IEB Grade 11 Circle Geometry and Measurement Cheat Sheet

Euclidean geometry and measurement are about 50 marks of Grade 11 Paper 2. Every statement needs a reason, so the accepted wording is given for each.

Chords and angles (reason in brackets)

• Perpendicular from the centre bisects a chord (line from centre $\perp$ chord).

• Centre to midpoint of a chord is perpendicular (line from centre to midpt of chord).

• $r^2=d^2+{\left(\frac{\text{chord}}{2}\right)}^2$

• Angle at the centre is double the angle at the circumference ($\angle$ at centre $=2\times \angle$ at circumference).

• Angles in the same segment ($\angle$s in the same seg); angle in a semicircle ($\angle$ in semi-circle); equal chords subtend equal angles.

Cyclic quadrilaterals and tangents

• Opposite angles supplementary (opp $\angle$s of cyclic quad); exterior angle = interior opposite (ext $\angle$ of cyclic quad).

• Prove cyclic using a converse, never 'cyclic quad'.

• Tangent $\perp$ radius (tan $\perp$ radius); two tangents from a point are equal (tans from common pt).

• Tan-chord theorem: angle between tangent and chord equals the angle in the alternate segment.

Riders

• Mark every given fact on the diagram first. Radii mean isosceles triangles; a diameter means 90°; a tangent means 90° to the radius and the tan-chord theorem.

• One statement, one reason, one line.

Measurement

• Right prism: $V=(\text{area of base})\times H$; $S=2(\text{area of base})+(\text{perimeter of base})\times H$.

• Cylinder: $V=\pi r^{2}h$; $S=2\pi r^2+2\pi rh$ (closed).

• Pyramid $V=\frac{1}{3}$ base $\times H$; cone $V=\frac{1}{3}\pi r^{2}H$, $S=\pi r^2+\pi rs$; sphere $V=\frac{4}{3}\pi r^3$, $S=4\pi r^2$.

• All dimensions $\times k$: area $\times k^2$, volume $\times k^3$.

• 1 m³ = 1 000 litres; 1 cm³ = 1 mℓ.

Know by heart

• All reasons in their short form

• The four examinable circle proofs (acute-angle case): line from centre to midpoint of chord, angle at centre, opposite angles of a cyclic quad, tan-chord

• Prism and cylinder formulas

• Pyramid, cone and sphere formulas (many papers print them in the question, but they are not on the information sheet)

On the formula sheet

• None of the geometry or measurement formulas

Common traps

• Statements without reasons.

• Using the slant height in a volume formula.

• Counting hidden faces in composite surface areas.

Related topics

• perpendicular from centre to chord
• angles in a circle
• cyclic quadrilaterals
• tangents to a circle
• tangent chord theorem
• circle geometry riders
• surface area and volume prisms
• surface area and volume cones spheres pyramids