Area of Rectangular Path (Frame) with Worked Examples
Area of Rectangular Path (Frame) is found by subtracting the area of the inner rectangle from the area of the outer rectangle. The formula changes according to whether the path lies outside or inside the rectangle. This page covers both cases, equal-width paths, solved calculations and useful shortcuts for aptitude problems.
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What Is the Area of a Rectangular Path (Frame)?
If a path of uniform width w surrounds a rectangle of length L and breadth B, the outer dimensions are L + 2w and B + 2w. Therefore, the outside path area is (L + 2w)(B + 2w) − LB. If the path is inside the rectangle, the inner dimensions become L − 2w and B − 2w, so the path area is LB − (L − 2w)(B − 2w).
Area of Rectangular Path (Frame) 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 Rectangular Path (Frame) Formula & Tricks
Important Formulas
Use this when a path of width w is constructed outside a rectangle of length L and breadth B.
Use this when a path of width w is made inside the boundary of a rectangle.
Since the inner rectangle perimeter is 2(L + B), these forms are equivalent to the standard formulas.
Quick Tricks
Find the outer and inner dimensions first, calculate both rectangle areas, and subtract the smaller area from the larger area. This avoids sign errors.
For a uniform path of width w, start with w times the perimeter of the original rectangle. Add 4w² for an outside path and subtract 4w² for an inside path.
Area of Rectangular Path (Frame) Concepts
Path Constructed Outside a Rectangle
The outer rectangle has length L + 2w and breadth B + 2w. Hence, area of the path = (L + 2w)(B + 2w) − LB. The width is added twice in each dimension because the path extends on both opposite sides.
Path Constructed Inside a Rectangle
The remaining inner rectangle has length L − 2w and breadth B − 2w. Therefore, area of the path = LB − (L − 2w)(B − 2w). The condition L > 2w and B > 2w must hold so that the remaining rectangle has positive dimensions.
Difference Between Outside and Inside Path Formulas
Expanding the outside expression gives (L + 2w)(B + 2w) − LB = 2w(L + B) + 4w². Expanding the inside expression gives LB − (L − 2w)(B − 2w) = 2w(L + B) − 4w². The sign depends on whether the corner squares are added outside or removed inside.
Finding Path Width from Areas
For an outside path, use A = 2w(L + B) + 4w². For an inside path, use A = 2w(L + B) − 4w². Solve the resulting equation and retain the positive width that is physically possible.
Area of Rectangular Path (Frame) Visual Guide & Data
Use the table and chart below to understand the data better.
Outside and Inside Rectangular Paths
Comparison of dimensions and area formulas for a uniform path of width w.
| Path position | Outer or remaining dimensions | Area of path |
|---|---|---|
| Outside rectangle | L + 2w, B + 2w | (L + 2w)(B + 2w) − LB |
| Inside rectangle | L − 2w, B − 2w | LB − (L − 2w)(B − 2w) |
Note: L and B are the original length and breadth; w is the uniform path width.
Rectangular Frame with an Outside Path
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Area of Rectangular Path (Frame) Revision Points
Use the difference between the outer and inner rectangular areas, taking care with the path position and units.
- Outside path dimensions are L + 2w and B + 2w.
- Inside path dimensions are L − 2w and B − 2w.
- Outside path area = 2w(L + B) + 4w².
- Inside path area = 2w(L + B) − 4w².
- A path area is expressed in square units, such as cm² or m².
- For an inside path, both L > 2w and B > 2w must be satisfied.
- Convert all dimensions to the same unit before applying the formula.
Area of Rectangular Path (Frame) FAQs
What is the formula for an outside rectangular path?
For length L, breadth B and path width w, area = (L + 2w)(B + 2w) − LB = 2w(L + B) + 4w².
What is the formula for an inside rectangular path?
Area = LB − (L − 2w)(B − 2w) = 2w(L + B) − 4w².
A rectangle is 25 m long and 15 m broad. Find the area of an outside path 2 m wide.
Outer dimensions are 29 m and 19 m. Path area = 29 × 19 − 25 × 15 = 551 − 375 = 176 m².
Why is 2w added to each dimension for an outside path?
The path extends by w on both opposite sides of the rectangle. Thus, the total increase in length and breadth is w + w = 2w.
Why is 4w² added in the outside formula but subtracted in the inside formula?
Expanding the outside area gives 2w(L + B) + 4w² because four corner squares are added. For an inside path, those four corner portions are removed, giving 2w(L + B) − 4w².
Can the length or breadth be smaller than twice the inside path width?
No. For an inside path, L − 2w and B − 2w must be positive. Therefore, both L > 2w and B > 2w are required.