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# How to use trigonometry in architecture

Updated July 19, 2017

Mathematics makes the design of buildings safer and more accurate. Trigonometry is especially important in architecture because it allows the architect to calculate distances and forces related to diagonal elements. Of the six functions in basic trigonometry, the sine, cosine and tangent are the most important to architecture because they allow the architect to easily find the opposite and adjacent values related to an angle or hypotenuse, translating a diagonal vector into horizontal and vertical vectors.

Calculate the horizontal span of a diagonal element by multiplying the total length of the element by the cosine of the diagonal element's angle.

Some diagonal elements that this is useful for include braces, bridge cables and rafters.

Calculate the vertical height of a diagonal element by multiplying the total length of the element by the sine of the diagonal element's angle.

Some diagonal elements that this is useful for include roofs, retaining walls, and landscape elevation changes.

Calculate the height of a structure by multiplying the length of its shadow by the tangent of the sun's angle.

Find the angle of an element by dividing the element's height by its span, and multiply this quotient by the inverse tangent. This is very useful for finding the slope of a roof or the ground.

Calculate the amount of force a diagonal element's support must hold by multiplying the total amount of load the element is carrying across its diagonal orientation by the sine of the diagonal element's angle.

Calculate the amount of horizontal force exerted through a diagonal element that must be retained by multiplying the total amount of load carried through the diagonal orientation by the cosine of the diagonal element's angle.

Calculate the distance to an object of known height by dividing the height of the object by the tangent of the angle measured from the objects bottom to top. The inverse of the answer is the object's distance.

#### Things You'll Need

• Calculator
• Pencil
• Paper
• Architectural design