Area of the orthogonal projection of a polygon
Use the theorem on the area of a projection, S′ = S · cos α, to find the area of a projection or of the polygon itself.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- Ukraine: grade 10
- Russia: grade 10
Guide
From the Ukrainian Grade 10 textbook (algebra and calculus with geometry), §5 'Perpendicularity in space', item 37 'Dihedral angle. The angle between planes'. The orthogonal projection of a polygon onto a plane is the polygon formed by the projections of all its points onto that plane.
Theorem 37.1 (area of the orthogonal projection of a polygon). The area of the projection of a convex polygon equals the product of its area and the cosine of the angle α between the polygon and its projection, where 0° ≤ α < 90°: S′ = S·cosα.
So the projection is never larger than the polygon itself. To find the polygon's area from the area of its projection, divide by the cosine: S = S′ : cosα.
Set the solution out as the textbook does: the area of the polygon, then the angle, then the formula, and finally the answer.