Wire Bending Problems in Mensuration with Perimeter Rules
Wire bending problems in mensuration involve changing a wire from one shape to another without changing its length. The wire length equals the perimeter of the original shape and the perimeter of the new shape. Use this conservation rule to find unknown sides, radii, areas or perimeters.
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What Are Wire Bending Problems in Mensuration?
If a wire of length L is first bent into a shape with perimeter P₁ and then into another shape with perimeter P₂, then L = P₁ = P₂. The shape may change, but the wire length does not. For example, a rectangle of length 18 cm and breadth 10 cm uses 2(18 + 10) = 56 cm of wire. If the same wire forms a square, each side is 56 ÷ 4 = 14 cm.
Wire Bending Problems in Mensuration Video Lessons
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Recasting Shapes with Same Perimeter
Understand how a wire or shape is recast into different forms while keeping the perimeter unchanged, and apply this relationship to solve mensuration problems.
Wire Bending Problems in Mensuration Formula & Tricks
Important Formulas
A wire is not stretched or cut during bending, so its length remains unchanged.
Here l and b are the rectangle's length and breadth, while a is the side of the square.
The perimeter of the square becomes the circumference of the circle.
For a wire bent into a polygon, add the lengths of all its sides.
Use π = 22/7 or 3.14 as directed, r is radius and d is diameter.
Wire length remains constant, but the enclosed area generally changes when the shape changes.
Quick Tricks
Do not equate areas in a wire-bending question. Equate the original and new perimeters because the wire length is conserved.
All lengths must be in the same unit before setting up the equation. Convert metres to centimetres or centimetres to metres first.
When the new shape is a circle, equate the wire length to 2πr or πd, not to the area of the circle.
Wire Bending Problems in Mensuration Concepts
Conserving Length During Reshaping
If the first shape has perimeter P₁ and the second shape has perimeter P₂, then P₁ = P₂. This rule applies to rectangles, squares, triangles, polygons and circles, provided the wire is neither cut nor stretched.
Changing a Rectangle into a Square
Dividing by 2 gives a = (l + b) ÷ 2. Thus, the side of the equivalent-perimeter square is the arithmetic mean of the rectangle's length and breadth.
Changing a Square into a Circle
The square's perimeter becomes the circle's circumference. If the diameter is required, use 4a = πd, giving d = 4a/π.
Area Can Change After Bending
For the same perimeter, different shapes can enclose different areas. A 44 cm wire forms a square of side 11 cm with area 121 cm², or a circle of radius 7 cm with area 154 cm², using π = 22/7.
Wire Bending Problems in Mensuration Visual Guide & Data
Use the table and chart below to understand the data better.
Common Perimeter Relations in Wire Bending
Use the appropriate perimeter formula for the original and reshaped wire.
| Shape | Perimeter or circumference formula | Wire-length equation |
|---|---|---|
| Square | 4a | L = 4a |
| Rectangle | 2(l + b) | L = 2(l + b) |
| Triangle | a + b + c | L = a + b + c |
| Circle | 2πr or πd | L = 2πr = πd |
Note: Use the same unit for every length before calculating.
Rectangle Wire Reshaped into a Square
Equivalent-Perimeter Square
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Wire Bending Problems: Revision Points
Use perimeter conservation to solve reshaping questions and calculate area only after finding the new dimensions.
- Wire length remains constant when it is bent without cutting or stretching.
- Equate the perimeters of the original and new shapes.
- Rectangle perimeter = 2(l + b); square perimeter = 4a.
- Circle circumference = 2πr = πd.
- Keep all measurements in the same unit.
- Area usually changes after reshaping, even though wire length remains constant.
- For a rectangle changed into a square, square side = (length + breadth) ÷ 2.
- Use π = 22/7 or 3.14 according to the question.
Wire Bending Problems in Mensuration FAQs
What is the wire bending problems in mensuration formula?
The basic formula is Perimeter of the original shape = Perimeter of the new shape. This follows because the wire length remains constant.
A 20 cm by 12 cm rectangular wire is bent into a square. What is the side of the square?
Rectangle perimeter = 2(20 + 12) = 64 cm. Square side = 64 ÷ 4 = 16 cm.
A wire of length 88 cm is bent into a circle. Find its radius using π = 22/7.
Using 2πr = 88, r = 88 ÷ (44/7) = 14 cm.
Does the area remain the same when a wire is reshaped?
No. Bending conserves the wire's length and therefore its perimeter, but the enclosed area can change.
A square wire has side 14 cm and is bent into a circle. What is the circle's diameter?
The wire length is 4 × 14 = 56 cm. Since πd = 56, d = 56 × 7 ÷ 22 = 56/ (22/7) = 17.82 cm approximately, using π = 22/7.
How are units handled in wire bending questions?
Convert every measurement to the same unit before equating perimeters. For example, 1.5 m must be written as 150 cm if the other dimensions are in centimetres.