96-calterWe are all familiar with the trigonometry textbook problem, the angle of elevation to the top of a building from a point 200 feet from... Find the height of the building. Here we describe a trigonometric method that not only measures heights of points on a building, but widths and depths of those points. The procedure involves readings with a theodolite, and computations of the readings by a computer. The method is suitable for sighting from sloping ground. It will work with walls that are leaning out of plumb, have offsets, are curved, or have projecting elements, like sills or cornices. The method was developed for the purpose of measuring historic monuments, where the erection of scaffolding to accomodate hand measuring is often impossible. This method also provides a low-cost alternative to stereogrammetric procedures, such as that used to measure Independence Hall in Philadelphia.

The Method

1. Study of facade and preparation of preliminary drawings to determine target points.
2. Determination of a base line, and establishment of two theodolite positions.
3. Set the theodolite at position A, sight a point T1 on the wall, and sight a plumb line over theodolite position B, setting horizontal scale to zero.
4. Sight each target point, recording horizontal angle and vertical angle.
5. Set the theodolite at position B, sight a point in line T2 vertically with T1, and measure the vertical distance between them.
6. Repeat step 4.
7. Enter all measurements into the computer spreadsheet and print out the x, y and z coordinate of each target point. Make a final scale drawing.

The method was tested by taking measurements of the facade of Green Academic Center at Vermont Technical College. Two equations are used to calculate the y coordinates, and the comparison of the two results provides a way to control the accuracy of the method. It appears that, with moderate care, accuracies within .5% are easily obtained. Theoretically, better accuracy may be obtained by taking a readings from a third theodolite setup point.

Paul A. Calter is a Visiting Scholar at Dartmouth and Professor Emeritus of Mathematics at Vermont Technical College. He has interests in both the fields of mathematics and art. He received his B.S. from Cooper Union and his M.S. from Columbia University, both in engineering, and his Masters of Fine Arts Degree at Vermont College of Norwich University. Calter has taught mathematics for over twenty-five years and is the author of ten mathematics textbooks and a mystery novel. He has been an active painter and sculptor since 1968, has participated in dozens of art shows, and has permanent outdoor sculptures at a number of locations in Vermont. For the "Mathematics Across The Curriculum" program, Calter developed the course "Geometry in Art & Architecture" and has taught it at Dartmouth and Vermont Technical College, as well as giving workshops and lectures on the subject. He is the author of "How to Construct a Logarithmic Rosette (Without Even Knowing it)" in the NNJ vol. 2, no. 2 (April 2000) and presented "Facade Measurement by Trigonometry" at Nexus '96, now available in Nexus: Architecture and Mathematics (1996).

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The correct citation for this paper is:
Paul Calter "Facade Measurement by Trigonometry", pp. 27-35 in Nexus: Architecture and Mathematics, ed. Kim Williams, Fucecchio (Florence): Edizioni dell'Erba, 1996.