IB MYP Mathematics: Geometry, area, volume and circles (MYP Mathematics) – Revision Notes

These notes condense the geometry strand of the spatial reasoning branch of the IB Mathematics guide (for use from September 2020/January 2021), first assessment 2022, across all three framework columns and the enrichment list. The full study guide has the explanations; then test yourself on the practice set.

Related pages: IB MYP Mathematics hub, the printable checklist, criteria in practice revision notes and the free diagnostics.

Angle facts

Fact	Rule
Straight line	angles add to 180°
Around a point	angles add to 360°
Triangle	interior angles add to 180°
Quadrilateral	interior angles add to 360°
n-sided polygon	interior sum = (n − 2) × 180°
Regular polygon	exterior angle = 360°/n
Vertically opposite	equal
Parallel lines	corresponding equal; alternate equal; co-interior add to 180°

Method for angle chasing:

1. Sketch the figure and write in each angle you are given.
2. Find any angle you can in one step, and write the reason beside it.
3. Continue until you reach the target. If an algebraic expression is involved, form an equation from the rule.

Reminder. The angles of a triangle are in the ratio 3 : 4 : 5. Twelve parts make 180°, so one part is 15° and the angles are 45°, 60° and 75°.

Area and perimeter

Shape	Area	Notes
Rectangle, square	l × w	square: side²
Parallelogram	b × h	h is perpendicular
Triangle	½bh	any base with its matching height
Trapezoid	½(a + b)h	a, b parallel sides
Rhombus, kite	½d₁d₂	diagonals cross at 90°
Circle	πr²	circumference 2πr or πd

Reminders.

• Trapezoid with parallel sides 9 and 15 and height 6: ½ × 24 × 6 = 72.
• Kite with diagonals 14 and 9: ½ × 14 × 9 = 63.
• Circumference 50 cm: r = 50 ÷ 2π = 7.96 cm, area = 199 cm² (3 s.f.).

Compound shapes in steps: split or subtract; find each part; add or subtract; for perimeter follow the outer boundary only.

Metric conversions

Quantity	Multiply or divide by
Length	km to m ×1000; m to cm ×100; cm to mm ×10
Area	the length factor squared (m² to cm² ×10 000)
Volume	the length factor cubed (m³ to cm³ ×1 000 000)
Capacity (extended)	1 mL = 1 cm³; 1 L = 1000 cm³; 1 m³ = 1000 L

Reminder. 7.5 L = 7500 cm³. Doubling every length multiplies area by 4 and volume by 8.

Solids

Solid	Volume	Total surface area
Prism	area of cross-section × length	2 × cross-section + perimeter × length
Cylinder	πr²h	2πr² + 2πrh
Pyramid	⅓ × base area × h	base + triangular faces
Cone	⅓πr²h	πr² + πrl

Slant height: l = √(r² + h²) for a cone. A cone with r = 15 and h = 8 has l = 17.

Nets: cylinder = 2 circles + a rectangle of length 2πr; triangular prism = 2 triangles + 3 rectangles; square pyramid = 1 square + 4 triangles; cuboid = 6 rectangles.

Method for compound solids:

1. Split into prisms, cylinders, pyramids or cones.
2. Find each volume and add (or subtract any hole).
3. For surface area, count only faces that are exposed; hidden joined faces are left out.

Reminder. Cylinder r = 2, h = 10: V = 40π = 126 (3 s.f.).

Short worked reminders

Cyclic quadrilateral with algebra. Opposite angles of a cyclic quadrilateral are (2y + 10)° and (3y − 20)°.

• Opposite angles add to 180°, so 5y − 10 = 180.
• y = 38, so the angles are 86° and 94°.
• Reason to write: opposite angles of a cyclic quadrilateral add to 180°.

Solid with a hole. A cuboid 10 cm by 6 cm by 4 cm has a cylindrical hole of radius 1.5 cm drilled through the 4 cm direction.

• Cuboid: 10 × 6 × 4 = 240 cm³.
• Hole: π × 1.5² × 4 = 9π = 28.3 cm³.
• Remaining volume = 240 − 28.3 = 212 cm³ (3 s.f.).

Segment. A circle of radius 6 cm has a chord that makes a 60° angle at the centre.

• Sector = (60/360) × π × 36 = 6π = 18.85 cm².
• The two radii and the chord form an equilateral triangle of side 6, with area ½ × 6 × (3√3) = 9√3 = 15.59 cm².
• Segment = 18.85 − 15.59 = 3.26 cm². Its perimeter is the arc 2π plus the chord 6, giving 12.3 cm.

Capacity of a cylinder. A water tank has radius 0.5 m and height 1.2 m.

• Volume = π × 0.25 × 1.2 = 0.942 m³.
• Capacity = 0.942 × 1000 = 942 L (3 s.f.).

Constructions in brief

• Angle bisector: draw an arc cutting both arms, then equal arcs from those points; join the vertex to where they meet.
• Perpendicular bisector: equal arcs from both ends, above and below the line; join the crossing points.
• Triangle from three sides: draw the base, then arcs from each end with the other two lengths.
• 60° angle: an arc from the vertex, then an arc of the same radius from where it cuts the arm.

Circle geometry

Theorem	Use
Angle at centre = 2 × angle at circumference	same arc
Angles in the same segment	equal
Angle in a semicircle	90°
Cyclic quadrilateral	opposite angles add to 180°
Tangent and radius	meet at 90°
Tangents from an external point	equal lengths
Alternate segment	tangent-chord angle matches the inscribed angle on the far side
Perpendicular from centre to chord	bisects the chord

Reminders.

• Tangent length with OP = 17 and radius 8: √(17² − 8²) = 15. (Tangent to a circle at T gives a right angle at T.)
• Triangle with a diameter as one side: if one base angle is 35°, the other is 55°, since the third angle is 90°.

Arc, sector, segment

Quantity	Degrees	Radians (enrichment)
Arc	(θ/360) × 2πr	rθ
Sector area	(θ/360) × πr²	½r²θ
Sector perimeter	2r + arc	2r + rθ
Segment area	sector − triangle	sector − triangle

Reminders.

• r = 15, θ = 48°: arc = (48/360) × 30π = 4π = 12.6; sector perimeter = 30 + 12.6 = 42.6.
• r = 5, θ = 1.2 rad: arc = 6, sector area = 15.
• Segment area uses sector minus triangle only when the angle at the centre is below 180°.

Enrichment. Inscribed square in a circle of radius r has diagonal 2r, so area 2r². Koch snowflake perimeter is multiplied by 4/3 at each stage.

Showing your work

Spatial reasoning questions often ask you to justify as well as calculate. Write the formula, substitute with units, then give the result with a sensible degree of accuracy. For angles, write the reason in words beside each line, for example “alternate angles, parallel lines”. For real-life tasks (criterion D), state the model you assumed, such as treating a tin as a perfect cylinder, and say whether your answer is reasonable. Draw a labelled sketch whenever a diagram is not given.

Must-know distinctions

• Slant height is not vertical height.
• Chord is a line segment; arc is a curve; tangent touches once.
• Sector is bounded by two radii; segment is bounded by a chord.
• Area scales by the factor squared; volume by the factor cubed.
• Capacity is liquid volume; volume is the space a solid occupies.

Quick self-test

1. What is the sum of interior angles of a hexagon?
2. Co-interior angles: one is 112°. Find the other.
3. Trapezoid with parallel sides 9 and 15, height 6. Find the area.
4. Convert 7.5 L to cm³.
5. Cylinder, radius 2 cm, height 10 cm. Find its volume to 3 s.f.
6. A cone has radius 15 cm and height 8 cm. Find the slant height.
7. A sector has radius 15 cm and angle 48°. Find its arc length to 3 s.f.
8. Points A, B and C lie on a circle, with AB passing through the centre. Angle BAC is 35°. Find angle ABC.
9. A tangent from a point 17 cm from the centre touches a circle of radius 8 cm. Find the tangent length.
10. Angles 3x, 4x and 5x form a triangle. Find the smallest angle.
11. A kite has diagonals 14 cm and 9 cm. Find its area.
12. A sector has radius 5 and angle 1.2 radians. Find its area (enrichment).

Answers

1. 720°
2. 68°
3. 72 square units
4. 7500 cm³
5. 126 cm³
6. 17 cm
7. 12.6 cm
8. 55°
9. 15 cm
10. 45°
11. 63 cm²
12. 15 square units

Where marks are usually lost

• Using a slanted side as the height of a parallelogram or triangle.
• Leaving diameter in place of radius in circle formulas.
• Converting area and volume with the length factor.
• Using the full cone height where the slant height is required.
• Counting hidden joined faces in compound surface area.
• Stating an angle with no reason or theorem name.
• Applying the angle-at-centre theorem to the wrong arc.
• Giving a segment area as a sector area.
• Rounding too early, or giving no units.
• Treating a perimeter of a segment as arc only; the chord is part of the boundary.
• In a “show that” or reasoning question, skipping the line that names the rule you used.

Official syllabus

IB Mathematics guide (for use from September 2020/January 2021), MYP framework, spatial reasoning: Geometry. International Baccalaureate Organization, first assessment 2022.