Place a protractor along a sloping line in a roof photograph: the result changes when the photographer walks around the house. How do we find the actual inclination? First choose the spatial angle, then the plane in which to measure it. One right triangle will suffice for our model.
Which angle are we looking for?
Prerequisites: right-triangle trigonometry, lines and planes, perpendicular, slant and projection.
The angle between two unoriented lines is the smaller angle between their directions, from 0° to 90°. Parallel or coincident lines have angle 0°. For skew lines, draw lines parallel to them through one point and measure the resulting angle. Translation preserves directions. Thus skew lines can also have angle 90° although they never meet; “perpendicular” below can refer to this perpendicularity of directions.
The vector angle in G021 is different: arrows specify orientation, so it can reach 180°. Lines have no arrows: 40° and 140° give the same smaller angle of 40°.
The angle between a line and a plane is the angle between the line and its orthogonal projection. It is 0° for a line contained in or parallel to the plane. It is 90° for a normal; its projection is a point, so we define that case separately. An arbitrary line on the floor cannot replace the projection.
A roof: choose the correct section
Define our own ideal roof: a horizontal eave and a planar roof surface. Moving 2.4 m horizontally from the eave, perpendicular to the eave, the height increases by 1.8 m. The corresponding sloping segment has length √(2.4²+1.8²)=3 m. This is a model, not a measurement of a real structure.
Two half-planes sharing their boundary line form a dihedral angle. Its edge is the eave. Cut both faces with a plane perpendicular to the edge: the angle between the resulting rays is the dihedral angle’s linear angle. Each ray must be perpendicular to the edge in its own face. An arbitrary oblique section will not work.
Section across the eave
The actual angle in the section plane
By the tangent definition in the correct section:
tanθ = (1.8)/(2.4) = 0.75, θ ≈ 36.87°.
This is both the roof plane’s inclination to the horizontal plane and the angle between the illustrated cross-slope segment and its horizontal projection. A 75% gradient means rise divided by horizontal run, multiplied by 100%; it is not an angle of 75°. Rise divided by sloping length gives the sine: 1.8/3=0.6.
For a symmetric gable roof, the interior angle below the ridge is 180°−2θ≈106.26°. The smaller angle between the two complete roof planes is about 73.74°. We must therefore specify the half-planes: a dihedral angle can be obtuse, while the angle between planes is conventionally from 0° to 90°.
When planes are perpendicular
Planes are perpendicular when their linear angle is 90°. A useful criterion: a plane containing a line perpendicular to another plane is perpendicular to that plane. Thus a vertical cross-section is perpendicular to a horizontal floor.
This does not make every line of one plane perpendicular to every line of the other. In two walls, choose lines parallel to their shared vertical edge: those lines are parallel to one another.
Distance means the shortest permitted path
For the lines and planes considered here, distance is the least length of a segment joining points of the figures. A shared point gives distance 0; enlarging the drawing cannot create a “gap”.
- From a point to a line or plane: the perpendicular length; 0 if the point belongs to the figure.
- Between parallel planes: a common perpendicular’s length. A floor and a parallel plane at height 2.2 m are 2.2 m apart.
- From a parallel line to a plane: the distance from any point of the line. A line meeting or lying in the plane has distance 0.
- Between parallel lines: a common perpendicular’s length; between intersecting lines: 0.
The figure’s limits matter. A perpendicular foot on an infinite line can fall outside a segment; then the nearest point on the segment is an endpoint. Similarly, distance to a plane need not equal distance to a bounded patch of roof.
Optional detailThe main route continues without this calculation.
Branch · Skew lines
Imagine a horizontal line a and another horizontal line b, 2 m higher, with direction perpendicular to a’s direction. Their floor projections meet at H. Point K above H belongs to b, and H belongs to a. Segment HK of length 2 m is perpendicular to both lines.
Move r from H along a and s from K along b. Two applications of Pythagoras give squared distance r²+s²+4≥4. The minimum 2 m occurs only at r=s=0. This is our own example of a common perpendicular to skew lines. In general it exists and is unique; an arbitrary segment joining the lines is not the common perpendicular.
Choose directions, projection or section first, then calculate. A photographic angle or arbitrary sloping distance cannot replace those constructions.
Sources and limits
Definitions and distance cases were checked against Ister’s grade-10 geometry textbook, §§10–12. Additional verification: Euclid XI, definitions 6–8, a section perpendicular to the edge; XI.18, the criterion for perpendicular planes. Numbers, roof and minimum argument are our own teaching models; loads, strength and building regulations are outside this lesson.