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3D distance formula?
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All Flashcards in Topic 3.1
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3.1.18 cards
3D distance formula?
d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²) — Pythagoras with a z-term.
3D midpoint formula?
((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) — average each coordinate.
Distance vs midpoint — what's the difference?
Distance squares the gaps and roots; midpoint averages the coordinates.
Does the order of points matter for distance?
No — each gap is squared, so the sign disappears.
Given midpoint M and endpoint A, how do you find B?
B = 2M − A (each coordinate).
Distance from origin to (2, 3, 6)?
√(4 + 9 + 36) = √49 = 7.
Space diagonal of a box with edges l, w, h?
√(l² + w² + h²).
How do you check a midpoint answer?
Average the two endpoints — you should recover the midpoint.
3.1.29 cards
Volume of a sphere?
V = ⁴⁄₃πr³.
Surface area of a sphere?
A = 4πr².
Volume of a cone?
V = ⅓πr²h — one third of the cylinder.
Curved surface area of a cone?
πrl, where l is the slant height = √(r² + h²).
Volume and surface area of a cylinder?
V = πr²h; A (closed) = 2πr² + 2πrh.
Volume of a pyramid or cone?
⅓ × base area × perpendicular height.
How do you find a composite solid's volume?
Add the volumes of the parts (subtract for a hole).
Composite surface area — what's the catch?
Don't count the join between two pieces; only exposed faces.
Sphere has volume 36π — find r.
⁴⁄₃πr³ = 36π ⇒ r³ = 27 ⇒ r = 3.
3.1.38 cards
How do you find an angle in a 3D solid?
Spot a right-angled triangle inside the solid and use SOH-CAH-TOA.
How is the angle between a line and a plane defined?
The angle between the line and its projection (shadow) on the plane.
What do you often need before the angle triangle is complete?
A face or base diagonal — found with Pythagoras.
Face diagonal of an a × a square?
a√2 (= √(a² + a²)).
Space diagonal of an a × a × a cube?
a√3 (= √(a² + a² + a²)).
What's the 'angle in a semicircle' fact?
A diameter subtends a right angle (90°) at any point on the circle.
Why redraw the triangle separately?
It's much easier to apply trig to a flat 2D triangle than to the 3D picture.
Typical 3D problem structure?
Two steps: a length by Pythagoras, then an angle by trig.
3.1.48 cards
A solid is given by coordinates. What's the first step?
Turn the coordinates into lengths — use the 3D distance and midpoint formulas to find edges, radii and heights.
How do you find the radius from a diameter [AB]?
Radius = ½ × the 3D distance AB; the centre is the midpoint of AB.
How do you find a cone or pyramid's height from coordinates?
It's the distance from the apex to the centre of the base (a vertical drop), not a slant edge.
Total surface area of a solid hemisphere, radius r?
3πr² — the curved dome 2πr² plus the flat base πr².
Volume of a hemisphere, radius r?
⅔πr³ — half of a sphere's ⁴⁄₃πr³.
Angle at a vertex between two edges, all three corners known?
Find the three side lengths with the distance formula, then use the cosine rule.
Angle between a slant edge and the base?
tan θ = height ÷ (horizontal distance from the base centre to that corner).
Exact or decimal?
Paper 1 usually wants exact (keep π and surds); Paper 2 round to 3 s.f.
Topic 3.1 study notes
Full notes & explanations for 3D geometry
Math AA exam skills
Paper structures, command terms & tips
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