Section Properties from Coordinates

The area of irregular closed polygons defined by coordinates may be conveniently calculated by summing the areas of adjacent trapeziums, as shown below.   The area of each trapezium is calculated from the X axis.  Note that with the apex coordinates defined in a clockwise direction the area between the bottom of the polygon and the x-axis is first added in (red trapeziums), then deducted (green trapeziums), leaving the correct area for the polygon, shaded blue. Other section properties, such as first and second moment of area, may be found in the same way, taking care to ensure that the formula gives a positive value for a positive XD, and negative for negative XD, for line segments above the X axis, and vice versa for segments below the X axis.
Area1

Arae1a

Voided shapes may be defined by connecting any external apex to one apex of the internal void, then listing the void apex coordinates in an anti-clockwise direction, then finally returning to the external apex.  This procedure is shown for a regular hollow octagon below, but may also be used for any irregular shape, with any number of voids.

Area2b

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2 Responses to Section Properties from Coordinates

  1. Pingback: Daily Download 20: Section Properties | Newton Excel Bach, not (just) an Excel Blog

  2. LAction says:

    Thanks doug!

    This is the same method I used to get the centroid of a polygon as part of an app I’m building since 2011. This process is the same one it’s used to calculate locations by modern GPS systems. My app does exactly the same, but applied to topography.

    Here’s part of the code of this method:

    Function Centroid(oRng As Range, Optional sType As String) As Variant

    Dim vCoords As Variant
    Dim vRow() As Variant
    Dim i As Long
    Dim Area As Double
    Dim xPos As Double, yPos As Double

    vCoords = oRng.Value

    Redim vRow(LBound(vCoords, 2) To UBound(vCoords, 2))

    For i = LBound(vCoords, 2) To UBound(vCoords, 2)
    vRow(i) = vCoords(1, i)
    Next i

    vCoords = AddRow(vCoords, vRow)

    Area = CalcArea(vCoords) ‘Note that this is a signed area; if the points are numbered in clockwise order then the area will have a negative sign

    xPos = CalcxPos(vCoords, Area)
    yPos = CalcyPos(vCoords, Area)

    If UCase(sType) = “X” Then
    Centroid = xPos
    ElseIf UCase(sType) = “Y” Then
    Centroid = yPos
    ElseIf UCase(sType) = “AREA” Then
    Centroid = Abs(Area)
    ElseIf UCase(sType) = “SAREA” Then
    Centroid = Area
    Else
    Centroid = “(” & Round(xPos, 3) & “,” & Round(yPos, 3) & “)”
    End If

    End Function
    Private Function CalcxPos(vCoords As Variant, Area As Double) As Double
    Dim i As Long

    For i = 1 To UBound(vCoords, 1) – 1
    CalcxPos = CalcxPos + (vCoords(i, 1) + vCoords(i + 1, 1)) * (vCoords(i, 1) * vCoords(i + 1, 2) – vCoords(i + 1, 1) * vCoords(i, 2))
    Next i

    CalcxPos = CalcxPos / (6 * Area)

    End Function
    Private Function CalcyPos(vCoords As Variant, Area As Double) As Double
    Dim i As Long

    For i = 1 To UBound(vCoords, 1) – 1
    CalcyPos = CalcyPos + (vCoords(i, 2) + vCoords(i + 1, 2)) * (vCoords(i, 1) * vCoords(i + 1, 2) – vCoords(i + 1, 1) * vCoords(i, 2))
    Next i

    CalcyPos = CalcyPos / (6 * Area)

    End Function
    Private Function CalcArea(vCoords As Variant) As Double
    Dim i As Long

    For i = 1 To UBound(vCoords, 1) – 1
    CalcArea = CalcArea + vCoords(i, 1) * vCoords(i + 1, 2) – vCoords(i + 1, 1) * vCoords(i, 2)
    Next i

    CalcArea = 0.5 * CalcArea

    End Function
    Private Function AddRow(InputArr As Variant, vRow As Variant) As Variant
    Dim vTemp As Variant
    Dim i As Long

    If LBound(vRow) LBound(InputArr, 2) Or UBound(vRow) UBound(InputArr, 2) Then AddRow = 0: Exit Function

    vTemp = TransposeArray(InputArr)

    Redim Preserve vTemp(LBound(vTemp, 1) To UBound(vTemp, 1), LBound(vTemp, 2) To UBound(vTemp, 2) + 1)

    vTemp = TransposeArray(vTemp)

    For i = LBound(vTemp, 2) To UBound(vTemp, 2)
    vTemp(UBound(vTemp, 1), i) = vRow(i)
    Next i

    AddRow = vTemp

    End Function
    Private Function TransposeArray(InputArr As Variant) As Variant

    Dim RowNdx As Long
    Dim ColNdx As Long
    Dim LB1 As Long
    Dim LB2 As Long
    Dim UB1 As Long
    Dim UB2 As Long
    Dim OutputArr() As Variant

    LB1 = LBound(InputArr, 1)
    LB2 = LBound(InputArr, 2)
    UB1 = UBound(InputArr, 1)
    UB2 = UBound(InputArr, 2)

    Redim OutputArr(LB2 To LB2 + UB2 – LB2, LB1 To LB1 + UB1 – LB1)

    For RowNdx = LB2 To UB2
    For ColNdx = LB1 To UB1
    OutputArr(RowNdx, ColNdx) = InputArr(ColNdx, RowNdx)
    Next ColNdx
    Next RowNdx

    TransposeArray = OutputArr

    End Function

    Like

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