SME
Geographic Information Systems (GIS)
Spatial analysis
Use this for “A parcel has corners (100, 50), (600, 150), (500, 450) and (200, 350) in metres. What is its area?”
Chart
x₁y₂ − x₂y₁ = 100 × 150 − 600 × 50
x₁y₂ − x₂y₁ = −15,000
x₂y₃ − x₃y₂
x₂y₃ − x₃y₂ = 600 × 450 − 500 × 150 = 195,000
x₃y₄ − x₄y₃
x₃y₄ − x₄y₃ = 500 × 350 − 200 × 450 = 85,000
x₄y₁ − x₁y₄ = 200 × 50 − 100 × 350
x₄y₁ − x₁y₄ = −25,000
Area = ½|−15,000 + 195,000 + 85,000 − 25,000| = 120,000 m².
This is an example. Type your own values to start a new problem.
x₁
x of corner 1
example
y₁
y of corner 1
example
x₂
x of corner 2
example
y₂
y of corner 2
example
x₃
x of corner 3
example
y₃
y of corner 3
example
x₄
x of corner 4
example
y₄
y of corner 4
example
A
Area
worked out (not typed)
Formulas
A
=
1
2
×
|
x₁
×
y₂
−
x₂
×
y₁
+
x₂
×
y₃
−
x₃
×
y₂
+
x₃
×
y₄
−
x₄
×
y₃
+
x₄
×
y₁
−
x₁
×
y₄
|
120,000
=
1
2
×
|100
×
150
−
600
×
50
+
600
×
450
−
500
×
150
+
500
×
350
−
200
×
450
+
200
×
50
−
100
×
350|
Assumptions
•
The corners go in order round the edge, and no two sides cross.
•
Projected coordinates in metres give the area in square metres; 1 hectare is 10,000 m².
•
Anticlockwise corners make the sum positive and clockwise ones negative; the area is its size.
Step-by-step
WE KNOW
x₁ = 100 m, y₁ = 50 m, x₂ = 600 m, y₂ = 150 m, x₃ = 500 m, y₃ = 450 m, x₄ = 200 m, y₄ = 350 m
FIND
area (A)
Step 1 · Find area (A)
Use A = ½ × |x₁ × y₂ − x₂ × y₁ + x₂ × y₃ − x₃ × y₂ + x₃ × y₄ − x₄ × y₃ + x₄ × y₁ − x₁ × y₄|
Cross-multiply each corner with the next one round, add the four terms, and halve the size of the sum.
A
=
1
2
×
|100
×
150
−
600
×
50
+
600
×
450
−
500
×
150
+
500
×
350
−
200
×
450
+
200
×
50
−
100
×
350|
100 × 150 − 600 × 50 = −15,000
600 × 450 − 500 × 150 = 195,000
500 × 350 − 200 × 450 = 85,000
200 × 50 − 100 × 350 = −25,000
195,000 + 85,000 − 15,000 − 25,000 = 240,000
A
=
120,000
m²
Check
Put the numbers back in. Both sides should match.
120,000
=
1
2
×
|100
×
150
−
600
×
50
+
600
×
450
−
500
×
150
+
500
×
350
−
200
×
450
+
200
×
50
−
100
×
350|
✓
Topic check
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