Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Geometry and Trigonometry — Topic 3.1
📋 Learning Intentions
- Calculate distances between two points in three-dimensional space using the 3D distance formula.
- Find the volume and surface area of three-dimensional solids including prisms, pyramids, cylinders, cones, and spheres using the IB formula booklet.
- Identify and measure angles between lines and between a line and a plane in 3D figures using right-triangle trigonometry.
- Apply Pythagoras' theorem in three dimensions to find lengths of diagonals and slant heights within composite solids.
- Interpret 3D diagrams to extract relevant lengths, angles, and cross-sections needed for calculation.
- Solve multi-step problems combining volume, surface area, and angle calculations in real-world contexts.
Real-World Introduction
Architects designing a glass pyramid entrance — like the one at the Louvre in Paris — must calculate the exact area of each triangular glass panel, the total volume enclosed, and the precise angle at which each panel meets the ground. These are not abstract exercises: an error of even one degree in the angle of inclination changes the structural load distribution and the amount of glass required.
Civil engineers planning water storage tanks, pharmaceutical companies designing capsule medications, and aerospace engineers modelling fuel tanks all rely on the same core toolkit: the volume and surface area formulas for standard 3D solids, combined with trigonometry to find angles in three-dimensional space. The mathematics in this topic is the foundation of every physical design problem.
🌍 Why This Matters The Great Pyramid of Giza has a square base of side 230.4 m and an original height of 146.5 m. Calculating the angle its sloping face makes with the base, the total surface area of stone cladding needed, and the volume of material inside are all direct applications of the skills in this topic — and they are exactly the type of multi-step problem that appears in IB Paper 2.
Core Concepts & Definitions
In two dimensions, the distance between two points is found using Pythagoras' theorem on a right triangle in the \(xy\)-plane. Extending this to three dimensions simply adds a third perpendicular direction: the \(z\)-axis. The three perpendicular displacements form the three edges of a rectangular box, and the space diagonal of that box is the distance between the two points.
📐 Key Definition — Distance Between Two Points in 3D The distance \(d\) between two points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\) in three-dimensional space is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] This is a direct extension of the 2D distance formula, with the additional \((z_2 - z_1)^2\)