Geometry and Trigonometry

Q: How do I find the centre and radius of a circle from its general equation?

Complete the square on the $x$-terms and $y$-terms separately to rewrite $x^2+y^2+Dx+Ey+F=0$ as $(x-a)^2+(y-b)^2=r^2$, giving centre $(-D/2,-E/2)$ and radius $\sqrt{(D/2)^2+(E/2)^2-F}$.

Coordinate geometry, circles, right-angle trig and the unit circle — the toolkit that shows up on every Paper 1 and Paper 2.

A coordinate plane showing a circle, a right triangle, and the unit circle side by side

Subject

Maths AA

Curriculum

IB Diploma Programme

Grade

Topic

Geometry and Trigonometry

Reading

7 min

Difficulty

Standard

Quick facts

Difficulty

★★★☆☆

Exam weight

~15-20% of AA SL teaching time

Appears on

Paper 1 (no GDC) and Paper 2 (GDC)

Prerequisites

Algebra, Pythagoras' theorem

You'll learn

Coordinate geometry, circles, right-angle trig, unit circle

Revision time

3-4 hours

Geometry and trigonometry in IB Maths AA blends two skill sets examiners love combining: coordinate geometry (distance, midpoint, gradient, circles) and trigonometry (SOHCAHTOA, the unit circle, exact values). The trap most students fall into is treating right-angle trig as universal — it only works when there's actually a 90° angle in the triangle you're solving, and spotting when to switch away from it is the single most examined skill in this topic. Meanwhile, coordinate geometry formulas like distance, midpoint and gradient aren't in your data booklet, so they need to be memorised cold. This teaser walks through the five ideas worth locking down first: coordinate geometry essentials, circles as loci, triangle area from coordinates, right-angle trig limits, and the unit circle with radians. The full revision notes go deeper with worked traps and mark-scheme wording.

What you’ll be able to do

✓Apply the distance, midpoint and gradient formulas from memory

✓Convert a circle's general equation into centre-radius form

✓Calculate the area of a triangle directly from its vertex coordinates

✓Recognise when SOHCAHTOA applies and when it doesn't

✓Use elevation and depression angles correctly in real-world problems

✓Define sine and cosine using the unit circle for any angle

✓Convert between degrees and radians and apply arc length and sector area

✓Recall exact trig values at 0°, 30°, 45°, 60°, 90° without a calculator

Coordinate Geometry: Distance, Midpoint, Gradient

The distance formula is just Pythagoras' theorem wearing a disguise — the segment between two points is the hypotenuse of a right triangle built from the horizontal and vertical differences. Midpoint is simply the average of the coordinates, giving you a point, not a single number. Gradient measures steepness, and it's the key to spotting parallel lines (equal $m$ ) and perpendicular lines ( $m_{1} m_{2} = - 1$ ). None of these three formulas live in the data booklet, so they need to be automatic.

Two points on a coordinate grid with distance, midpoint and gradient labelled

Exam tip

If a question mixes 'triangle' language with coordinate points instead of side lengths, expect to compute distances first before applying any trig rule.

Mini summary

Distance, midpoint and gradient are unlisted in the booklet — memorise them and know the perpendicular condition $m_{1} m_{2} = - 1$ .

Circles as a Locus: Standard vs General Form

A circle is the set of every point sitting exactly distance $r$ from a fixed centre — that single locus idea gives the standard form $(x - a)^{2} + (y - b)^{2} = r^{2}$ directly from the distance formula. Exams usually hand you the expanded general form $x^{2} + y^{2} + D x + E y + F = 0$ instead, and you recover the centre and radius by completing the square. A very common slip is reading the centre straight off as $(D, E)$ rather than $(- D /2, - E /2)$ , or losing a sign mid-calculation.

A circle on a coordinate grid transitioning from general equation to standard form

Exam tip

Complete the square from scratch every time instead of quoting the shortcut formula — the algebra self-corrects sign errors that a memorised formula won't catch.

Common mistake

Reading the centre as $(D, E)$ instead of $(- D /2, - E /2)$ , or dropping a sign while completing the square.

Mini summary

General form → complete the square → standard form gives centre $(- D /2, - E /2)$ and radius $(D /2)^{2} + (E /2)^{2} - F$ .

Triangle Area From Coordinates — Watch for Two Answers

When a triangle is defined only by three coordinate points, the shoelace-style formula gives the area directly without ever finding a height. The modulus in the formula matters because swapping vertex order flips the sign — and it means that any 'area equals a given number' equation typically splits into two valid solutions, not one.

A triangle with three labelled vertices on a coordinate grid

Common mistake

Solving only one branch of an 'area = k' equation and stopping — this throws away half the available marks.

Mini summary

Area from coordinates always carries a modulus; expect a $\pm$ pair of solutions when area is given as a fixed value.

SOHCAHTOA — and Knowing When It Breaks Down

SOHCAHTOA only ever applies inside a right-angled triangle: sine, cosine and tangent as ratios of opposite, adjacent and hypotenuse, underpinned by Pythagoras. The instant a triangle isn't right-angled, you must switch to a different rule — recognising that switch is the most heavily examined skill in this whole topic. Elevation and depression angles are always measured from the horizontal, and they're equal to each other only because the two horizontal lines involved are parallel.

A right triangle with opposite, adjacent, hypotenuse labelled, alongside a cliff and boat elevation/depression diagram

Exam tip

If the question gives an angle using π, switch your calculator to radians; otherwise, use degrees. Sanity-check your final answer against physical plausibility.

Common mistake

Leaving the calculator in the wrong angle mode (RAD instead of DEG, or vice versa) — always check the mode indicator before calculating.

Mini summary

SOHCAHTOA is right-triangle only — the moment the triangle isn't right-angled, a different approach is required.

The Unit Circle, Radians and Exact Values

The unit circle generalises trig beyond 0°-90°: $cos θ$ is the $x$ -coordinate and $sin θ$ is the $y$ -coordinate of the point reached by rotating angle $θ$ anticlockwise from the positive $x$ -axis, valid for any real angle. Radians measure the same rotation as arc length divided by radius, and arc length and sector area formulas only work correctly when $θ$ is in radians. The Pythagorean identity $sin^{2} θ + cos^{2} θ = 1$ is simply $x^{2} + y^{2} = 1$ restated, since $(cos θ, sin θ)$ lies on the unit circle.

Unit circle diagram with angle theta and point (cos theta, sin theta) marked, radian arc shown

Exam tip

If a question says 'without a calculator' or wants an answer 'in exact form', recall $sin 30° = \frac{1}{2}$ and similar values from the 30-60-90 and 45-45-90 triangles — there's no lookup table in the data booklet.

Mini summary

The unit circle extends sin/cos to all angles as coordinates; know exact values and radian conversions cold.

Quick formula sheet

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Distance between two points — It's Pythagoras with dx and dy as the legs

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Midpoint of a segment

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Gradient of a line

m_{1} m_{2} = - 1

Condition for two lines to be perpendicular

(x - a)^{2} + (y - b)^{2} = r^{2}

Standard (centre-radius) form of a circle

x^{2} + y^{2} + D x + E y + F = 0, centre = (- \frac{D}{2}, - \frac{E}{2}), r = (\frac{D}{2})^{2} + (\frac{E}{2})^{2} - F

General form of a circle and how to extract centre/radius

Area = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

Area of a triangle given three vertices

sin θ = \frac{opp}{hyp}, cos θ = \frac{adj}{hyp}, tan θ = \frac{opp}{adj}

SOHCAHTOA — right triangles only

cos θ = x, sin θ = y

Unit circle definitions, valid for any real angle

sin^{2} θ + cos^{2} θ = 1

Pythagorean identity

s = r θ

Arc length (

θ

in radians)

A = \frac{1}{2} r^{2} θ

Sector area (

θ

in radians)

Practice questions

Easy

Find the distance between the points $(1, 2)$ and $(4, 6)$ .
Find the midpoint of the segment joining $(- 3, 5)$ and $(7, - 1)$ .
State the exact value of $sin 30°$ and $cos 60°$ .

Medium

Find the gradient of the line through $(2, 1)$ and $(5, - 4)$ , then find the gradient of a line perpendicular to it.
Find the centre and radius of the circle $x^{2} + y^{2} - 6 x + 4 y - 3 = 0$ .
The angle of elevation to the top of a tower from a point 40 m from its base is 35°. Find the height of the tower.

Challenge

Points $A (1, 2)$ , $B (5, 6)$ , $C (3, k)$ form a triangle of area $8$ . Find both possible values of $k$ .
A circle has general equation $x^{2} + y^{2} - 4 x + 2 y - 11 = 0$ . Find the perpendicular distance from its centre to the line $3 x - 4 y + 1 = 0$ , and state whether the line intersects the circle.
Convert $\frac{5 π}{6}$ radians to degrees, then find the arc length and sector area for a circle of radius 8 cm subtending this angle.

Frequently asked questions

Do I need to memorise the distance, midpoint and gradient formulas for IB Maths AA?+

Yes — these three formulas are not included in the data booklet, so you need to know them from memory, along with the perpendicular condition $m_{1} m_{2} = - 1$ .

How do I find the centre and radius of a circle from its general equation?+

Complete the square on the $x$ -terms and $y$ -terms separately to rewrite $x^{2} + y^{2} + D x + E y + F = 0$ as $(x - a)^{2} + (y - b)^{2} = r^{2}$ , giving centre $(- D /2, - E /2)$ and radius $(D /2)^{2} + (E /2)^{2} - F$ .

When can I use SOHCAHTOA instead of another trig rule?+

Only when the triangle you're solving has a right angle. If it doesn't, SOHCAHTOA won't work and you need a different approach for non-right triangles.

Why does an area-from-coordinates question sometimes have two answers?+

The triangle area formula uses a modulus, so an equation like 'area = 9' typically produces a $\pm$ pair, giving two valid values for the unknown.

What's the difference between using degrees and radians on my calculator?+

Radians and degrees measure angles differently, and using the wrong mode gives wildly incorrect results. Use radians when the angle is given with π, and check your calculator's mode indicator before every trig calculation.

What exact trig values should I memorise for IB Maths AA?+

You should know sin, cos and tan at 0°, 30°, 45°, 60° and 90° without a calculator, since there's no lookup table for these in the data booklet.

Get the Full Geometry and Trigonometry Revision Notes

Complete worked examples with common traps flagged, including circle-and-line and coordinate-triangle problems Full breakdown of when to use each trig approach, with the decision rule spelled out Exact-value tables, identity derivations and mark-scheme style tips for 'show that' questions Original mock papers and exam-style questions to test yourself under timed conditions

Get the Geometry and Trigonometry notes on RevisionPrep →

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