Geometry turns a sketch into something measurable. The main habit in this part of the course is to break a 3D object into one or more 2D right-angled triangles. Spot the right triangle inside the solid, and the algebra is usually brief.

Distance and midpoint in three-dimensional space

A point in three-dimensional space is a location given by three coordinates. These may represent left-right, forward-back and up-down, or any equivalent set of three perpendicular directions. Consider two points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$. Here, $x_1$, $y_1$ and $z_1$ are the coordinates of point $A$ in units of length, while $x_2$, $y_2$ and $z_2$ are the coordinates of point $B$ in the same units. The midpoint is the point on a line segment that is equally distant from both endpoints.

The distance between two points is the length of the straight line segment that joins them. In 3D, apply Pythagoras twice: once in the horizontal rectangle, then once more up into space.

Volume and surface area of solids

Volume measures the amount of three-dimensional space occupied by a solid and is given in cubic units. Surface area measures the total area of the solid’s outer boundary and is given in square units.

| $S=4\pi r^2$ | | Hemisphere | $V=\frac{2}{3}\pi r^3$ | curved area $=2\pi r^2$, total $S=3\pi r^2$ | | Right cone | $V=\frac{1}{3}\pi r^2h$ | $S=\pi r^2+\pi rl$ | | Right pyramid | $V=\frac{1}{3}B_{\text{base}}h$ | base area plus the triangular face areas |

A right cone has its apex directly above the centre of its circular base. In a right pyramid, the apex lies directly above the centre of the base. A sphere consists of all points that are a fixed distance from a centre point. A hemisphere is half of a sphere, formed by cutting it with a plane through its centre.

Combination solids are formed by joining familiar solids or removing parts from them. When pieces are joined, add their volumes. When a hole or missing part is removed, subtract its volume. Surface area needs a more physical approach: count only the surfaces you could touch from outside. For example, the internal circular join between a cone and a hemisphere doesn’t belong to the outside surface.

Right-angled triangles inside 3D objects

Pythagoras's theorem relates the side lengths of a right-angled triangle: the square of the hypotenuse equals the sum of the squares of the other two sides.

In SL examinations, 3D trigonometry is assessed using right-angled triangles. You aren’t expected to invent advanced 3D vector methods. Instead, identify the correct triangular slice. It might contain a diagonal across a base, a height to an apex, a slant height or a projection onto a plane.

The angle between two intersecting lines is the smaller angle formed at the point where the lines meet. The angle between a line and a plane is measured between the line and its perpendicular projection onto the plane. Focus on that projection; don’t measure the angle against an arbitrary line drawn on the plane.

Architecture and design provide natural settings for this geometry. Roofs, ramps, towers and domes can all be reduced to lengths, angles, volumes and exposed areas. The same spatial reasoning appears in physics when estimating stellar volumes or working with inverse-square relationships. There is also a quiet TOK point here: Euclidean geometry starts from axioms, and something that feels self-evident in one geometry may not be self-evident in every geometry.

Trigonometry deals with relationships: side lengths and angle sizes determine one another. Here, the diagram isn’t decoration; it forms part of the argument. A clear, labelled diagram often does half the work before you even reach for the calculator.

Right-angled triangle ratios

A right-angled triangle contains one angle of $90^\circ$. The hypotenuse lies opposite the right angle and is always the longest side. Choose an acute angle: the opposite side is directly across from it, while the adjacent side touches it but is not the hypotenuse.

Use sine, cosine or tangent for a right-angled triangle when you know one side together with either another side or an acute angle. If you need to find an angle, use the inverse functions $\arcsin$, $\arccos$ or $\arctan$. This course avoids the notation $\sin^{-1}$ for inverse sine because it can be mistaken for a reciprocal. The idea links directly to inverse functions in algebra: reverse the trig ratio to recover the angle.

The sine rule

For a non-right triangle, label each side opposite its corresponding angle.

The sine rule is usually the quickest choice when you know two angles and a side, or two sides and a non-included angle. The ambiguous case of the sine rule is not included in this syllabus point, so these problems are designed to produce a single triangle.

The cosine rule

In any triangle, the cosine rule connects three side lengths with one angle. Using the same notation as above,

Use this version when two sides and the included angle are known, and you need the opposite side. To find an angle, rearrange it as

This form is used when all three sides are known. As a quick check, the largest angle must sit opposite the longest side, so the answer should match the shape of the triangle.

Area of a triangle

The included angle is the angle formed by two known sides of a triangle. If two sides and their included angle are known, the area is

This is the familiar base-height area formula expressed with trigonometry. One side acts as the base, while the sine of the included angle gives the perpendicular height. Keep exact values for as long as possible, then round only at the end; exact and rounded representations do not contain the same information.

Applications usually aren't about learning new formulas. The real skill is turning a sentence into the correct right or non-right triangle. When you see words such as north, horizontal, line of sight, distance apart, or bearing, pause and sketch the situation.

Angles of elevation and depression

An angle of elevation is measured upward from a horizontal line to a line of sight. An angle of depression is measured downward from a horizontal line to a line of sight. The horizontal lines in these problems are usually parallel. Alternate interior angles can therefore explain why the angle of depression at one point equals the angle of elevation at another.

Right-angled trigonometry and Pythagoras's theorem are common here. A vertical height and horizontal distance, for instance, naturally make a right triangle, with the line of sight usually forming the hypotenuse. The same geometry appears in field studies, surveying and physics when measuring inaccessible heights, resolving forces, or separating scalar distances from directed quantities.

Bearings

A bearing gives direction as an angle measured clockwise from north, usually written with three digits. For example, $050^\circ$ is a bearing rather than simply $50^\circ$; its leading zero shows that it is a three-figure bearing. North is the starting ray. East is $090^\circ$, south is $180^\circ$, and west is $270^\circ$.

For bearing problems, draw a north line at every point where a direction is given or required. If an object travels from point $P$ on a bearing, measure the angle at $P$. If you're asked for the bearing of $A$ from $C$, measure the angle at $C$, not at $A$.

Constructing labelled diagrams from words

A labelled diagram is a mathematical drawing that shows the given points, lengths, angles, directions and unknowns from a written statement. It doesn't need to look artistic, but there must be no ambiguity. A useful routine is:

• mark the fixed point or starting point first;
• add north lines or horizontal lines when directions or elevation angles are involved;
• place known lengths on the correct segments;
• mark the required unknown with a clear label;
• decide whether the triangle is right-angled or non-right-angled before choosing a method.

This skill underpins triangulation, map-making, navigation, radio transmission and parallax. Measurements taken from different positions are combined to locate something that may not be directly reachable. Pythagoras's theorem and early trigonometry also appear historically in several mathematical traditions, including early Chinese, Indian and Greek work. This raises a fair TOK question about mathematical credit: when a result is rediscovered, proved in many ways, or known before receiving its famous name, what should that name recognise?

Euclidean triangle problems in this course assume that the angles in a triangle sum to $180^\circ$. In other geometries, triangle angle sums may be less than, equal to, or greater than $180^\circ$. Mathematical truth, then, is truth within a stated system of assumptions.

The difference between degrees and radians becomes clearer when you look at a circle. Degrees split one full turn into $360$ equal parts. Radians tie the angle directly to arc length, making the formulas in this section much cleaner.

Radian measure

A radian is a unit of angle measure. It is the central angle that subtends an arc whose length equals the radius of the circle. One full turn is $2\pi$ radians, the same angle as $360^\circ$.

To change degrees into radians, multiply by $\frac{\pi}{180}$. For radians to degrees, multiply by $\frac{180}{\pi}$. A radian measure can be exact, such as $\frac{3\pi}{4}$, or decimal, such as $2.36$. Exact multiples of $\pi$ keep all the information, while decimals are usually approximations unless stated otherwise.

In examination papers, assume radian measure unless the question includes a degree symbol or indicates degrees in another way.

Arc length

An arc is a connected part of a circle’s circumference. When the central angle is measured in radians, the arc length is

The formula has this simple form only when $\theta$ is in radians.

This is why radians are more than just another angle unit. From the definition of a radian, the formula follows almost automatically: the angle gives a fraction of a full turn, while the arc length gives the corresponding fraction of the circumference.

Area of a sector

A sector is the region of a circle enclosed by two radii and the arc between them. If the central angle is measured in radians, then

The formula reflects the full-circle area $\pi r^2$. A sector takes up a fraction of the circle, determined by its angle out of the full turn of $2\pi$ radians. For the same reason, radians fit naturally into circular motion and diffraction patterns in physics: both the angle and the circular distance come from the same radius.