SRV-30

Trapezoid area

A = (h/2)(a + b). Two parallel offsets and the interval h.

Reading speed

AreaTrapezoidal rule

Governing equation

A=\tfrac12 h(a+b)

$h \in \left[1, 50\right]$

where

h: Interval (m)
a: Offset a (m)
b: Offset b (m)
A: Area (m²)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

A=\tfrac12 h(a+b)

$h \in \left[1, 50\right]$

$h = 1$ $A = 9.8$

Along the curve

②

Domain of the variable — blue

h_{\min} = 1,\quad h_{\max} = 50

$h \in \left[1, 50\right]$

$h = 1$ $h = 1$

Along the curve

③

Response curve — green

A=\tfrac12 h(a+b)

$h \in \left[1, 50\right]$

$h = 1$ $A = 9.8$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}A}{\mathrm{d}h}

$h \in \left[1, 50\right]$

$h = 1.308$ $A^{\prime} = 9.8$

Along the curve

Lecture brief

Historical brief

From chain and theodolite through Bowditch’s compass-rule (1802) to GPS PDOP, surveying is the propagation of small errors across a network. These sheets close traverses, reduce stadia and state map scale. This sheet (SRV-30 — Trapezoid area) is the form associated with Trapezoidal rule. Working symbols: $h$ , $a$ , $b$ $\rightarrow$ $A$ . The trapezoidal rule for a cross-section or a chain of offsets. Simpson is the next order.

Purpose

Purpose: compute

A

from

h

a

b

in Surveying via

A=\tfrac12 h(a+b)

A = (h/2)(a + b). Two parallel offsets and the interval h. Use it when a real surveying question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

h = 10.000\,\mathrm{m}

a = 8.400\,\mathrm{m}

b = 11.200\,\mathrm{m}

, the governing relation

A=\tfrac12 h(a+b)

yields

A = 98.00\,\mathrm{m^{2}}

. Two offsets, a spacing, an area. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Interval h10 mOffset a a8.4 mOffset b b11.2 m

Outputs

Area A98.00 m²

Reading speed

Watch on YouTube

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Narration of this film

Two offsets, a spacing, an area.

The trapezoidal rule for a cross-section or a chain of offsets. Simpson is the next order.

Reading speed

Watch on YouTube