IB Maths AA HL 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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Study SL 3.1—Three-dimensional geometry conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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The points A and B lie on a circle, with centre O and radius 19.5 cm , such that $\mathrm{BO} \widehat{\mathrm{O}}=210^{\circ}$ .
A piece of paper is cut into the shape of the sector BOA .
A hollow cone with no base is constructed from the sector by joining the points A and B . The sector forms the curved surface of the cone.
This is shown in the following diagrams.

Figure for Question IB Maths AA HL Sl 3 1 Three Dimensional Geometry Questions question 1 — IB Maths AA HL

Find

the radius of the cone.

EITHER
arc length $=19.5\left(\frac{7 \pi}{6}\right)$ OR $=\frac{210}{360}(2 \pi(19.5))\left(=\frac{91 \pi}{4}=71.4712 \ldots\right)$
attempt to set $2 \pi r$ equal to arc length
$2 \pi r=71.4712 \ldots$

OR
attempt to set $\pi r l$ equal to their area from (a)
$19.5 \pi r=696.844 \ldots$

THEN
$r=11.4\left(=\frac{91}{8}=11.375\right)(\mathrm{cm})$

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IB Maths AA HL Sl 3 1 Three Dimensional Geometry Questions question 1