SME
Surveying · Problem type
Area from coordinates
Use this for “A parcel’s corners are (1000, 2000), (1180, 1950), (1240, 2130) and (1050, 2210) m, east then north. Find its area.”
Chart
Area = ½|−410,000 + 95,400 + 503,900 − 110,000| = 39,650 m².
This is an example. Type your own values to start a new problem.
E₁
East of corner 1
example · m
N₁
North of corner 1
example · m
E₂
East of corner 2
example · m
N₂
North of corner 2
example · m
E₃
East of corner 3
example · m
N₃
North of corner 3
example · m
E₄
East of corner 4
example · m
N₄
North of corner 4
example · m
A
Area
calculated
Formulas
A
=
1
2
×
|
E₁
×
N₂
−
E₂
×
N₁
+
E₂
×
N₃
−
E₃
×
N₂
+
E₃
×
N₄
−
E₄
×
N₃
+
E₄
×
N₁
−
E₁
×
N₄
|
39,650
=
1
2
×
|1,000
×
1,950
−
1,180
×
2,000
+
1,180
×
2,130
−
1,240
×
1,950
+
1,240
×
2,210
−
1,050
×
2,130
+
1,050
×
2,000
−
1,000
×
2,210|
Assumptions
•
The corners are adjusted coordinates, after the traverse is balanced.
•
They go in order round the edge, either way, and no two sides cross.
•
Plane coordinates: a small parcel’s area needs no correction for the earth’s curve.
Step-by-step
WE KNOW
E₁ = 1,000 m, N₁ = 2,000 m, E₂ = 1,180 m, N₂ = 1,950 m, E₃ = 1,240 m, N₃ = 2,130 m, E₄ = 1,050 m, N₄ = 2,210 m
FIND
area (A)
Step 1 · Find area (A)
Use A = ½ × |E₁ × N₂ − E₂ × N₁ + E₂ × N₃ − E₃ × N₂ + E₃ × N₄ − E₄ × N₃ + E₄ × N₁ − E₁ × N₄|
Cross-multiply each corner with the next one round, add the four terms, and halve the size of the sum.
A
=
1
2
×
|1,000
×
1,950
−
1,180
×
2,000
+
1,180
×
2,130
−
1,240
×
1,950
+
1,240
×
2,210
−
1,050
×
2,130
+
1,050
×
2,000
−
1,000
×
2,210|
1,000 × 1,950 − 1,180 × 2,000 = −410,000
1,180 × 2,130 − 1,240 × 1,950 = 95,400
1,240 × 2,210 − 1,050 × 2,130 = 503,900
1,050 × 2,000 − 1,000 × 2,210 = −110,000
95,400 + 503,900 − 410,000 − 110,000 = 79,300
A
=
|79,300|
÷
2
A
=
39,650
m²
Check
Put the numbers back in. Both sides should match.
39,650
=
1
2
×
|1,000
×
1,950
−
1,180
×
2,000
+
1,180
×
2,130
−
1,240
×
1,950
+
1,240
×
2,210
−
1,050
×
2,130
+
1,050
×
2,000
−
1,000
×
2,210|
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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