1984 AIME Problem 9

Question

In tetrahedron

ABCD,

edge

AB

has length

3

cm. The area of face

ABC

is

15\text{ cm}^2

and the area of face

ABD

is

12\text{ cm}^2.

These two faces meet each other at a

30^\circ

angle. Find the volume of the tetrahedron in

\text{cm}^3.

Answer

Find the altitudes from

C

and

D

to the common edge

AB

Express the tetrahedron’s volume using the common edge, the two altitudes, and the sine of the dihedral angle

Let

h_C

and

h_D

be the perpendicular distances from

C

and

D

to

AB.

From the two face areas,

\begin{aligned} h_C&=\frac{2\cdot15}{3}=10,\\ h_D&=\frac{2\cdot12}{3}=8. \end{aligned}

The angle between these two perpendicular directions is the

30^\circ

dihedral angle. Hence the scalar triple product gives

\begin{aligned} V&=\frac16(AB)h_Ch_D\sin30^\circ\\ &=\frac16\cdot3\cdot10\cdot8\cdot\frac12\\ &=20. \end{aligned}

Problem 9 in other years