CC - The Starburst Pavilion Design - Hugo G.
Problem Statement:
An architectural firm is designing the roof structure for an open-air park pavilion. The geometric blueprint, created in GeoGebra, forms a large four-pointed star with a hollow center.
The structure is perfectly symmetrical and consists of:
- A central square opening (the white space in the middle).
- Four identical trapezoids (the shaded interlocking sections) that wrap around the center. For each trapezoid, the shorter parallel side is 3 meters, the longer parallel side is 5 meters, and the perpendicular height between them is 2 meters.
- Four identical isosceles triangles (the blue points of the star) attached to the outer edges of the trapezoids. The base of each triangle matches the outer edge of the trapezoid, and the height from the base to the outer tip is 4 meters.
1. GeoGebra Replication
Recreate this exact 2D geometric pattern using GeoGebra. Start by constructing the central square, then use the polygon tool to construct the four interlocking trapezoids, and finally attach the four outer triangles. Ensure all vertices align perfectly to maintain the 4-fold rotational symmetry.
2. Total Solid Material Area
The shaded and blue sections represent the physical roof, while the central white square is open to the sky. Calculate the total surface area of the solid roof structure (the area of the 4 trapezoids plus the 4 triangles) to determine how much roofing material needs to be ordered.
3. The Central Skylight
The design team needs to calculate the area of the central square opening to see how much sunlight will reach the center of the pavilion. If the inner sides of the trapezoids form a perfect square with a side length equal to the shorter parallel side 3 meters, find the area of this central square.