Name: shoelace formula for traverse area
Uploaded: 2025-03-26T15:21:51.000Z
Duration: 1 min 24 s
Channel: JoVE

The alternative coordinate method computes the area within the traverse using its x and y coordinates, which are arranged sequentially around the shape. Each corner's coordinates are written as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator, listed horizontally in order. The first corner's fraction is repeated at the end of the sequence.Draw solid diagonal lines connecting each x-coordinate to the next y-coordinate in sequence. Similarly, draw dashed diagonal lines connecting each y-coordinate to the next x-coordinate in the same sequence.Calculate the solid-line contributions by multiplying each x-coordinate with the next y-coordinate.Similarly, multiply each y-coordinate by the next x-coordinate to compute the dashed-line contributions. Twice the area within the traverse is calculated by subtracting the total of the dashed-line products from the total of the solid-line products. Take the absolute value of this difference and divide it by two to determine the exact area within the traverse. This method is widely used for its simplicity, accuracy, and adaptability for manual and computer-based calculations in surveying.

area computation by the alternative coordinate method

The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially around the traverse, ensuring the first corner&#8217;s coordinate is repeated at the end of the sequence to close the shape.Solid diagonal lines are drawn from each x-coordinate to the subsequent y-coordinate, while dashed diagonal lines connect each y-coordinate to the subsequent x-coordinate. The products of these coordinates are calculated for both the solid and dashed lines:<ul><li>Solid-line contributions: Multiply each x-coordinate by the y-coordinate of the next point in the sequence.</li><li>Dashed-line contributions: Multiply each y-coordinate by the x-coordinate of the subsequent vertex.</li></ul>The area of the traverse is obtained by subtracting the total of dashed-line products from the total of solid-line products and dividing the absolute difference by two:<figure class="image image_resized image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;width:64.17%;"><img style="aspect-ratio:2646/204;" src="https://cloudfront.jove.com/files/ftp_upload/18111/18111_Equation_1_1__editor_18111.svg" alt="Equation for area within traverse; &#931; calculus, solid vs. dashed lines, formula analysis." data-altupdatedat="1762173717000" data-alt="18111_Equation_1_1_.svg" /></figure>The alternative coordinate method is preferred in surveying for its simplicity, adaptability, and compatibility with manual and computer-assisted computations. Its systematic structure minimizes computational errors and offers a reliable way to handle complex traverses in geospatial analysis.

Area Computation by the Alternative Coordinate Method

The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. ...

area-computation-by-the-alternative-coordinate-method

Education

JoVE Core

Civil Engineering

Unit 1

Traverse and Topographic Surveying

KEY TERMS AND CONCEPTS

shoelace formula for traverse area

The Shoelace Formula is a coordinate method for finding the area of a traverse. It uses Cartesian coordinates, which are x and y values on a graph, to measure the shape accurately and with a simple step-by-step setup.

The points of the traverse are listed in order around the shape. Each corner is written as a fraction, with the x-coordinate on top and the y-coordinate on the bottom. The first corner is repeated at the end of the list to close the shape.

Next, diagonal lines are used to organize the calculations. Solid lines connect each x-coordinate to the next y-coordinate. Dashed lines connect each y-coordinate to the next x-coordinate. The products from the solid lines are added together, and the products from the dashed lines are added together.

The area is found by subtracting the dashed total from the solid total. The absolute difference is then divided by two. This method is often used in surveying because it is simple, works well for manual or computer-assisted calculations, and helps reduce errors when dealing with complex traverses.

Shoelace Formula for Traverse Area

adjusting a traverse

Correcting Traverse Angle Errors

Correcting traverse angle errors is an important step in site surveying. In a four-sided traverse, the internal angles must add up to 360 degrees. Survey data first showed a total of 359 degrees and 48 minutes, which was 12 minutes short of the expected sum. That gap pointed to a likely measurement error from fieldwork. The correction was made by spreading the 12-minute shortfall evenly across the four internal angles. Three minutes were added to each angle, and the values were recalculated.

latitudes and departures

Balancing Traverse Latitude and Departure

Latitudes and departures are key parts of surveying that help break a traverse line into north-south and east-west parts. These projections let surveyors work with line direction, length, bearings, and area in a clear way. They are also useful for measuring and mapping land accurately. Latitude is the north-south projection of a line. It is found by multiplying the line length by the cosine of its bearing. Departure is the east-west projection. It is found by multiplying the line length by the...

The Shoelace Formula is a coordinate method for finding the area of a traverse. It uses Cartesian coordinates, which are x and y values on a graph, to measure the shape accurately and with a simple step-by-step setup. The points of the traverse are listed in order around the shape. Each corner is written as a fraction, with the x-coordinate on top and the y-coordinate on the bottom. The first corner is repeated at the end of the list to close the shape. Next, diagonal lines are used to...

areas within irregular boundaries

Measuring Irregular Land Areas

Measuring irregular land areas often starts with a traverse. A traverse is a connected series of straight lines that follows the boundary as closely as possible. Surveyors use the coordinates of each traverse point to help calculate the enclosed area. One common method is the double meridian distance formula. It uses the vertex coordinates of the traverse to find the area inside the straight-line boundary. This gives a practical base for later calculations. The space between the traverse and...

design example: marking boundaries of a site using a compass

design-example-marking-boundaries-of-a-site-using-a-compass

Compass Traverse for Site Boundary Marking

Marking a site boundary with a compass uses a traverse, which is a line-by-line surveying path. The method relies on the bearings and lengths of each boundary side. It helps surveyors place corner points with accuracy. The process begins with the site details. These include the bearing and length of every boundary line. Surveyors first calculate the latitude and departure of each side. Latitude is the north-south part of a line, and departure is the east-west part. This check shows whether the...

topographic surveying and contours

Reading Terrain with Topographic Maps

Topographic surveying records the shape of land by measuring elevation, slope, and natural and man-made features. Surveyors use these measurements to build a topographic map, which shows the terrain in a visual way. These maps are important for construction planning, water resource management, and land-use analysis. Contour lines are the key feature of a topographic map. A contour line connects points with the same elevation, so it helps show the relief of the land. The spacing between contour...

plotting of topographic maps

Drawing Contour Lines from Elevation Data

Topographic maps use contour lines to show the shape of the land from elevation data. A contour line connects points with the same elevation. This makes a two-dimensional map that shows three-dimensional terrain in a clear way. To build the map, start with a scaled grid. Mark the intersections that match the survey’s elevation data points. Then assign elevation values at those points to create the base map. Next, choose a contour interval, which is the vertical change between contour lines.

methods of obtaining topography

Topographic Surveying: Old and New Methods

Topographic surveying uses land measurements to map elevations, natural features, and artificial structures. The goal is to create an accurate picture of the terrain. Surveyors have used both traditional and modern methods to do this work, and each approach has its own strengths and limits. Traditional surveying methods included transit stadia surveys and plane table surveys. These methods used tools such as theodolites and stadia rods to measure distances and angles. Surveyors then wrote the...

design example: analyzing capacity contours for flood risk assessment

design-example-analyzing-capacity-contours-for-flood-risk-assessment

Setting Bund Height to Reduce Flood Risk

Capacity contours help engineers check flood risk when a bund is built across a small valley. A bund is a raised barrier used here to store stormwater. The contour map shows how far water could spread at a chosen level, which helps judge whether nearby homes are safe. The process starts by placing the proposed bund in the valley and estimating the high flood level. Engineers then use leveling instruments to find and mark points along the upstream side that match this water level. Those points...

key terms and concepts

Learn how the Shoelace Formula uses ordered x and y coordinates to find traverse area with a simple subtraction step, ideal for surveying and geometry.

Shoelace Formula for Traverse Area | JoVE K12

JoVE Core Civil Engineering offers a foundational exploration of essential construction materials such as bricks, stones, steel, wood, and concrete, along with fluid mechanics topics like flow over immersed bodies, open channel flow, and dimensional modeling. It also covers key surveying concepts, including leveling, traverse adjustments, curve design, topographic mapping, and geospatial technologies (GPS & GIS). Science-in-action videos showcase laboratory research on materials, fluids, and surveying practices in real-world engineering systems.

civil engineering

Shoelace Formula for Traverse Area

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