r = d/2.
The path width is the difference between the outer and inner radii.
Quick Tricks
Calculate R² − r² as (R − r)(R + r). This reduces multiplication when the two radii are close.
When the radius or diameter contains a factor of 7, π = 22/7 often gives a quick exact calculation. Otherwise, use the value specified in the question.
Area of Circular Path (Ring) Concepts
Area from Inner and Outer Radii
The outer circle has area πR² and the inner circle has area πr². Hence, the ring area is πR² − πr² = π(R² − r²).
Area When Path Width Is Given
Substitution gives area = π[(r + w)² − r²]. Expanding the squares gives area = π(2rw + w²) = πw(2r + w).
Circular Path Inside or Outside a Circle
If the original radius is r and the path width is w, an outside path has outer radius r + w and inner radius r. An inside path has outer radius r and inner radius r − w. In both cases, subtract the inner-circle area from the outer-circle area.
Using Diameters Instead of Radii
Convert each diameter to radius, or apply the diameter formula directly. The factor 1/4 appears because R² = D²/4 and r² = d²/4.
Units and Conversion in Ring Problems
Convert lengths before squaring them. For example, 1 m² = 10,000 cm², while 1 m = 100 cm. Do not convert an area using a linear conversion factor.
Area of Circular Path (Ring) Visual Guide
Use the diagram or chart below to understand the concept clearly.
Outer and Inner Circles Forming a Circular Ring
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Area of Circular Path (Ring) Revision Points
Use these formulas and rules to revise ring-area calculations.
- Area of ring = π(R² − r²), where R is outer radius and r is inner radius.
- If path width is w, then R = r + w and area = πw(2r + w).
- Using diameters: area = (π/4)(D² − d²).
- For an outside path, add the width to the original radius; for an inside path, subtract it.
- Use consistent length units before squaring; the final unit is square units.
- Use R² − r² = (R − r)(R + r) for faster calculation.
Area of Circular Path (Ring) FAQs
What is the area of circular path (ring) formula?
The formula is Area = π(R² − r²), where R is the outer radius and r is the inner radius.
How do you find the area of a ring if its width is given?
If the inner radius is r and path width is w, use Area = πw(2r + w). For r = 5 cm and w = 2 cm, area = π × 2 × 12 = 24π cm².
What is the area of a circular path with outer radius 14 cm and inner radius 10 cm?
Area = π(14² − 10²) = π(196 − 100) = 96π cm², approximately 301.59 cm².
What is the ring-area formula when diameters are given?
If D is the outer diameter and d is the inner diameter, Area = (π/4)(D² − d²).
A circular path is outside a circle of radius 7 m and has width 2 m. What is its area?
The outer radius is 9 m. Area = π(9² − 7²) = 32π m², approximately 100.53 m².
How is the area of a circular ring different from its circumference?
Ring area measures the region between two circles and is π(R² − r²). The circumference of a circle measures its boundary and is 2πr; they have different units and formulas.