Three-dimensional Pythagoras and line-plane angles — 0580
Project a line onto the plane to identify the right triangle.
Extended · Syllabus E6.6
Notes and key methods
For a rectangular box, first find a diagonal in a face using Pythagoras. Then use that face diagonal and the perpendicular height to find the space diagonal. State which plane and which endpoints define the requested line.
The angle between a line and a plane is the acute angle between the line and its perpendicular projection onto the plane. For a base-to-opposite-top box diagonal, the projection is the base diagonal. The vertical height is opposite this angle.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Three-dimensional Pythagoras and line-plane angles
A rectangular box has horizontal base dimensions 18 cm and 24 cm and vertical height 6 cm. Consider the line from one base corner to the opposite top corner.
Find the base diagonal in cm.
Find the space diagonal in cm, to 3 significant figures.
Find the angle between the space diagonal and the base plane in degrees, to 1 decimal place.
Find the angle between the space diagonal and a vertical line in degrees, to 1 decimal place.
Show the reasoning
(a)
Apply Pythagoras within the horizontal base first.
The horizontal projection has length \(\displaystyle \sqrt{324+576}=30 \mathrm{cm}\).
Answer: \(30\).
(b)
Use the base diagonal and perpendicular height as legs of a second right triangle.
The projection and vertical height give \(6 \sqrt{26}\), rounding to \(30.6\) cm.
Answer: \(30.6\).
(c)
Use the angle between the line and its horizontal projection.
The angle has tan θ=height/base-diagonal=\(\displaystyle \frac{1}{5}\), giving \(11.3\)°.
Answer: \(11.3\).
(d)
Use the complementary acute angle in the same projection triangle.
This angle complements the line-plane angle, giving \(78.7\)°.
Answer: \(78.7\).
Example 2 · Three-dimensional Pythagoras and line-plane angles
A rectangular box has horizontal base dimensions 3 cm and 4 cm and vertical height 7 cm. Consider the line from one base corner to the opposite top corner.
Find the base diagonal in cm.
Find the space diagonal in cm, to 3 significant figures.
Find the angle between the space diagonal and the base plane in degrees, to 1 decimal place.
Find the angle between the space diagonal and a vertical line in degrees, to 1 decimal place.
Show the reasoning
(a)
Apply Pythagoras within the horizontal base first.
The horizontal projection has length \(\displaystyle \sqrt{9+16}=5 \mathrm{cm}\).
Answer: \(5\).
(b)
Use the base diagonal and perpendicular height as legs of a second right triangle.
The projection and vertical height give \(\sqrt{74}\), rounding to \(8.60\) cm.
Answer: \(8.60\).
(c)
Use the angle between the line and its horizontal projection.
The angle has tan θ=height/base-diagonal=\(\displaystyle \frac{7}{5}\), giving \(54.5\)°.
Answer: \(54.5\).
(d)
Use the complementary acute angle in the same projection triangle.
This angle complements the line-plane angle, giving \(35.5\)°.
Answer: \(35.5\).
Example 3 · Three-dimensional Pythagoras and line-plane angles
A rectangular box has horizontal base dimensions 6 cm and 8 cm and vertical height 8 cm. Consider the line from one base corner to the opposite top corner.
Find the base diagonal in cm.
Find the space diagonal in cm, to 3 significant figures.
Find the angle between the space diagonal and the base plane in degrees, to 1 decimal place.
Find the angle between the space diagonal and a vertical line in degrees, to 1 decimal place.
Show the reasoning
(a)
Apply Pythagoras within the horizontal base first.
The horizontal projection has length \(\displaystyle \sqrt{36+64}=10 \mathrm{cm}\).
Answer: \(10\).
(b)
Use the base diagonal and perpendicular height as legs of a second right triangle.
The projection and vertical height give \(2 \sqrt{41}\), rounding to \(12.8\) cm.
Answer: \(12.8\).
(c)
Use the angle between the line and its horizontal projection.
The angle has tan θ=height/base-diagonal=\(\displaystyle \frac{4}{5}\), giving \(38.7\)°.
Answer: \(38.7\).
(d)
Use the complementary acute angle in the same projection triangle.
This angle complements the line-plane angle, giving \(51.3\)°.
Answer: \(51.3\).
Common mistakes
Keep intermediate values exact until the final step.