Bounds: Rectangle Perimeter & Area

Statement

For a rectangle with length \( \ell \) and width \( w \), if each side is only known within bounds \( \ell \in [L_{\ell},\,U_{\ell}] \) and \( w \in [L_{w},\,U_{w}] \), then the bounds for the perimeter \(P=2(\ell+w)\) and the area \(A=\ell w\) follow by pushing the interval endpoints through the formulas (both are increasing in each variable for non-negative lengths):

\[ P \in \big[\,2(L_{\ell}+L_{w}),\;2(U_{\ell}+U_{w})\,\big], \qquad A \in \big[\,L_{\ell}L_{w},\;U_{\ell}U_{w}\,\big]. \]

We normally leave answers in exact form (no premature rounding). Add the correct units: perimeter in the base unit (cm, m, …), and area in squared units (cm\(^2\), m\(^2\), …).

Why it’s true (short reason)

• \(P(\ell,w)=2(\ell+w)\) is linear and increasing in both \( \ell \) and \( w \) for non-negative inputs. Therefore the smallest perimeter occurs at the smallest allowed pair \((L_{\ell},L_{w})\) and the largest at \((U_{\ell},U_{w})\).
• \(A(\ell,w)=\ell w\) is also increasing in each variable when the other is non-negative. Thus \(A_{\min}=L_{\ell}L_{w}\) and \(A_{\max}=U_{\ell}U_{w}\).
• This is the monotonicity principle: increasing functions map interval endpoints to the corresponding extreme outputs.

Recipe (how to use it)

1. Convert the wording to bounds. From “nearest unit / \(d\) d.p. / \(s\) s.f.” build half-step intervals:
   • Nearest cm: \(x\) cm \(\Rightarrow [x-0.5,\,x+0.5)\) cm.
   • \(d\) decimal places: step \(=0.5\times 10^{-d}\).
   • \(s\) significant figures: identify the place value of the last significant digit; the half-step is half that place value.
2. Perimeter. Compute \(P_{\min}=2(L_{\ell}+L_{w})\) and \(P_{\max}=2(U_{\ell}+U_{w})\). Attach the base unit.
3. Area. Compute \(A_{\min}=L_{\ell}L_{w}\) and \(A_{\max}=U_{\ell}U_{w}\). Attach squared units.
4. Keep results exact until the end; round only if the question demands a decimal with a stated accuracy.
5. Present cleanly. Finish with an interval statement such as \(A\in[12.13,12.88]\;\text{cm}^2\).

Spotting it

These bounds appear whenever a question gives rounded rectangle dimensions and asks for a “maximum” or “minimum” possible perimeter/area, or directly asks for bounds. Typical triggers:

• “\(\ell=9.4\) cm to 1 d.p., \(w=2.6\) cm to 1 d.p.—find bounds for area.”
• “A decking board is rectangular; length and width are measured to the nearest metre—give the greatest possible perimeter.”
• “A poster is \(42.0\) cm by \(29.7\) cm (3 s.f.)—find the smallest possible area.”

Common pairings

• Rounding to bounds. Translating measurement statements into intervals before any substitution.
• Percentage/relative error. Comparing the spread of perimeter vs area when side errors are small.
• Unit conversions. Converting mm–cm–m either before or after bounds (stay consistent!).
• Compound rectangles. Splitting into rectangles, bounding each area, then combining (add areas, add perimeters when appropriate).

Mini examples

1. Nearest integers: \(\ell=5\) cm, \(w=3\) cm (nearest cm). Bounds: \(\ell\in[4.5,5.5)\), \(w\in[2.5,3.5)\). \(P\in[2(4.5+2.5),\,2(5.5+3.5)]=[14,18]\) cm; \(A\in[4.5\cdot2.5,\,5.5\cdot3.5]=[11.25,19.25]\) cm\(^2\).
2. 1 d.p.: \(\ell=9.4\) cm, \(w=2.6\) cm. \(\ell\in[9.35,9.45)\), \(w\in[2.55,2.65)\). \(A\in[9.35\cdot2.55,\,9.45\cdot2.65]\approx[23.84,25.04]\) cm\(^2\).
3. 2 s.f.: \(\ell=2.0\) m, \(w=1.5\) m. \(\ell\in[1.95,2.05)\), \(w\in[1.45,1.55)\). \(P\in[2(1.95+1.45),\,2(2.05+1.55)]=[6.8,7.2]\) m.

Working carefully with rounding

Remember that rounded intervals usually have an open upper end (e.g., \([11.5,12.5)\)), because values at the upper midpoint would round up to the next integer/decimal. When you pass such an interval through a continuous increasing function (like \(2(\ell+w)\) or \(\ell w\) on non-negative inputs), the openness is preserved in principle. GCSE marking rarely hinges on the bracket style, but it is good mathematical hygiene to be consistent.

Reverse and mixed information

Sometimes only one side is rounded and the other is exact; sometimes a semi-perimeter or area is given. A few patterns:

• One side exact. If \( \ell \) is exact and \( w \in [L_w,U_w] \), then \(P\in[2(\ell+L_w),\,2(\ell+U_w)]\) and \(A\in[\ell L_w,\,\ell U_w]\).
• Given area bounds, find a side. If \(A\in[L_A,U_A]\) and \(w\) is exact and positive, then \( \ell \in \big[\,\tfrac{L_A}{w},\,\tfrac{U_A}{w}\,\big] \) (division by a positive number preserves order).
• Given perimeter bounds, find a side. If \(P\in[L_P,U_P]\) and \(w\) is exact, then \( \ell \in \big[\,\tfrac{L_P}{2}-w,\,\tfrac{U_P}{2}-w\,\big] \).

Percentage uncertainty: perimeter vs area

For small relative side errors, the perimeter error is roughly the same size (perimeter depends linearly on each side), whereas area error mixes multiplicatively. If both sides are a little high simultaneously, the area can be disproportionately high. This is built into the bounds \(L_{\ell}L_w\) and \(U_{\ell}U_w\): using the extreme pairings captures the “worst-case” combination.

Pitfalls

• Skipping the bounds step. Never substitute rounded values directly—first write \([L_{\ell},U_{\ell}]\) and \([L_w,U_w]\).
• Using mismatched endpoints for maxima/minima. For both perimeter and area (with non-negative sides), use lower with lower for the minimum and upper with upper for the maximum.
• Unit slips. Perimeter uses the base unit; area uses the squared unit. Do not mix cm with m without converting.
• Rounding too early. Keep intermediate values exact; round once at the end to the required precision.
• Negative or impossible dimensions. In measurement contexts sides are non-negative; if a calculation would make a side negative, you’ve misapplied a bound.

Exam strategy

• Underline the accuracy phrase and immediately write the two side intervals.
• Compute \(P_{\min},P_{\max}\) or \(A_{\min},A_{\max}\) with one neat line each.
• Give a final interval with units and—if requested—a decimal to the correct precision.
• Sanity-check: if \( \ell\approx 10 \) m and \( w\approx 4 \) m, expect \(P\) around \(28\) m and \(A\) around \(40\) m\(^2\); your bounds should straddle these estimates.

Extended micro-examples

1. Decking: \(\ell=10\) m, \(w=7\) m (nearest m). \(\ell\in[9.5,10.5)\), \(w\in[6.5,7.5)\). \(A_{\min}=9.5\cdot6.5=61.75\) m\(^2\); \(A_{\max}=10.5\cdot7.5=78.75\) m\(^2\). \(P_{\min}=2(9.5+6.5)=32\) m; \(P_{\max}=2(10.5+7.5)=36\) m.
2. Poster (3 s.f.): \(42.0\) cm by \(29.7\) cm. \([41.95,42.05)\) and \([29.65,29.75)\). \(P_{\max}=2(42.05+29.75)=143.6\) cm. \(A_{\min}=41.95\cdot29.65\approx1243.7\) cm\(^2\).
3. 1 d.p. both sides: \(\ell=3.6\) m, \(w=1.8\) m. \([3.55,3.65)\), \([1.75,1.85)\). \(A_{\max}=3.65\cdot1.85=6.75\) m\(^2\).

Summary

To bound a rectangle’s perimeter or area from rounded side lengths, first convert each measurement into a numerical interval. Because \(P=2(\ell+w)\) and \(A=\ell w\) are increasing in \( \ell \) and \( w \) for non-negative values, plug the lower endpoints to get the minimum and the upper endpoints to get the maximum: \[ P \in [\,2(L_{\ell}+L_w),\;2(U_{\ell}+U_w)\,],\qquad A \in [\,L_{\ell}L_w,\;U_{\ell}U_w\,]. \] Keep units correct, delay rounding until the end, and present a clean final interval. With this workflow you can produce quick, reliable bounds for any rectangle problem.