What is an orthogonal projection?
An orthogonal projection is the 'shadow' of an object cast onto a flat plane using rays that are all parallel to one another and perpendicular (at 90°) to the plane. Because the rays strike the plane squarely, the projection keeps the true shape and size of any part of the object that lies flat against the plane.
Key idea / Notation
An edge parallel to the plane projects to its full length. An edge perpendicular to the plane projects to a single point (length 0).
Projecting a solid
Think of a cuboid sitting on a table. Shine light straight down: the top face flattens exactly onto the base, so the projection onto the horizontal plane is a rectangle with the base's length and width. The vertical edges point straight at the plane, so each shrinks to a point. Projecting sideways onto a vertical wall instead uses the height and one of the horizontal dimensions.
Worked example
A cuboid measures 6 cm (length) × 4 cm (width) × 3 cm (height). Find the orthogonal projection onto the horizontal (base) plane and its area.
The base is parallel to the plane, so it projects to its true size: a 6 cm × 4 cm rectangle. The four vertical edges (3 cm) are perpendicular to the plane, so they project to points. Area = 6 × 4 = 24 cm².
Why it matters
Orthogonal projection is the rule behind plans and elevations: the plan is the projection onto a horizontal plane, while the front and side elevations are projections onto vertical planes.
Remember
- Projection rays are parallel and hit the plane at 90°.
- Parallel to plane → true length kept; perpendicular to plane → becomes a point.
- A flat face against the plane keeps its exact area.