A Plot That Will Not Close: Heron's Formula and Measurement Strategy
A Class X mathematics group must find the area of a triangular plot ABC beside the school wall. Using a 30 m steel tape they record AB = 7.00 m, BC = 8.00 m and AC = 15.10 m. When they apply Heron's formula the calculation fails: the semi-perimeter is 15.05 m, one factor (s − AC) is negative, and the square root cannot be taken. A second measurement of AC gives 14.95 m and an area of about 6.4 m²; a third gives 15.05 m and again fails. The plot is in fact a long, narrow strip, and the perpendicular distance from B to the line AC is clearly less than one metre. The group has a metre scale, a plumb line and a right-angle cross-staff, but only one practical period left. They must decide how to obtain a defensible area and how to report the uncertainty, instead of simply averaging the three readings of AC.
1. Explain why Heron's formula gave an impossible result, and what that tells the group about the measurements.
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The failure is not an arithmetic slip; it is a signal that the readings are geometrically inconsistent.
A triangle with sides 7 and 8 can have a third side of any length strictly between 1 and 15. At exactly 15 the three vertices lie on a straight line and the area is zero. The readings 15.10 m and 15.05 m both violate the triangle inequality (7 + 8 = 15), which is why (s − c) turns negative and the square root is undefined. In Heron's expression,
A = √[s(s − a)(s − b)(s − c)]
the factor (s − c) becomes negative exactly when c exceeds a + b.
The deeper point is that the plot is nearly degenerate. As AC approaches 15.00 m, the area approaches zero and the altitude from B to AC approaches zero. In that region a change of a few centimetres in the measured diagonal causes a very large change in the computed area. Averaging the three readings would hide this; the group must recognise that the long diagonal is the wrong quantity to build the calculation on.
2. Compare the sensitivity of the two possible methods and justify a choice of measurement plan.
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Method 1 - three sides and Heron's formula. Near the degenerate limit the sensitivity dA/dc grows without bound. Using the readings: at AC = 14.90 m the area is about 6.4 m², at AC = 14.95 m it falls to about 4.6 m², and at 15.00 m it is zero. A 5 cm error in a 15 m length has changed the area by roughly 2 m², and near 15.00 m a 1 cm error can change it by more than 1 m². A tape readable to 1 cm therefore cannot give a reliable area by this route.
Method 2 - base × height. Measure the base along AC (≈15 m) and measure the perpendicular height h from B to AC directly. Since A = ½ × base × h and h ≈ 0.85 m, a 1 cm error in h changes the area by only about 0.075 m², roughly 1 per cent. The error in the long base contributes similarly little.
Choice: Method 2, and report it honestly. The reason is not that the triangle is special but that the area formula is well-conditioned when the small quantity is measured directly instead of being inferred by subtraction.
3. Set out the procedure for the final practical period, and state the area with a realistic uncertainty.
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Procedure:
- Stretch the tape along AC and record the length to the nearest centimetre, e.g. 15.00 m; keep the tape taut and horizontal along the ground.
- From B, drop a plumb line or use the cross-staff to locate the foot of the perpendicular D on line AC. Check squareness with a 3–4–5 test (3 m and 4 m along the two directions should give 5 m between the marks) rather than trusting the eye.
- Measure BD with the metre scale, keeping the scale perpendicular to AC, and repeat from the other side.
- If D falls outside segment AC, the triangle is obtuse at A or C and the base should be extended along the wall line before measuring; note this in the record.
Result: with base 15.00 m and h = 0.85 m, A = ½ × 15.00 × 0.85 ≈ 6.4 m².
Uncertainty: a 1 cm error in h gives about 0.08 m², and 2 cm in the base about 0.01 m²; allowing for the plumb-line setting, report 6.4 ± 0.1 m². The three original readings of AC should be recorded as rejected, with the reason (violation of the triangle inequality), not averaged.