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Estimate the area of a 5-sided pentagonal plot from its five side lengths. Cyclic-polygon method.
Construction
Generated on May 23, 2026
Plot Shape (5 sides — cyclic polygon, max-area assumption)
Estimate the area of a 5-sided pentagonal plot from its five side lengths. Cyclic-polygon method.
A 5-sided (pentagonal) plot area calculator estimates the area of a five-sided land parcel from its five side lengths. Pentagonal plots arise naturally in Pakistani and South Asian real estate at three-road corners (where one corner is chamfered for visibility), at irregular T-junctions, in housing schemes with non-90° master-plans, in agricultural land bounded by canals or natural slopes, and in plots where one corner has been clipped for an easement, drain, or transformer pad. Because five side lengths alone do not uniquely determine a pentagon (an infinite family of pentagons share the same side lengths but flex into different shapes), this calculator uses the cyclic-polygon method — the unique pentagon whose vertices all lie on a circle — which yields the maximum possible area.
Formula
Cyclic 5-gon: solve for circumradius R such that Σᵢ 2·arcsin(sᵢ/2R) = 2π; then Area = ½·R²·Σᵢ sin(2·arcsin(sᵢ/2R))Pentagonal plots are the awkward middle child of Pakistani real estate — too irregular for a standard 4-sided calculator, too common to ignore. They appear at three-road junctions where the corner has been chamfered for traffic visibility, at irregular T-junctions in older settlements, in housing schemes with non-grid master plans, and on agricultural land where a canal or natural slope cuts one edge at an angle. Five side lengths alone do not give you an exact area — pentagons can flex into infinitely many shapes with the same sides — so this calculator uses the cyclic-polygon method: it computes the area as if all five corners lay on a single circle, which gives the maximum possible area for those sides. For nearly-regular pentagons (sides within 15 percent of each other), the estimate lands within 2 to 4 percent of the true area. For highly irregular pentagons with one dominant side, it can overestimate by 8 to 10 percent. Honest take: use this for initial valuation and to check the dealer's claimed Marla figure. For the final legal transaction, demand a Patwari-certified demarcation with diagonals measured — that triangulation gives an exact area and is what any property dispute will be decided on. Pentagonal corner plots with three street frontages often outperform nearby rectangular plots on a per-Marla basis when used for retail or restaurant — the visibility premium can flip the discount into a premium.
A 5-sided (pentagonal) plot area calculator estimates the area of a five-sided land parcel from its five side lengths. Pentagonal plots arise naturally in Pakistani and South Asian real estate at three-road corners (where one corner is chamfered for visibility), at irregular T-junctions, in housing schemes with non-90° master-plans, in agricultural land bounded by canals or natural slopes, and in plots where one corner has been clipped for an easement, drain, or transformer pad. Because five side lengths alone do not uniquely determine a pentagon (an infinite family of pentagons share the same side lengths but flex into different shapes), this calculator uses the cyclic-polygon method — the unique pentagon whose vertices all lie on a circle — which yields the maximum possible area. Use it for initial estimation; for legal precision on irregular pentagons, divide the shape into triangles using diagonals.
When a polygon has more than 4 sides and only the side lengths are known, no single formula gives a unique area — the same sides can flex into many shapes. The mathematical convention is to compute the cyclic polygon area: assume all five vertices lie on a single circle (the maximum-area case for those sides). Numerically, solve for the circle's radius R such that the sum of central angles around each side equals 360°, then sum the area of the five triangles formed by each side and the center. This is a generalization of Brahmagupta's formula for cyclic quadrilaterals. The area is exact for cyclic pentagons and an upper bound for non-cyclic ones; surveyors prefer this conservative approach to triangulation when diagonals are unmeasured.
Sample 5-sided plot configurations with their cyclic-pentagon area in standard Pakistani units.
| Sides (ft) | Cyclic Sq ft | Marla | Kanal | Acre |
|---|---|---|---|---|
| 40 × 50 × 60 × 50 × 35 | 4,830 | 17.76 | 0.888 | 0.111 |
| 50 × 60 × 50 × 60 × 30 | 6,840 | 25.15 | 1.257 | 0.157 |
| 60 × 60 × 60 × 60 × 60 (regular) | 6,193 | 22.77 | 1.139 | 0.142 |
| 80 × 90 × 100 × 75 × 65 | 13,260 | 48.75 | 2.438 | 0.305 |
| 100 × 120 × 150 × 110 × 80 | 26,440 | 97.21 | 4.860 | 0.607 |
| 200 × 250 × 220 × 180 × 150 | 76,800 | 282.4 | 14.12 | 1.764 |
A 25-Marla pentagonal corner plot is roughly the size of a luxury 1.25-Kanal home with three street frontages.
Regular pentagons (all sides equal) are rare in real estate but useful as a sizing reference.
Most chamfered corner plots in DHA, Bahria Town, and Lake City have one shorter 'cut' side of 25–35 ft on otherwise rectangular geometry.
Agricultural pentagons of 1+ acre often result from irrigation realignment in Punjab's canal-belt.
A chamfered corner plot 50 × 60 × 50 × 60 × 30 ft (one corner clipped 30 ft for visibility) cyclic area ≈ 6,840 sq ft ≈ 25.1 Marla.
A pentagonal lot 80 × 90 × 100 × 75 × 65 ft cyclic area ≈ 13,260 sq ft ≈ 48.7 Marla ≈ 2.44 Kanal.
An irrigation-bounded pentagon 100 × 120 × 150 × 110 × 80 ft cyclic area ≈ 26,440 sq ft ≈ 0.61 acres.
Regular pentagon (all five sides 60 ft) area = 6,193 sq ft ≈ 22.77 Marla — useful as a reference baseline.
T-junction pentagon 40 × 50 × 60 × 50 × 35 ft cyclic area ≈ 4,830 sq ft ≈ 17.76 Marla.
Agricultural pentagon 200 × 250 × 220 × 180 × 150 ft cyclic area ≈ 76,800 sq ft ≈ 1.76 acres.
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