Leaving Cert Geometry (Higher Level): Theorems & Proofs

Geometry on the Higher course covers the examinable theorems, circle theorems, enlargements and formal proofs. It is examined on Paper 2.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

A set list of theorems is examinable, and you may be asked to prove specific ones.
Circle theorems are a reliable source of marks on Paper 2.
Enlargements involve a centre and a scale factor that changes area by the factor squared.
Geometry is examined on Paper 2.

Geometry explained

Points and the Plane

In geometry, a plane is a flat, two-dimensional surface that extends infinitely in all directions.

A point is an exact location in space with no size or dimension. We name points with capital letters: $A$ , $B$ , $C$ .

Key concept: Through any two distinct points, there is exactly one line.

Lines and Line Segments

A line is a straight path extending infinitely in both directions. We write it as $\overleftrightarrow{AB}$ or $\ell$ .

A line segment is part of a line with two endpoints. We write it as $\overline{AB}$ and its length as $AB$ .

A ray is part of a line that starts at one point and extends infinitely in one direction. Written as $\overrightarrow{AB}$ .

Parallel and Perpendicular Lines

Parallel lines never intersect and are always the same distance apart. Notation: $\ell_1 \parallel \ell_2$

Perpendicular lines intersect at right angles ( $90°$ ). Notation: $\ell_1 \perp \ell_2$

Important: In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to both.

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Key formulas

Name	Formula	Description
Area of Rectangle	$A = l \times w$	Area equals length times width
Area of Triangle	$A = \frac{1}{2}bh$	Area equals half base times height
Area of Circle	$A = \pi r^2$	Area of circle with radius r
Circumference	$C = 2\pi r$	Circumference of circle with radius r
Arc Length	$s = r\theta$	Arc length with angle θ in radians
Sector Area	$A = \frac{1}{2}r^2\theta$	Area of sector with angle θ in radians

Worked examples

Worked example 1

Find the area of a circle with radius 3 cm

Area of circle = $π r^2 = π × 3^2 = 9π$ cm².

Answer:

9π cm²

Worked example 2

Find the perimeter of a rectangle with length 8 cm and width 5 cm

Perimeter = 2(l + w) = 2(8 + 5) = 2(13) = 26 cm.

Answer:

26 cm

Where students lose marks

Not stating the reason for each step in a proof.
Forgetting that an enlargement scales area by the square of the scale factor.
Misreading which circle theorem applies to a diagram.

Frequently asked questions

Which theorems do I need to know for the Leaving Cert?

There is a defined list of examinable theorems, and a smaller set you may be asked to prove in full. The syllabus on ncca.ie lists exactly which ones.

Are circle theorems on Higher Level?

Yes, circle theorems are part of the Higher Level geometry course and appear regularly on Paper 2.

Is geometry on Paper 1 or Paper 2?

Geometry is examined on Paper 2.

Authoritative sources

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