This SL mini-lesson covers 3D geometry (distances, angles, volumes & surface areas of solids), right-angled trig (SOH-CAH-TOA), the sine and cosine rules and the area of a triangle, arcs & sectors, and the AI-only topic of Voronoi diagrams.
AI flavour: geometry here is applied — surveying, navigation, packaging and the 'nearest facility' logic of Voronoi diagrams.
Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Every number on the calculation screens has been re-derived and checked. Press Start when you are ready.
In a right-angled triangle with hypotenuse H, opposite O and adjacent A:
Use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) to find an angle from a ratio.
For any triangle with sides a, b, c opposite angles A, B, C:
A = 40°, side a = 8, and B = 65°. Find side b.
b = a · sin B ÷ sin A = 8 × sin65° ÷ sin40° = 11.28 (2 d.p.)
Sides a = 7, b = 10 with included angle C = 45°.
c² = 7² + 10² − 2×7×10×cos45° = 149 − 98.99 = 50.01, c = 7.07
When two sides and the included angle are known:
a = 7, b = 10, C = 45°.
Area = ½ × 7 × 10 × sin45° = 35 × 0.7071 = 24.75 square units
For a sector of radius r with angle θ in radians:
r = 6, θ = 1.2 rad.
ℓ = 6 × 1.2 = 7.2
area = ½ × 6² × 1.2 = 21.6
In degrees: ℓ = 2πr × (θ°⁄360) and area = πr² × (θ°⁄360). For r = 5, 60°: ℓ = 5.24, area = 13.09.
The space diagonal of a cuboid with edges a, b, c is √(a²+b²+c²). Angles between a line and a plane are found by dropping to the base and using right-angled trig.
A box 4 × 6 × 3. Space diagonal = √(16+36+9) = √61 = 7.81.
Base diagonal = √(16+36) = √52 = 7.21.
Angle of the space diagonal to the base = tan⁻¹(3 ÷ 7.21) = 22.6°
Tap a description, then the trig tool it belongs to.
A Voronoi diagram partitions a plane around a set of sites so that every point is assigned to its nearest site. Each cell is bounded by perpendicular bisectors of the lines joining neighbouring sites.
Sites at (2, 3) and (6, 7).
Midpoint = ((2+6)⁄2, (3+7)⁄2) = (4, 5)
Segment gradient = (7−3)⁄(6−2) = 1, so bisector gradient = −1
Edge: y − 5 = −1(x − 4) ⇒ y = −x + 9
Toxic-waste / largest-empty-circle: the point furthest from all sites sits at a Voronoi vertex — used to place a new facility as far as possible from existing ones.
Tap an item on the left, then its match on the right.
AI expects volumes and surface areas of prisms, cylinders, cones, spheres and pyramids from the formula booklet — e.g. sphere V = 4⁄3 πr³, cone V = 1⁄3 πr²h.
Model real objects (silos, tanks, packaging) as combinations of these solids.
Right-angle trig: sin=O⁄H, cos=A⁄H, tan=O⁄A; Pythagoras a²+b²=c²
Sine rule: a⁄sinA = b⁄sinB — a side–angle pair plus one more
Cosine rule: c² = a²+b²−2ab cosC — two sides + included angle
Triangle area: ½ab sinC
Arc & sector (radians): ℓ = rθ ; area = ½r²θ
Voronoi: edges are perpendicular bisectors → nearest-site regions
You have worked through the whole topic. Press Finish to see your score.
You have covered SL Geometry & Trigonometry for AI. 🎉
Your stars: 0 / 0
Next: test yourself in the Evaluate stage, then lock it in with Verify.