The 3D Geometry of a Hexagonal Roof
Geometry & Trigonometry
A hexagonal summerhouse roof is built from six identical triangular panels meeting at a central apex. Find the panel angles and the roof's surface area and volume in terms of the base width and apex height.
Introduction
Consider a garden summerhouse with a regular hexagonal floor plan and a pyramid-shaped roof: six triangular panels rising from the six sides of the hexagon to meet at a central apex. The shape of each panel, and the roof's overall pitch, depends on two choices: the hexagon's side length and the height of the apex above the base. This exploration models the roof in 3D coordinates to find the angle each panel makes with the horizontal, the total area of roofing material needed, and how these change as the two design variables change.
Guiding Questions
- Set up 3D coordinates for the hexagon's base vertices and the apex. What is the angle between one roof panel and the horizontal base?
- How does the panel angle change as you increase the apex height while keeping the hexagon's side length fixed?
- Find the total surface area of the roof as a function of side length and apex height.
- Is there a height that minimises the roofing material needed for a fixed enclosed volume? Is that optimum practical to build?