Distance, midpoint and volume in three dimensions, and the one angle question that catches almost everybody.
A school hall in Bangkok is 6 m wide, 8 m long and 5 m to the ceiling. A lighting cable runs from one bottom corner to the opposite top corner. What angle does that cable make with the floor?
Press the button. The first triangle is the one most people draw.
Three triangles, one cable. Only the third one is standing on the floor the whole way.
A right-angled triangle needs all three of its sides to be real lines in the solid, and the one you want has to lie flat on the base for its whole length. An edge does not reach the corner under the cable. The base diagonal does.
So it is two steps, never one.
Both wrong answers are too big, and that is not a coincidence. An edge is shorter than the base diagonal, so dividing by it gives a bigger tangent every time. If your angle comes out larger than you expected, you have probably measured along an edge.
The same two steps give the length of the cable, and the second Pythagoras absorbs the first:
√(6² + 8² + 5²) = √125 = 11.18
That three-term version is worth knowing, but only because you understand where it came from. Written down cold it is a formula; derived, it is the base diagonal with the height added on.
For P(2, 3, 1) and Q(8, 11, 6):
PQ = √(6² + 8² + 5²) = 11.18, the same arithmetic as the cuboid because the gaps are the same.
Midpoint = (5, 7, 3.5). Average each coordinate. Nothing is squared.
| Solid | Volume | Surface |
|---|---|---|
| Cone, r = 3, h = 4 | 12π | curved 15π, slant 5 |
| Sphere, r = 3 | 36π | 36π |
| Hemisphere, r = 3 | 18π | |
| Cone on a hemisphere | 30π |
A sphere of radius 3 has volume 36π and surface area 36π. The same number, in different units. It happens only at r = 3, and it catches people who think they have found a rule.
Type it as one expression. Working out the base diagonal, writing it down, and typing it back in is where the rounding creeps in, and here it is not needed at all.
When you may use it. Applications allows a calculator in every paper. Analysis does not allow one in Paper 1, so on that paper an answer like this is left as tanθ = 1/2, or θ = arctan(1/2). Only Papers 2 and 3 ask you to turn it into 26.6°.
The mark people lose. A machine left in radians gives 0.46. That does not look like a wrong angle, it looks like a wrong method, so it rarely gets spotted. Check the mode indicator before the first question of every paper, and if an angle comes back smaller than 1.6 when you expected tens of degrees, that is what happened.
1. A box is 9 by 12 by 8. Find the angle its space diagonal makes with the base, to 1 decimal place.
2. For the same box, how long is the space diagonal? Give it to 2 decimal places.
3. A student answers question 1 with 33.7°. Without calculating, what did they almost certainly do?
1 markFinding the base diagonal first, and showing it.
1 markA correct trigonometric statement using it.
1 markThe angle, to the accuracy asked for.
A diagram with the triangle marked on it earns the first mark on its own, even when the arithmetic later goes wrong. Draw it.
These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.
Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.
Already have a full draft? Have the whole thing reviewed against all five criteria, $99.