IB Maths AA SL 3.1 Three Dimensional Geometry Questions

Practise finding 3D distances, midpoints, lengths, angles, volumes and surface areas from coordinates or solid diagrams using geometry and correct units.

Exam points

Calculate 3D distances and midpoints from coordinates or labelled solid diagrams, identifying the relevant coordinate differences.
Use right-angled triangles within a 3D solid to find an edge, diagonal, height or angle with Pythagoras or trigonometry.
Determine volumes and surface areas of common 3D solids or combined solids, selecting the correct faces, radii, heights and units.
Use coordinate or geometric constraints in a 3D diagram to justify a length, midpoint, perpendicularity or other spatial result.

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All lengths in this question are in centimetres.
A solid metal ornament is in the shape of a right pyramid, with vertex V and square base ABCD . The centre of the base is X . Point V has coordinates (1,5,0) and point A has coordinates (-1,1,6).

Figure for Question IB Maths AA SL 3.1 Three Dimensional Geometry Questions question 1 — IB Maths AA SL

(a)

Find AV .

[ 2 ]

attempt to use the distance formula to find AV

\begin{aligned} & \sqrt{(1-(-1))^{2}+(5-1)^{2}+(0-6)^{2}} \\ & =7.48331 \ldots \\ & =7.48(\mathrm{~cm})(=\sqrt{56} \text { or } 2 \sqrt{14}) \end{aligned}

(b)

Find the height of the pyramid, VX.

A second ornament is in the shape of a cuboid with a rectangular base of length $2 x \mathrm{~cm}$ , width $x \mathrm{~cm}$ and height $y \mathrm{~cm}$ . The cuboid has the same volume as the pyramid.

[ 3 ]

METHOD 1 equating volume of pyramid formula to 57.2
$\frac{1}{3} \times 5.11 \ldots{ }^{2} \times h=57.2$
(A1)
$h=6.54886 \ldots$ $h=6.55(\mathrm{~cm})$
METHOD 2
Let M be the midpoint of AB ]
$A V^{2}=A M^{2}+M X^{2}+X V^{2}$ $\Rightarrow X V=\sqrt{7.48 \ldots^{2}-\left(\frac{5.11 \ldots}{2}\right)^{2}-\left(\frac{5.11 \ldots}{2}\right)^{2}}$ $h=6.54886 \ldots$ $h=6.55(\mathrm{~cm})$

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IB Maths AA SL 3.1 Three Dimensional Geometry Questions question 1

Question (a)

Question (b)