A pyramid hip roof follows a specific set of criteria. The sides of a hip style roof slope downward to meet the exterior walls of the house. Pyramid hip roofs consist of four equal-sized triangular sections converging in a single point. Roofingkey.com asserts that pyramid style roofs give increased resistance to damaging winds. After taking a quick inventory of the roof's dimensions, calculating the inner and exterior angles turns into a simple matter of mathematics.
Measure each side of the roof. Attach one end of the tape measure to the outermost end of one side of the base, and measure the length. Measure all four sides to account for errors in calculation caused by "out of square" walls.
Measure the intended height of the roof. For existing structures, set up OSHA-approved scaffolding to go up through the rafters, and place one end of the tape at the highest interior point of the roof. Measure the distance from that point to the top base plate of the house. If you plan to build the structure, the roof height will appear on the blueprints. Write down all measurements.
Set your calculator to degree mode. For scientific calculators, turn the calculator on and press the "DRG" button until the upper right-hand display reads "DEG." For graphing calculators, press the "MODE" button, and change the setting to "RADIAN."
Calculate the angles. The sides of the roof make triangles, consisting of a base (the side attached to the house) and two sides that angle up to a point. Measure the distance from the middle of one side's base to the tip of its point. To find the slant angle of one side, use the following formula:
sin-1(height of roof / height of side)
Use the base angle to compute the angle between two sides at the vertex (top point) of the pyramid. Use the following formula to compute the top angle:
2*(90° - slant angle) = top angle
Choose a side. Calculate one angle of the side using the following formula:
sin-1(height of triangle / length of the side closest to the angle)
Draw a triangle with its hypotenuse on the line formed when two sides meet. Measure the length of the hypotenuse. Calculate the inside corner angle with the following formula:
sin-1(height of roof / hypotenuse of triangle)
Calculate the corner angle of the triangle. Measure the diagonal length of the roof (from corner to corner). Find the corner angle with this formula:
tan-1(height of roof / one-half diagonal length of roof)
Calculate the dihedral angle. Measure the roof's other diagonal base length. Compute the dihedral angle with this formula:
2*tan-1(first diagonal / second diagonal)
Check local permit laws and building codes before doing any type of construction. All calculators contain "Sin-1", "Cos-1" and a "Tan-1" buttons; this describes the inverse sine of an angle, but it gives the angle rather than the sine, cosine or tangent of the angle.
Always follow all safety precautions and be extremely careful when around construction.