## Introduction:

Paving and streetmasonry can, at times, seem littered with references to barely-remembered geometric terminology, but most of it is comparatively simple and based around the geometry of circles and right-angled triangles.

Much of that geometry is most easily explained using kerbs as an example, as they are often the closest physical manifestation we have to the geometry created by a designer, specifier or architect. However, the same geometric terms and relationships apply to all aspects of paving, streetmasonry and hardscapes. Just take a look at the work done in preparing a flagstone fan radius, if further proof is needed!

This page sets out to define some of the most commonly encountered terms. ## Circle and Arc Definitions

### Origin

The centre of a circle. The point from which a radius is swung. All points on a circle's outer edge (circumference) are the same distance from the origin.

This has to be the most commonly used term in streetmasonry, which tends to focus on kerblines of a given radius and fancy set-outs, but it is also used with more simple paving projects.

The radius is the distance from the centre of a circle (the origin) to any point on its outside edge (the circumference)

The plural is sometimes referred to as radii (ray-di-eye)

A physical line following the radius, from the origin out to the circumference, may also be known as a radial The kerbline follows the circumference of a circle (red) and each kerb unit is the same distance from the origin, shown as pink radial lines

### Diameter

The diameter is the span across a full circle, from one side to the other passing through the origin.

Any line touching the circle circumference in two places and passing through the origin MUST be a diameter.

The diameter of a circle is always equal to two radii The 'step' running across this feature kerb circle is a diameter, as it touches both sides of the circle and passes through the origin (lighting column)

### Tangent

A line or other geometrical construction (could be another circle, an arc or and ellipse) is said to be a tangent, or is tangential to, when it meets or 'kisses' a circle or arc at only one point.

### Tangent Point

The point where a tangent meets or 'kisses' the circle

This term is most commonly used to describe the point on a kerbline where a straight line becomes a curve. The kerbs of the tangent line will be straight units, but once past the tangent point, the kerbs may be radius units or be laid to an arc. ### Arc

An arc is any part of the circumference of a circle.

It may be relatively short and shallow, or it could be almost a full circle.

Bends and curves are formed using one or more arcs An s-curve formed by linking an internal and external arc by a short length of straight

A quadrant is the area under an arc that is exactly one-quarter of a full circle.

In classical geometry, the area under an arc is known as a 'sector'. A quadrant is a special type of sector.

The term 'Quadrant' may refer to the arc or to the sector enclosed by two radial lines set at 90°

Single-piece quarter-circle kerbs are often known as "Quadrants" or "Cheeses" ## Chord

Whereas a tangent only touches the circumference of a circle or arc at one point, a chord touches at two points.

When such a line passes through the origin, it is a diameter, but any other line touching the same circle or arc at two points is a chord.

Chords can be very useful when it's necessary to determine the radius of an arc from a single kerb or short section of arc...see drawing opposite.

This technique is explained in greater detail on a separate page...

Create a chord, here labelled C - D, and measure its length. Divide by 2 and refer to as "A"

Measure perpendicular distance from arc to mid-point of chord. Refer to as "B"

Add together the square of A plus the square of B

Divide the result by 2B (twice what was measured as a perpendicular)

### Segment

As previously described, the area of a circle bounded by an arc and two radii is referred to as a sector. Similarly, the area bounded by an arc and a chord is known as a segment.

Segments aren't widely used in paving and streetmasonry but are included here for completeness. ## Circle and Arc Equations

### Calculating circumference

The circumference of a circle is calculated using Pi (Π) which can be as a fraction, 22/7, or as a decimal number, roughly 3.142

C = 2 x Π x r

Where
C = circumference
Π = 3.142
r = radius of the circle

So: a circle with a measured radius of, say, 2.4m has a circumference of...

2 x 3.142 x 2.4 = 15.08m

For the length of an arc, the calculated circumference is simply multiplied by the ratio of the angle of the arc over 360°...

arc = 2 x Π x r x (∠ ÷ 360)

Where C = circumference Π = 3.142 r = radius of the circle ∠ = angle of arc

So: in our 2.4m radius circle described above, an arc of 72° has a circumference of...

2 x 3.142 x 2.4 x (72 ÷ 360)

15.08 x 0.2 = 3.02m ### Calculating area

Generally speaking, the ability to calculate the area of a circle or arc is more useful to the typical paving contractor than circumference, and once again, the calculation depends on that magic figure, Π

A = Π x r²

Where
A = Area
Π = 3.142
r = radius of the circle

So: the now-familiar circle with a measured radius of 2.4m has an area of...

3.142 x (2.4 x 2.4)

3.142 x 5.76 = 18.1m²

For the area under an arc (more correctly known as a 'sector'), the calculated whole circle area is simply multiplied by the ratio of the angle of the arc over 360°...

A = Π x r²: (∠ ÷ 360)

Where
A = Area of sector
Π = 3.142
r = radius of the circle
∠ = angle of arc

So: in our 2.4m radius circle described above, an arc of 72° has a sector with an area of...

3.142 x (2.4 x 2.4) x (72 ÷ 360)

3.142 x 5.76 x 0.2 = 3.62m² ## Angles:

There are three basic 'angles' that are discussed when considering the setting-out or measuring-up of paving works:

Acute: - an angle that is LESS than 90°

Right Angle: - an angle that is exactly 90°

Obtuse or Reflex: - an angle that is MORE than 90° 