Midpoint of (x1,y1,z1) and (x2,y2,z2)?

\left(\frac{x1+x2}{2},\frac{y1+y2}{2},\frac{z1+z2}{2}\right)

Volume of a right pyramid?

\frac13\timesbase area\timesheight

Curved surface area of a cone?

\pi rl, where l is the slant height.

Volume and curved surface area of a hemisphere?

V=\frac23\pi r^3, curved surface 2\pi r^2

Total surface area of a solid hemisphere?

3\pi r^2, the curved surface plus the flat circular face.

What is the angle between a line and a plane?

The angle between the line and its projection on the plane.

What kind of trigonometry is set for 3D shapes at SL?

Right-angled trigonometry only, in a right-angled triangle you identify in the solid.

What is hidden when two solids are joined?

The joined faces; they are not part of the exposed surface area.

3.1 Three-dimensional geometryIB Maths: Applications and Interpretation HL: Flashcards

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Distance between $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$?

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Distance between $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ ?: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
Midpoint of $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ ?: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$
Volume of a right pyramid?: $\frac13\times\text{base area}\times\text{height}$
Volume of a cone?: $\frac13\pi r^2h$
Curved surface area of a cone?: $\pi rl$ , where $l$ is the slant height.
How do you find the slant height of a cone?: $l=\sqrt{r^2+h^2}$
Volume and surface area of a sphere?: $V=\frac43\pi r^3$ , $A=4\pi r^2$
Volume and curved surface area of a hemisphere?: $V=\frac23\pi r^3$ , curved surface $2\pi r^2$
Total surface area of a solid hemisphere?: $3\pi r^2$ , the curved surface plus the flat circular face.
Space diagonal of a cuboid $a\times b\times c$ ?: $\sqrt{a^2+b^2+c^2}$
What is the angle between a line and a plane?: The angle between the line and its projection on the plane.
What kind of trigonometry is set for 3D shapes at SL?: Right-angled trigonometry only, in a right-angled triangle you identify in the solid.
What is hidden when two solids are joined?: The joined faces; they are not part of the exposed surface area.

Exam questions on 3.1 Three-dimensional geometry

The points $P(2,-1,4)$ and $Q(8,3,-8)$ lie in three-dimensional space, with all lengths in metres.
The point $R$ is such that $Q$ is the midpoint of $[PR]$ . Find the coordinates of $R$ .2 marks
A toy is made from a solid hemisphere of radius 6 cm and a solid cone of base radius 6 cm and vertical height 8 cm. The base of the cone is fixed exactly onto the flat circular face of the hemisphere.
Find the angle between the slant height of the cone and its circular base.2 marks
A tent is in the shape of a right pyramid with square base $ABCD$ of side 6 m. The vertex $V$ is vertically above the centre $M$ of the base, and $VM=4$ m.
Find the length of the edge $VA$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).