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Grade 12 / High 3 (age 17-18)

high school Space Figures

Space Figures

On flat paper, two lines can only meet or run parallel. But in the real 3D space we live in there are also 'skew lines' that neither meet nor stay parallel, and a line and a plane, or two planes, can be parallel, intersecting, or perpendicular. The dihedral angle, which measures how far two planes are spread apart, is the same idea as the angle of an open book. Rotate the view with the slider and the solid sits in space; switch the position relation and the dihedral angle and the lines and planes shift.

Space Coordinate Axes

30°
👀 3D Space
①Adding a y-axis to the plane (2D) gives space (3D)
②x-axis (red), y-axis (green), z-axis (blue) are mutually perpendicular
③Rotate the view to feel the depth
④Every point is described by three coordinates (x, y, z)

Position of Lines and Planes

∥
📐 Position Relations
①0: Two planes are parallel (no intersection)
②1: Two planes meet to form an intersection line
③2: A line is perpendicular to a plane (perpendicular to every line in it)
④In space there are also 'skew lines' — neither intersecting nor parallel

Dihedral Angle

60°
📖 What is a Dihedral Angle?
①Two half-planes share a common edge (intersection line)
②From a point on the edge, draw rays in each plane perpendicular to the edge
③The angle between those rays is the dihedral angle
④Same idea as the angle of an open book

Perpendicularity and Parallelism Theorems

Three Perpendiculars Theorem
If PH ⊥ α then (PO ⊥ ℓ ⟺ HO ⊥ ℓ). P is off the plane, H is the foot, O is on ℓ
Determine perpendicularity to a line in the plane via the foot of perpendicular
Parallel Test
ℓ ∥ α ⟺ ℓ ∥ m (some line m ⊂ α exists)
If parallel to some line in the plane, the line is parallel to the plane
🔍 Key Theorems
①Line ⊥ plane: perpendicular to every line in the plane
②Two planes ⊥: a line perpendicular to one is contained in the other
③Three perpendiculars theorem: a frequent test topic
④Drawing diagrams while tracing the proofs speeds understanding
⑤Distance between skew lines is in the next chapter

Work It Out

Example 1 — Three Perpendiculars Theorem
From a point P not on line ℓ in plane α, let H be the foot of the perpendicular from P to α, and O the foot of the perpendicular from H to ℓ. If PH=3 and HO=4, find the length PO.
1
By the three-perpendiculars theorem PO⊥ℓ, and triangle PHO is right-angled at H.
PH ⊥ α, HO ⊥ ℓ ⇒ PO ⊥ ℓ
2
In right triangle PHO, use the Pythagorean theorem to find the hypotenuse PO.
PO = √(PH² + HO²) = √(9 + 16) = √25
▸ PO = 5
Since PH is perpendicular to the plane, PH is also perpendicular to HO, so triangle PHO is a right triangle.
Example 2 — Orthogonal Projection
A figure of area 12 lies in plane β, which forms a 60° dihedral angle with plane α. Find the area of its orthogonal projection onto α.
1
The projected area equals the original area times the cosine of the angle between the planes.
S' = S · cosθ
2
Substitute S=12 and θ=60°.
S' = 12 · cos60° = 12 · (1/2)
▸ S' = 6
A larger angle gives a smaller cosine, so the projection shrinks — at 90° the projected area becomes 0.

Summary

Projection Area
S' = S cosθ
Projected area = original area × cos(dihedral angle)
2021 KICE mock exam Math (Geometry) type, adapted
A square with side length 4 forms a 45° angle with plane α. What is the area of its orthogonal projection onto α?
①8
②8√2
③16
④4√2
⑤4
▸ ② 8√2
1
The area of the square is 4 × 4 = 16.
2
Substitute S=16 and θ=45° into the projection-area formula.
S' = 16 · cos45° = 16 · (√2/2) = 8√2
🎯 Exam Points
①Identify the 5 position relations of lines/planes in space
②Three perpendiculars theorem: state conditions and conclusion exactly
③Find dihedral angle — two rays perpendicular to the edge
④Projected area = original × cosθ
⑤Distance between skew lines
← Previous
Components and Dot Product of Plane Vectors
Next →
Space Coordinates
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