Geometry & Trigonometry (AI HL)
This topic takes right-angled trigonometry beyond SOHCAHTOA into triangles with no right angle, using the sine rule and cosine rule to find missing sides and angles. It also covers real-world applications - angles of elevation and depression, bearings, and 3D problems like the angle between a line and a plane - along with the volume and surface area of common solids.
What the syllabus says
This topic draws on three points in the official IB Applications & Interpretation syllabus - content shared with Analysis & Approaches at SL, and examined further at HL.
| Code | Syllabus content |
|---|---|
| SL3.1 | The distance between two points in three-dimensional space, and their midpoint. Volume and surface area of solids including right-pyramids, right cones, spheres, hemispheres and combinations of these. The size of an angle between two intersecting lines, or between a line and a plane. |
| SL3.2 | Sine, cosine and tangent ratios for right-angled triangles. The sine rule \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) and cosine rule \(c^2=a^2+b^2-2ab\cos C\) (this section excludes the ambiguous case). Area of a triangle as \(\tfrac12 ab\sin C\). |
| SL3.3 | Applications of right- and non-right-angled trigonometry, including Pythagoras' theorem, angles of elevation and depression, bearings, and constructing labelled diagrams from written statements. |
The ambiguous case of the sine rule (two possible triangles for the same data) is an AHL extension of this content.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is the sine rule?
The sine rule, \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\), relates each side of any triangle to the sine of its opposite angle. Use it when you know an angle and its opposite side, plus one more side or angle.
e.g. For \(a=9\), \(A=35^\circ\), \(b=12\): \(\sin B = \dfrac{12\sin35^\circ}{9} \approx 0.7648 \Rightarrow B\approx 49.9^\circ\).
What is the cosine rule?
The cosine rule, \(c^2=a^2+b^2-2ab\cos C\), finds a missing side when you know two sides and the included angle, or a missing angle when you know all three sides.
e.g. For \(b=7\), \(c=10\), \(A=68^\circ\): \(a^2 = 149-140\cos68^\circ \approx 96.56 \Rightarrow a\approx 9.83\).
What is the area formula for a triangle?
When you know two sides and the included angle, the area is \(\tfrac12 ab\sin C\) - no need to know the height.
e.g. For \(a=8\), \(b=5\), \(C=30^\circ\): Area \(=\tfrac12(8)(5)\sin30^\circ = 20\times0.5=10\).
What is an angle of elevation or depression?
Both are measured from the horizontal. An angle of elevation looks up at an object; an angle of depression looks down at one - the two angles are equal for the same line of sight (alternate angles).
e.g. From a 50 m cliff, an angle of depression of \(18^\circ\) gives a horizontal distance \(=\dfrac{50}{\tan18^\circ}\approx 153.9\) m.
What is a bearing?
A bearing is a direction given as an angle measured clockwise from north, always written with three digits, e.g. \(070^\circ\). The angle between two bearings taken from the same point is found by subtraction.
e.g. A bearing of \(070^\circ\) then \(130^\circ\) gives an included angle of \(130-70=60^\circ\).
Key formulas
Three triangle formulas cover most of this topic, alongside the volume and surface area formulas for common 3D solids.
Formula reference
The sine rule, cosine rule and area formula are all on the formula booklet, as are the standard volume and surface area formulas for solids - any non-standard formula (like a frustum) is given in the question itself.
| Formula | Used for | Booklet? |
|---|---|---|
| \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) | Sine rule | ✓ Yes |
| \(c^2=a^2+b^2-2ab\cos C\) | Cosine rule (side) | ✓ Yes |
| \(\text{Area} = \tfrac12 ab\sin C\) | Area of a triangle | ✓ Yes |
| Volume/surface area of cones, spheres, pyramids | Standard 3D solids | ✓ Yes |
| e.g. \(V=\tfrac13\pi h(R^2+Rr+r^2)\) (frustum) | Non-standard solids | Not in the formula booklet - given in the question |
Right-angled vs non-right-angled trigonometry
Whether a triangle has a right angle determines which toolkit you reach for.
| Feature | Right-angled triangle | Non-right-angled triangle |
|---|---|---|
| Main tool | SOHCAHTOA, Pythagoras' theorem | Sine rule and cosine rule |
| Area formula | \(\tfrac12 \times \text{base} \times \text{height}\) | \(\tfrac12 ab\sin C\) |
| When it applies | One angle is known to be \(90^\circ\) | No right angle is given |
The sine and cosine rules
Choosing the right rule depends entirely on which three pieces of information you're given.
Sine rule
\[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\]
Use with an angle and its opposite side, plus one more known side or angle.
✓ In the formula bookletCosine rule
\[c^2=a^2+b^2-2ab\cos C\]
Use with two sides and the included angle (to find the third side), or all three sides (to find an angle).
✓ In the formula bookletArea with sine
\[\text{Area}=\tfrac12 ab\sin C\]
Works for any triangle where two sides and the included angle are known - no height needed.
✓ In the formula bookletApplications in context
Most exam questions dress these formulas up in a real-world scenario - the maths is the same, but you first have to extract the triangle from the words.
Elevation and depression
Draw the horizontal line first, then mark the angle up (elevation) or down (depression) from it - the two triangles formed are often solved with simple tangent ratios.
Diagram techniqueBearings
Always draw a north line at each point mentioned; the angle between two bearings at the same point comes from subtracting them.
Diagram techniqueConstructing a diagram
Sketching a clear, labelled diagram from the written description is often the hardest part of a bearings or elevation question - and it's explicitly examined.
Syllabus skill (SL3.3)3D and volume applications
The same right-angled and non-right-angled tools extend into three dimensions, plus the standard volume and surface area formulas.
Distance in 3D
Find the distance between two points, or from a point to the centre of a shape, before applying trigonometry to the resulting right-angled triangle.
SL3.1Angle between a line and a plane
Identify the right-angled triangle formed by the line, its projection onto the plane, and the perpendicular height - then use a tangent ratio.
SL3.1Volume and surface area
Cones, spheres, hemispheres, pyramids and combinations of these are all on the formula booklet - non-standard solids like a frustum have their formula given in the question.
✓ Standard solids in the formula bookletWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
In triangle \(ABC\), \(b=7\) cm, \(c=10\) cm and \(A=68^\circ\).
(a) Find side \(a\).
(b) Find the perimeter.
Worked solution
(a) \(a^2=7^2+10^2-2(7)(10)\cos68^\circ\) M1
\(=149-140\cos68^\circ\approx96.56\) A1
\(a\approx9.83\) cm. A1
(b) \(\approx26.8\) cm. A1
A triangular plot has area \(150\) m². Two of its sides are \(20\) m and \(18\) m.
(a) Find the included angle (acute value).
(b) Find the length of the third side.
Worked solution
(a) \(150 = \tfrac12(20)(18)\sin C \Rightarrow \sin C = 0.8333 \Rightarrow C\) M1
\(\approx 56.4^\circ.\) A1
(b) \(c^2 = 400 + 324 - 720\cos56.4^\circ \approx 326.0 \Rightarrow c\) M1
\(\approx 18.1\) m. A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Using the sine rule when the cosine rule is needed. If you only know three sides, or two sides and the angle between them, the sine rule has no opposite pair to work with - reach for the cosine rule instead.
- Measuring bearings anticlockwise, or forgetting the leading zeros. Bearings are always measured clockwise from north and written as three digits, e.g. \(070^\circ\) not \(70^\circ\).
- Mixing up elevation and depression. Both angles are measured from the horizontal - elevation looks up, depression looks down - never from the vertical or from the ground directly beneath the observer.
- Forgetting a right-angled triangle inside a 3D problem. Angles between a line and a plane, or distances to a centre point, almost always reduce to a right-angled triangle once you identify the correct base and height.
Using your GDC
There's no single dedicated GDC function for sine rule or cosine rule questions on this topic - once you've set up the correct equation by hand, your calculator does the arithmetic. Enter the expression exactly as derived (for example \(\sqrt{149-140\cos68^\circ}\) or \(\sin^{-1}(0.7648)\)) directly on the home screen or in Run-Matrix, making sure the angle mode (usually degrees, unless the question states radians) is set correctly first. For 3D and bearings questions, sketching the triangle clearly before you calculate is more valuable than any calculator shortcut - the GDC only helps once the setup is right.
See the full GDC guide for calculator models and other topics.
Ready to practise properly?
Geometry & trigonometry questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
When do I use the sine rule vs the cosine rule?
Use the sine rule when you know an angle and its opposite side (plus one more piece of information). Use the cosine rule when you know all three sides, or two sides and the angle between them - situations the sine rule can't handle.
What's the difference between an angle of elevation and depression?
An angle of elevation is measured upward from the horizontal, looking up at something. An angle of depression is measured downward from the horizontal, looking down at something. Both are measured from the horizontal, never from the vertical.
How do bearings work?
A bearing is an angle measured clockwise from north, always given as three digits, e.g. \(060^\circ\). To find the angle between two bearings from the same point, subtract them; to reverse a bearing, add or subtract \(180^\circ\).
Can I use my GDC for sine rule and cosine rule questions?
Yes - once you've set up the correct equation, your GDC evaluates the trig functions, inverse trig functions and square roots directly. Just make sure it's in the right angle mode (usually degrees for this topic) before you start.
Sub-topics
Geometry & Trigonometry broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AI HL syllabus unit, in case you want to keep going.