Much of what we have learned about finding the length of sides and measure of angles in triangles can be extended to shapes in three dimensions. The sine rule and the cosine rule can be applied without modification to the faces of 3D shapes, and the 3D version of Pythagoras' theorem is very similar to the familiar 2D version.
The sine rule and cosine rule
Previously we have been using the sine rule and the cosine rule to find properties of triangles that live in the 2D cartesian plane. Since the faces of 3D shapes are also flat 2D planes, we can apply these rules to triangular faces as well.
In the tetrahedron below, the triangle $ACD$ACD lies in the plane and the vertex at $B$B lies above the plane. The face that consists of the triangle $ABC$ABC has angles named $A$A, $B$B and $C$C, and their opposite sides have lengths $a$a, $b$b and $c$c respectively.
The sine rule states that
$\frac{\sin A}{a}$sinAa$=$=$\frac{\sin B}{b}$sinBb$=$=$\frac{\sin C}{c}$sinCc.
This can also be written as
$\frac{a}{\sin A}$asinA$=$=$\frac{b}{\sin B}$bsinB$=$=$\frac{c}{\sin C}$csinC.
The cosine rule states that
$c^2=a^2+b^2-2ab\cos C$c2=a2+b2−2abcosC.
This can also be written as
$\cos C=\frac{a^2+b^2-c^2}{2ab}$cosC=a2+b2−c22ab.
Pythagoras' theorem
Consider a rectangular prism with side lengths $a$a m, $b$b m, and $c$c m. How can we determine the diagonal length of the prism? In the figure below we have labelled an intermediary length, $x$x m, which is the diagonal length of the bottom face.
We can use Pythagoras' theorem to find the value of $x$x as follows:
$x^2=a^2+b^2$x2=a2+b2.
If we now look at the triangle with side lengths $x$x m, $c$c m, and $d$d m, we can use Pythagoras' theorem again to relate these three lengths:
$d^2=x^2+c^2$d2=x2+c2.
But notice that this expression contains the term $x^2$x2, which we had previously found is equivalent to $a^2+b^2$a2+b2. So we can now write the length $d$d in terms of $a$a, $b$b, and $c$c only:
$d^2=a^2+b^2+c^2$d2=a2+b2+c2.
Looking back at the image of the prism we can see that each of the three side lengths correspond to segments that are perpendicular to the other two side lengths. Just as Pythagoras' theorem in 2D only works for right-angled triangles, where there is a pair of perpendicular sides, so the corresponding theorem in 3D only works with three mutually perpendicular segments.
Given the three lengths $a$a, $b$b, and $c$c, of three mutually perpendicular sides of a rectangular prism, the length $d$d of the main diagonal is given by
$d=\sqrt{a^2+b^2+c^2}$d=√a2+b2+c2
Practice questions
Question 1
A pyramid has a square base with side length $14$14 m and a vertical height of $24$24 m.
Find the length of the edge $\overline{AE}$AE. Round your answer to one decimal place.
If $\theta$θ is the measure of the surface angle $BEA$BEA, find $\theta$θ correct to the nearest minute.
Question 2
Two hot air balloons are moored to level ground below, each at a different location. An observer at each location measures the angle of elevation to the opposite balloon. The observers are $1600$1600 m apart.
Determine the difference in height between the two balloons. Round your answer to the nearest metre.
Use your result from part (a) to determine the distance separating the two balloons. Round your answer to the nearest metre.
Question 3
Roald is standing at point $P$P on the horizontal ground and observes two poles, $\overline{AB}$AB and $\overline{CD}$CD, of different heights. $P$P, $B$B, and $D$D are in the same horizontal plane. From $P$P the angles of inclination to the top of the poles $A$A and $C$C have measures $29^\circ$29° and $18^\circ$18° respectively. Roald is $16$16 m from the base of pole $\overline{AB}$AB. The height of pole $\overline{CD}$CD is $7$7 m.
Calculate the distance from Roald to the top of pole $\overline{CD}$CD. Round your answer to two decimal places.
Calculate the distance from Roald to the top of pole $\overline{AB}$AB. Round your answer to two decimal places.
Calculate the distance between the tops of the poles. Round your answer to two decimal places.