Cantilevered Boardwalks For Motorway Bridges?

Bridge over M9 Kilcullen Bypass on Sunnyhill Road. Image coiurtesy Google.

When Kildare County Council, or whoever was responsible, produced the spec for this bridge, (and the one over the M7 at the Bundle of Sticks Roundabout outside Naas), they left out foothpaths. Maybe it was lack of common sense and an attitude of "Sure, only cars go out there". Or perhaps it was due to penny pinching and lack of foresight and a narrow bridge was cheaper than one a couple of metres wider? Anyway, I'm wondering could boardwalks/cantilevered foothpaths be fixed onto the sides, exterior to the parapets?
It would obviously be impractical to install foothpaths on the inside of the parapets due to to both lanes being reduced in width. As it is, cars can't pass pedestrians safely without partially moving into the other lane. An article by Paul MacDonald, Kildare National Roads Office, on the construction of the M7 Newbridge Bypass and M9 Kilcullen Bypass is available on the Kilcullen Diary here.

The section about the earthing of the steelwork of the underbridge for the River Liffey is interesting—High voltage pylons can induce voltages and currents in conductors, potentially leading to different and hazardous potentials between isolated elements of a structure. Bonding them together eliminates these voltages and earthing a structure pulls the voltage down to ground (similar to the equipotential bonding in bathrooms, where metal objects such as drains, radiators, towel rails, water pipes and metal baths are bonded together):

Bridge over M9 Kilcullen Bypass on Sunnyhill Road. Image coiurtesy Google.

How to Calculate Bolt Circle Diameter (BCD) for Chainrings and Bash Guards

Bolt circle diameter (BCD). © Eugene Brennan

What Is a Bolt Circle Diameter (BCD) or Pitch Circle Diameter (PCD)?

As you may be aware, if you've found this guide, a bolt circle is an imaginary circle that passes through the centre of the bolt or screw holes in a round pattern. This is typically a wheel hub on a vehicle, chainring on a bicycle or flange on a pipe. The diameter of this circle is known as the bolt circle diameter (BCD) or pitch circle diameter (PCD).

Typically, BCD is quoted for chainrings on bikes and the bash guards that cover them, and you can calculate it by measuring the distance between the mounting holes of the chainring on the spider. If you need to order a part for a bicycle or check the BCD of a flange and can only measure the bolt hole spacing, you need to know the formula for working it out. Alternatively you can skip to the end of this guide where there are tables for various bolt spacings.

Imaginary bolt circle. © Eugene Brennan

Working Out the Formula for Bolt Circle Diameter

I've given the formula further down this guide, but the math for working it out is fairly straightforward if you want to follow it through. To do the calculations, we first need to understand what sine and cosine mean.

A right-angled triangle has one angle that measures 90 degrees. The longest side opposite this angle is called the hypotenuse. Sine and cosine are trigonometric functions of an angle and are the ratios of the lengths of the other two sides to the hypotenuse of a right-angled triangle.

Let's represent one of the angles by the Greek letter θ (theta) as shown in the diagram below.

Side a is called the "opposite" side and side b is called the "adjacent" side to the angle θ.

"|" around the words below means "length of"

So:

sine θ = |opposite side| / |hypotenuse|

cosine θ = |adjacent side| / |hypotenuse|

Sine and cosine are sometimes abbreviated to sin and cos.

Sine and cosine for a right-angled triangle.. © Eugene Brennan

Angle Subtended by the Arc

Since there are 360 degrees in a full circle, then if there are N equally spaced mounting holes, the angle subtended by an arc (produced by joining the ends of the arc to the centre) on an imaginary circle passing through the bolt holes is:

θ = 360 / N

In the diagram below, there are six equally spaced holes, so:

θ = 360 / 6 = 60º

Angle subtended by arc. In this example, the angle is 360/6 = 60 degrees. © Eugene Brennan

Use Sine and Cosine to Work Out the Radius

We can draw triangles and use sine and cosine to solve the triangles and find the length of the radius R and, therefore the diameter D.

Working out bolt circle diameter (BCD). © Eugene Brennan

The simplest case is when holes are equally spaced around in a circle.

If we can find the radius R of the bolt circle, then this gives us the diameter D.

We calculated that:

θ = 360 / N

Also, A is the spacing between holes.

D is the bolt circle diameter.

Referencing the diagram above, a triangle can be drawn between the two ends of an arc on the bolt circle and the centre of the circle. This can be split up into two equal right-angled triangles. The length of the chord or line joining the two ends of the arc is the hole spacing A. (For more info on circles, see this guide.)

So from the diagram, we see that:

Sin (θ/2) = |opposite| / |hypotenuse| = (A/2) / R

Rearranging:

R = (A/2) / Sin (θ/2)

But D = 2R

So, D = 2 x (A/2) / Sin (θ/2) = A/Sin (θ/2)

But we worked out θ = 360 / N

So substituting for θ gives

D = A/Sin (θ/2) = A/Sin ((360 / N)/2) = A/Sin (180 / N)

Our final equation is:

D = A/Sin (180 / N)

Working Out the BCD for a Chainring or Bash Guard

All you need to do to calculate the BCD is measure the hole spacing A between adjacent holes, count the number of holes N and plug them into the equation above to calculate the BCD.

Tables of Bolt Circle Diameters

The tables below give the BCD for hole spacings from 10 to 200 mm, 1 to 12 inches and 3 to 8 holes.

Table 1: Bolt Circle Diameters for Hole Spacings From 10 to 200 mm

Table 1: Bolt Circle Diameters for Hole Spacings From 10 to 200 mm

Hole spacing	Number of holes
	3	4	5	6	7	8
10	11.5	14.1	17.0	20	23.0	26.1
11	12.7	15.6	18.7	22	25.4	28.7
12	13.9	17.0	20.4	24	27.7	31.4
13	15.0	18.4	22.1	26	30.0	34.0
14	16.2	19.8	23.8	28	32.3	36.6
15	17.3	21.2	25.5	30	34.6	39.2
16	18.5	22.6	27.2	32	36.9	41.8
17	19.6	24.0	28.9	34	39.2	44.4
18	20.8	25.5	30.6	36	41.5	47.0
19	21.9	26.9	32.3	38	43.8	49.6
20	23.1	28.3	34.0	40	46.1	52.3
21	24.2	29.7	35.7	42	48.4	54.9
22	25.4	31.1	37.4	44	50.7	57.5
23	26.6	32.5	39.1	46	53.0	60.1
24	27.7	33.9	40.8	48	55.3	62.7
25	28.9	35.4	42.5	50	57.6	65.3
26	30.0	36.8	44.2	52	59.9	67.9
27	31.2	38.2	45.9	54	62.2	70.6
28	32.3	39.6	47.6	56	64.5	73.2
29	33.5	41.0	49.3	58	66.8	75.8
30	34.6	42.4	51.0	60	69.1	78.4
31	35.8	43.8	52.7	62	71.4	81.0
32	37.0	45.3	54.4	64	73.8	83.6
33	38.1	46.7	56.1	66	76.1	86.2
34	39.3	48.1	57.8	68	78.4	88.8
35	40.4	49.5	59.5	70	80.7	91.5
36	41.6	50.9	61.2	72	83.0	94.1
37	42.7	52.3	62.9	74	85.3	96.7
38	43.9	53.7	64.6	76	87.6	99.3
39	45.0	55.2	66.4	78	89.9	101.9
40	46.2	56.6	68.1	80	92.2	104.5
41	47.3	58.0	69.8	82	94.5	107.1
42	48.5	59.4	71.5	84	96.8	109.8
43	49.7	60.8	73.2	86	99.1	112.4
44	50.8	62.2	74.9	88	101.4	115.0
45	52.0	63.6	76.6	90	103.7	117.6
46	53.1	65.1	78.3	92	106.0	120.2
47	54.3	66.5	80.0	94	108.3	122.8
48	55.4	67.9	81.7	96	110.6	125.4
49	56.6	69.3	83.4	98	112.9	128.0
50	57.7	70.7	85.1	100	115.2	130.7
51	58.9	72.1	86.8	102	117.5	133.3
52	60.0	73.5	88.5	104	119.8	135.9
53	61.2	75.0	90.2	106	122.2	138.5
54	62.4	76.4	91.9	108	124.5	141.1
55	63.5	77.8	93.6	110	126.8	143.7
56	64.7	79.2	95.3	112	129.1	146.3
57	65.8	80.6	97.0	114	131.4	148.9
58	67.0	82.0	98.7	116	133.7	151.6
59	68.1	83.4	100.4	118	136.0	154.2
60	69.3	84.9	102.1	120	138.3	156.8
61	70.4	86.3	103.8	122	140.6	159.4
62	71.6	87.7	105.5	124	142.9	162.0
63	72.7	89.1	107.2	126	145.2	164.6
64	73.9	90.5	108.9	128	147.5	167.2
65	75.1	91.9	110.6	130	149.8	169.9
66	76.2	93.3	112.3	132	152.1	172.5
67	77.4	94.8	114.0	134	154.4	175.1
68	78.5	96.2	115.7	136	156.7	177.7
69	79.7	97.6	117.4	138	159.0	180.3
70	80.8	99.0	119.1	140	161.3	182.9
71	82.0	100.4	120.8	142	163.6	185.5
72	83.1	101.8	122.5	144	165.9	188.1
73	84.3	103.2	124.2	146	168.2	190.8
74	85.4	104.7	125.9	148	170.6	193.4
75	86.6	106.1	127.6	150	172.9	196.0
76	87.8	107.5	129.3	152	175.2	198.6
77	88.9	108.9	131.0	154	177.5	201.2
78	90.1	110.3	132.7	156	179.8	203.8
79	91.2	111.7	134.4	158	182.1	206.4
80	92.4	113.1	136.1	160	184.4	209.1
81	93.5	114.6	137.8	162	186.7	211.7
82	94.7	116.0	139.5	164	189.0	214.3
83	95.8	117.4	141.2	166	191.3	216.9
84	97.0	118.8	142.9	168	193.6	219.5
85	98.1	120.2	144.6	170	195.9	222.1
86	99.3	121.6	146.3	172	198.2	224.7
87	100.5	123.0	148.0	174	200.5	227.3
88	101.6	124.5	149.7	176	202.8	230.0
89	102.8	125.9	151.4	178	205.1	232.6
90	103.9	127.3	153.1	180	207.4	235.2
91	105.1	128.7	154.8	182	209.7	237.8
92	106.2	130.1	156.5	184	212.0	240.4
93	107.4	131.5	158.2	186	214.3	243.0
94	108.5	132.9	159.9	188	216.6	245.6
95	109.7	134.4	161.6	190	219.0	248.2
96	110.9	135.8	163.3	192	221.3	250.9
97	112.0	137.2	165.0	194	223.6	253.5
98	113.2	138.6	166.7	196	225.9	256.1
99	114.3	140.0	168.4	198	228.2	258.7
100	115.5	141.4	170.1	200	230.5	261.3
101	116.6	142.8	171.8	202	232.8	263.9
102	117.8	144.2	173.5	204	235.1	266.5
103	118.9	145.7	175.2	206	237.4	269.2
104	120.1	147.1	176.9	208	239.7	271.8
105	121.2	148.5	178.6	210	242.0	274.4
106	122.4	149.9	180.3	212	244.3	277.0
107	123.6	151.3	182.0	214	246.6	279.6
108	124.7	152.7	183.7	216	248.9	282.2
109	125.9	154.1	185.4	218	251.2	284.8
110	127.0	155.6	187.1	220	253.5	287.4
111	128.2	157.0	188.8	222	255.8	290.1
112	129.3	158.4	190.5	224	258.1	292.7
113	130.5	159.8	192.2	226	260.4	295.3
114	131.6	161.2	193.9	228	262.7	297.9
115	132.8	162.6	195.6	230	265.0	300.5
116	133.9	164.0	197.4	232	267.4	303.1
117	135.1	165.5	199.1	234	269.7	305.7
118	136.3	166.9	200.8	236	272.0	308.3
119	137.4	168.3	202.5	238	274.3	311.0
120	138.6	169.7	204.2	240	276.6	313.6
121	139.7	171.1	205.9	242	278.9	316.2
122	140.9	172.5	207.6	244	281.2	318.8
123	142.0	173.9	209.3	246	283.5	321.4
124	143.2	175.4	211.0	248	285.8	324.0
125	144.3	176.8	212.7	250	288.1	326.6
126	145.5	178.2	214.4	252	290.4	329.3
127	146.6	179.6	216.1	254	292.7	331.9
128	147.8	181.0	217.8	256	295.0	334.5
129	149.0	182.4	219.5	258	297.3	337.1
130	150.1	183.8	221.2	260	299.6	339.7
131	151.3	185.3	222.9	262	301.9	342.3
132	152.4	186.7	224.6	264	304.2	344.9
133	153.6	188.1	226.3	266	306.5	347.5
134	154.7	189.5	228.0	268	308.8	350.2
135	155.9	190.9	229.7	270	311.1	352.8
136	157.0	192.3	231.4	272	313.4	355.4
137	158.2	193.7	233.1	274	315.8	358.0
138	159.3	195.2	234.8	276	318.1	360.6
139	160.5	196.6	236.5	278	320.4	363.2
140	161.7	198.0	238.2	280	322.7	365.8
141	162.8	199.4	239.9	282	325.0	368.5
142	164.0	200.8	241.6	284	327.3	371.1
143	165.1	202.2	243.3	286	329.6	373.7
144	166.3	203.6	245.0	288	331.9	376.3
145	167.4	205.1	246.7	290	334.2	378.9
146	168.6	206.5	248.4	292	336.5	381.5
147	169.7	207.9	250.1	294	338.8	384.1
148	170.9	209.3	251.8	296	341.1	386.7
149	172.1	210.7	253.5	298	343.4	389.4
150	173.2	212.1	255.2	300	345.7	392.0
151	174.4	213.5	256.9	302	348.0	394.6
152	175.5	215.0	258.6	304	350.3	397.2
153	176.7	216.4	260.3	306	352.6	399.8
154	177.8	217.8	262.0	308	354.9	402.4
155	179.0	219.2	263.7	310	357.2	405.0
156	180.1	220.6	265.4	312	359.5	407.6
157	181.3	222.0	267.1	314	361.8	410.3
158	182.4	223.4	268.8	316	364.2	412.9
159	183.6	224.9	270.5	318	366.5	415.5
160	184.8	226.3	272.2	320	368.8	418.1
161	185.9	227.7	273.9	322	371.1	420.7
162	187.1	229.1	275.6	324	373.4	423.3
163	188.2	230.5	277.3	326	375.7	425.9
164	189.4	231.9	279.0	328	378.0	428.6
165	190.5	233.3	280.7	330	380.3	431.2
166	191.7	234.8	282.4	332	382.6	433.8
167	192.8	236.2	284.1	334	384.9	436.4
168	194.0	237.6	285.8	336	387.2	439.0
169	195.1	239.0	287.5	338	389.5	441.6
170	196.3	240.4	289.2	340	391.8	444.2
171	197.5	241.8	290.9	342	394.1	446.8
172	198.6	243.2	292.6	344	396.4	449.5
173	199.8	244.7	294.3	346	398.7	452.1
174	200.9	246.1	296.0	348	401.0	454.7
175	202.1	247.5	297.7	350	403.3	457.3
176	203.2	248.9	299.4	352	405.6	459.9
177	204.4	250.3	301.1	354	407.9	462.5
178	205.5	251.7	302.8	356	410.2	465.1
179	206.7	253.1	304.5	358	412.6	467.7
180	207.8	254.6	306.2	360	414.9	470.4
181	209.0	256.0	307.9	362	417.2	473.0
182	210.2	257.4	309.6	364	419.5	475.6
183	211.3	258.8	311.3	366	421.8	478.2
184	212.5	260.2	313.0	368	424.1	480.8
185	213.6	261.6	314.7	370	426.4	483.4
186	214.8	263.0	316.4	372	428.7	486.0
187	215.9	264.5	318.1	374	431.0	488.7
188	217.1	265.9	319.8	376	433.3	491.3
189	218.2	267.3	321.5	378	435.6	493.9
190	219.4	268.7	323.2	380	437.9	496.5
191	220.5	270.1	324.9	382	440.2	499.1
192	221.7	271.5	326.6	384	442.5	501.7
193	222.9	272.9	328.4	386	444.8	504.3
194	224.0	274.4	330.1	388	447.1	506.9
195	225.2	275.8	331.8	390	449.4	509.6
196	226.3	277.2	333.5	392	451.7	512.2
197	227.5	278.6	335.2	394	454.0	514.8
198	228.6	280.0	336.9	396	456.3	517.4
199	229.8	281.4	338.6	398	458.6	520.0
200	230.9	282.8	340.3	400	461.0	522.6

Disclaimer

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2018 Eugene Brennan

Thomas McGarry of McGarry's Lane

McGarry's Lane. © Eugene Brennan

I think I may have found the person who lent his name to the lane.

The Primary Valuation, also known as Griffith's Valuation was a mid-19th property valuation in Ireland. Published between 1847 and 1864, it was the first full-scale survey of its kind, and collected information about the occupiers of land and buildings, their landlords and the amount and value of property held. Griffith's Valuation is searchable here on the Ask About Ireland Website.

In the valuation, a Thomas McGarry appears, with an address in Nicholastown. The Nicholastown townland extends out as far as Yellowbogcommon townland to the south and New Abbey townland to the east. I was aware of the presence of a building, visible on the first edition, c. 1837 6" Ordnance Survey map. This was located across the road from the cottage in McGarry's Lane. By the time the c. 1900 last-edition 25" OSI map was drafted, this building had disappeared. It appears from the map and record below that Thomas McGarry was listed as an occupier of the 8 acres and 3 roods of land on which the building stood, with Robert Brereton as lessor (The owner of New Abbey House). 

Valuation book showing the entry for Thomas McGarry. Image courtesy Ask About Ireland.

Map showing corresponding parcel of land leased by Thomas McGarry (16). Map attribution: Ask About Ireland.

The Tithe Applotment Books, however, show no record of a Thomas McGarry. These books were records of tithes, a form of religious tax paid to the established church in Ireland—the Anglican Church of Ireland. Tithes were extremely unpopular, as the Roman Catholic majority also had to pay them, leading to the Tithe War of the 1830s and eventual repeal of the tax by The Irish Church Act 1869, which disestablished the Church of Ireland. In the records, there are three entries for a John McGarry, and he's recorded as having paid 3/3½  (three shillings, three and a half pence) in one of these records. No information is available on whether John was a brother or father of Thomas. I found a death record for Thomas McGarry on the Irish Genealogy website. He died on 14th March, 1889, with the location indicated as "New Abbey".
It's possible he's buried in New Abbey Cemetery.

Tithe payment by a John McGarry. Image attribution: The National Library of Ireland

The Tithe Applotment database search page is available here.  Image attribution: National library of Ireland

Death record for Thomas McGarry. Image attribution: Irish Genealogy.

The structure shown on the map was located behind this fence. No trace remains.  © Eugene Brennan

Cycle Paths, Moone High Cross and Poor Road Signage

© Eugene Brennan

Yesterday's Sunday cycle took me to the area around Moone, via a more roundabout but pleasant traffic-free route up Brewel Hill, then through Colbinstown and Timolin. I returned home via Ballitore, Crookstown and Narraghmore Bog, a 30-mile round trip. I wanted to see the infamous Timolin-Moone cycle paths on the R448. As per usual, the path layout includes those ubiquitous grass verges. I'm not sure whether this feature is purely functional to reduce water run-off onto the carriageways or for growing wildflowers in the grass keep to the environmental lobby happy. I dropped in to see the High Cross of Moone. I hadn't been there for a few years. The cross has been protected by a corrugated-Perspex-roofed canopy for over 15 years. Such constructions help to protect carvings on heritage stonework from being eroded by the elements. Perhaps their use could be more widespread, as inevitably more detail will be lost from stonework over the coming centuries?

Unusually, I met lots of drivers on the narrow L-roads I normally travel on. Many of these are only the width of a car. It wasn't until a woman stopped her car and asked me for directions, while I was taking a cycling break at a bridge on one of these L-roads that branches off the Timolin-Baltinglass road, that I realised what was happening: Blockades on the M9 had forced drivers to resort to their sat-navs to find alternative routes. The woman was on her way to Dublin and the sat-nav had sent her down this boreen. Unfortunately, the lack of signage on many minor junctions, which is pretty typical in Kildare and Wicklow and presumably most other counties, meant that she was lost and had to resort to technology. Coupled with the obscuring of signs because landowners neglect to cut back vegetation, drivers who aren't familiar with roads must find it difficult. As a 9-year-old child, I was given the job of "navigator" by my father when we got lost on a Sunday drive near Avoca and I noticed we had taken a wrong turn on the journey home. My father told me to go to John Joe Dowling's newsagent and buy a Sheet 16 map of Kildare Wicklow to use for navigation (he was a navigation officer in the LDF during the Emergency, or perhaps later in the 50s as member of the FCA). From then on I kept track of our progress on drives, making sure we didn't get lost. We always had paper road maps in the car—something that people may no longer do, relying on technology to get them to where they want to go.

New Moone–Timolin cycle lanes. © Eugene Brennan

Moone High Cross. © Eugene Brennan

Carving detail on Moone High Cross. © Eugene Brennan

Carving detail on Moone High Cross. © Eugene Brennan

The L8286, off the Timolin–Baltinglass road. © Eugene Brennan

Obscured road near Boleybeg. © Eugene Brennan

Missing Clip on My Quick-Release Skewer

Quick-release axle skewer from a bike. © eugene Brennan 

The cam lever fell out of the housing when I took my front wheel off to change a puncture, half-way between Ballymore Eustace and Harristown last Sunday. I hadn't noticed that the retaining clip had disappeared, possibly rusting off over the last 29 years.

On the Couch With ChatGPT

Created by Grok

Eugene (transcript of me incoherently expressing myself in voice mode):

Sometimes when I'm on the verge of dozing and falling asleep at the table during the daytime, it's as if my mind goes back to maybe 40 years ago and I feel like I did then.

How to Find the Probability of an Event and Calculate Odds, Permutations and Combinations

Learn about probabilities, odds, permutations and combinations. blickpixel, public domain image via Pixabay

What Is Probability Theory?

Probability theory is an interesting area of statistics concerned with the odds or chances of an event happening in a trial, e.g., getting a six when a dice is thrown or drawing an ace of hearts from a pack of cards.

17th Century Dunlavin Headstones

A 17th century headstone in the old cemetery in Dunlavin village, Co. Wicklow. © Eugene Brennan

I visited the little park adjacent to the market house in Dunlavin on my Sunday cycle yesterday. I've cycled past it dozens of times over the last thirty years, but never actually went in. This was the location of a 17th-century church and graveyard, the graveyard now having been “recycled”, with the headstones stacked up against the back wall of the space. It’s a practice I’m not too fond of, and it's something that has occurred in several locations in Dublin—Cemeteries have been turned into parks, such as at the rear of St Mary’s Church (foundation stone laid in 1700) on Mary Street, and at St Kevin’s Church, Camden Row (behind the now-demolished Kevin Street College of Technology, my alma mater). It was a shock to discover on Street View that the college had gone, but it was an ugly building.
The cemetery in Dunlavin has a couple of old 17th-century headstones, something which isn’t commonplace, as the inscriptions on headstones older than the 18th century are normally eroded and illegible unless they’ve been sheltered from the elements.

Edit: Dunlavin local historian Chris Lawlor has kindly sent me a link to his thesis, The Establishment and Evolution of an Irish Village: The Case of Dunlavin, County Wicklow 1600 -1910, which includes some details about the cemetery on p. 44:

Map courtesy Tailte Éireann.

Another 17th century headstone in Dunlavin old cemetery. © Eugene Brennan 

The location of the cemetery in Dunlavin village. Image courtesy Tailte Éireann.

Chimney Stacks and Pots in Kilcullen

Chimney stacks, some topped with pots, on buildings along Kilcullen’s Upper Main Street. 

I've been reading the cover article from the April edition of The Bridge about the planned streetscape survey of Kilcullen's Upper Main Street. Zooming in on a c. 1900 image of the street, it appears that many of the chimney stacks were devoid of chimney pots—the same is true for Lower Main Street. A chimney pot serves several functions beyond mere aesthetics. It improves airflow by effectively raising the chimney's height without requiring the entire stack to be built taller. Since a chimney pot is narrower than the flue inside the stack (these chimney stacks were often wide and sometimes flue-less; in fact, children used to climb the stacks to sweep the chimneys), it creates a venturi effect, increasing airflow speed and improving suction.

Chimney pots became more common in the 18th century, with the peak of the fashion occurring in the 19th century.

Another feature of the chimney stacks is that many of them appear to have been constructed from brick, rather than rough stone. Over the intervening 130 years or so since the Lawrence Collection photo was taken, most of the stacks have been rendered with cement or lime mortar.

Images courtesy The National Library of Ireland.

Brick-constructed chimney stacks on buildings along Kilcullen's upper main street.

 

Lawrence Collection photo, c. 1900, of Kilcullen's Upper Main Street, 

 

How to Calculate Arc Length of a Circle, Segment and Sector Area

Circumference, diameter and radius. © Eugene Brennan

 

What You'll Learn

In this tutorial you'll learn about:

• names for different parts of a circle
• degrees and radians and how to convert between them
• chords, arcs and secants
• sine and cosine
• how to work out the length of an arc and chord
• how to calculate the area of sectors and segments
• the equation of a circle in the Cartesian coordinate system

What Is a Circle?

"A locus is a curve or other figure formed by all the points satisfying a particular equation."

A circle is a single-sided shape, but it can also be described as a locus of points where each point is equidistant (the same distance) from the centre.

Angle Formed by Two Rays Emanating From the Center of a Circle

An angle is formed when two lines or rays that are joined together at their endpoints, diverge or spread apart. Angles range from 0 to 360 degrees. We often "borrow" letters from the Greek alphabet to use in math and science. So for instance, we use the Greek letter "p" which is π (pi) and pronounced "pie" to represent the ratio of the circumference of a circle to the diameter. We also use the Greek letter θ (theta) and pronounced "the - ta", for representing angles.

An angle is formed by two rays diverging from the centre of a circle. This angle ranges from 0 to 360 degrees. © Eugene Brennan

360 degrees in a full circle. © Eugene Brennan

Parts of a Circle

• A sector is a portion of a circular disk enclosed by two rays and an arc.
• A segment is a portion of a circular disk enclosed by an arc and a chord.
• A semi-circle is a special case of a segment, formed when the chord equals the length of the diameter.

Arc, sector, segment, rays and chord. © Eugene Brennan

What Is Pi (π) ?

Pi represented by the Greek letter π is the ratio of the circumference to the diameter of a circle. It's a non-rational number which means that it can't be expressed as a fraction in the form a/b where a and b are integers.

Pi is equal to 3.1416 rounded to 4 decimal places.

What's the Length of the Circumference of a Circle?

If the diameter of a circle is D and the radius is R.

Then the circumference C = πD

But D = 2R

So in terms of the radius R

C = πD = 2πR

What's the Area of a Circle?

The area of a circle is A = πR ²

But R = D/2

So the area in terms of the radius R is

Scroll to Continue

A = πR ² = π (D/2)² = πD ²/4

What Are Degrees and Radians?

Angles are measured in degrees, but sometimes to make the mathematics simpler and elegant it's better to use radians which is another way of denoting an angle. A radian is the angle subtended by an arc of length equal to the radius of the circle. ( "Subtended" means produced by joining two lines from the end points of the arc to the center).

An arc of length R where R is the radius of a circle, corresponds to an angle of 1 radian.

So if the circumference of a circle is 2πR i.e 2π times R, the angle for a full circle will be 2π times 1 radian or 2π radians.

And 360 degrees = 2π radians.

How to Convert From Degrees to Radians

1. 360 degrees = 2π radians
2. Dividing both sides by 360 gives
3. 1 degree = 2π /360 radians
4. Then multiply both sides by θ
5. θ degrees = (2π/360) x θ = θ(π/180) radians
6. So to convert from degrees to radians, multiply by π/180

A radian is the angle subtended by an arc of length equal to the radius of a circle. © Eugene Brennan

How to Convert From Radians to Degrees

1. 2π radians = 360 degrees
2. Divide both sides by 2π giving
3. 1 radian = 360 / (2π) degrees
4. Multiply both sides by θ, so for an angle θ radians
5. θ radians = 360/(2π) x θ = (180/π)θ degrees
6. So to convert radians to degrees, multiply by 180/π

How to Find the Length of an Arc

You can work out the length of an arc by calculating what fraction the angle is of the 360 degrees for a full circle.

1. A full 360 degree angle has an associated arc length equal to the circumference C
2. So 360 degrees corresponds to an arc length C = 2πR
3. Divide by 360 to find the arc length for one degree:
4. 1 degree corresponds to an arc length 2πR/360
5. To find the arc length for an angle θ, multiply the result above by θ:
6. 1 x θ = θ corresponds to an arc length (2πR/360) x θ

So arc length s for an angle θ is:

s = (2πR/360) x θ = πRθ /180

The derivation is much simpler for radians:

By definition, 1 radian corresponds to an arc length R

So if the angle is θ radians, multiplying by θ gives:

Arc length s = R x θ = Rθ

Arc length is Rθ when θ is in radians. © Eugene Brennan

What Are Sine and Cosine?

A right-angled triangle has one angle measuring 90 degrees. The side opposite this angle is known as the hypotenuse and it is the longest side. Sine and cosine are trigonometric functions of an angle and are the ratios of the lengths of the other two sides to the hypotenuse of a right-angled triangle.

In the diagram below, one of the angles is represented by the Greek letter θ.

The side a is known as the "opposite" side and side b is the "adjacent" side to the angle θ.

sine θ = length of opposite side / length of hypotenuse

cosine θ = length of adjacent side / length of hypotenuse

Sine and cosine apply to an angle, not necessarily an angle in a triangle, so it's possible to just have two lines meeting at a point and to evaluate sine or cos for that angle. However sine and cos are derived from the sides of an imaginary right angled triangle superimposed on the lines. In the second diagram below, you can imagine a right angled triangle superimposed on the purple triangle, from which the opposite and adjacent sides and hypotenuse can be determined.

Over the range 0 to 90 degrees, sine ranges from 0 to 1 and cos ranges from 1 to 0

Remember sine and cosine only depend on the angle, not the size of the triangle. So if the length a changes in the diagram below when the triangle changes in size, the hypotenuse c also changes in size, but the ratio of a to c remains constant.

Sine and cosine are sometimes abbreviated to sin and cos.

Sine and cosine of angles. © Eugene Brennan

How to Calculate the Area of a Sector of a Circle

The total area of a circle is πR ² corresponding to an angle of 2π radians for the full circle.

If the angle is θ, then this is θ/2π the fraction of the full angle for a circle.

So the area of the sector is this fraction multiplied by the total area of the circle

or

(θ/2π) x (πR ²) = θR ²/2 with θ in radians.

Area of a sector of a circle knowing the angle θ in radians. © Eugene Brennan

How to Calculate the Length of a Chord Produced by an Angle

The length of a chord can be calculated using the Cosine Rule.
For the triangle XYZ in the diagram below, the side opposite the angle θ is the chord with length c.

From the Cosine Rule:

c ² = R ² + R ² -2RRcos θ

Simplifying:

c ² = R ² + R ² -2R ²cos θor c ² = 2R ² (1 - cos θ)

But from the half-angle formula (1- cos θ)/2 = sin ² (θ/2) or (1- cos θ) = 2sin ² (θ/2)

Substituting gives:

c² = 2R ² (1 - cos θ) = 2R ²2sin ² (θ/2) = 4R ²sin ² (θ/2)

Taking square roots of both sides gives:

c = 2Rsin(θ/2) with θ in radians.

A simpler derivation arrived at by splitting the triangle XYZ into 2 equal triangles and using the sine relationship between the opposite and hypotenuse, is shown in the calculation of segment area below.

The length of a chord. © Eugene Brennan

How to Calculate the Area of a Segment of a Circle

To calculate the area of a segment bounded by a chord and arc subtended by an angle θ , first work out the area of the triangle, then subtract this from the area of the sector, giving the area of the segment. (see diagrams below)

The triangle with angle θ can be bisected giving two right angled triangles with angles θ/2.

sin(θ/2) = a/R

So a = Rsin(θ/2) (cord length c = 2a = 2Rsin(θ/2)

cos(θ/2) = b/R

So b = Rcos(θ/2)

The area of the triangle XYZ is half the base by the perpendicular height so if the base is the chord XY, half the base is a and the perpendicular height is b. So the area is:

ab

Substituting for a and b gives:

Rsin(θ/2)Rcos(θ/2)

= R ²sin(θ/2)cos(θ/2)

But the double angle formula states that sin(2θ) = 2sin(θ)cos(θ)

Substituting gives:

Area of the triangle XYZ = R ²sin(θ/2)cos(θ/2) = R ² ((1/2)sin θ) = (1/2)R ²sin θ

Also, the area of the sector is:

R ²(θ/2)

And the area of the segment is the difference between the area of the sector and the triangle, so subtracting gives:

Area of segment = R ²(θ/2) - (1/2)R ²sin θ

= (R ²/2)( θ - sin θ ) with θ in radians.

To calculate the area of the segment, first calculate the area of the triangle XYZ and then subtract it from the sector.. © Eugene Brennan

Area of a segment of a circle knowing the angle. © Eugene Brennan

Equation of a Circle in Standard Form

If the centre of a circle is located at the origin, we can take any point on the circumference and superimpose a right angled triangle with the hypotenuse joining this point to the centre.
Then from Pythagoras's theorem, the square on the hypotenuse equals the sum of the squares on the other two sides. If the radius of a circle is r then this is the hypotenuse of the right angled triangle so we can write the equation as:

x ² + y ² = r ²

This is the equation of a circle in standard form in Cartesian coordinates.

If the circle is centred at the point (a,b), the equation of the circle is:

(x - a)² + (y - b)² = r ²

The equation of a circle with a centre at the origin is r² = x² + y². © Eugene Brennan

Equation of a Circle in Parametric Form

Another way of representing the coordinates of a circle is in parametric form. This expresses the values for the x and y coordinates in terms of a parameter. The parameter is chosen as the angle between the x-axis and the line joining the point (x,y) to the origin. If this angle is θ, then:

x = cos θ

y = sin θ

for 0 < θ < 360°

Summary of Equations for a Circle

Table 1. Circle formulas. θ is in radians.

Quantity	Equation
Circumference	πD
Area	πR²
Arc Length	Rθ
Chord Length	2Rsin(θ/2)
Sector Area	R²θ/2
Segment Area	(R²/2) (θ - sin(θ))
Perpendicular distance from circle centre to chord	Rcos(θ/2)
Angle subtended by arc	arc length / (Rθ)
Angle subtended by chord	2arcsin(chord length / (2R))

Example

Here's a practical example of using trigonometry with arcs and chords. A curved wall is built in front of a building. The wall is a section of a circle. It's necessary to work out the distance from points on the curve to the wall of the building (distance "B"), knowing the radius of curvature R, chord length L, distance from chord to wall S and distance from centre line to point on curve A. See if you can determine how the equations were derived. Hint: Use Pythagoras's Theorem.

© Eugene Brennan

References

Harris, J., & Stöcker Horst. (1998). Handbook of Mathematical Formulas and Computational Science. Springer.

This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.

© 2018 Eugene Brennan

 

Q&A From Readers 

George Dimitriadis from Templestowe on May 18, 2018:

Hi.

A good introduction to the basics of circle properties.

Diagrams are clear and informative.

Just a couple of points.

You have So C = πD = πR/2, which should be C = πD = 2πR

and A = πR^2 = π (D/2)2 = πD^2/2

should be A = πR^2 = π (D/2)2 = πD^2/4

Eugene Brennan (author) from Ireland on May 18, 2018:

Thanks George, I should have proof read before publishing, instead of beta testing on the readers !!

Larry Rankin from Oklahoma on May 19, 2018:

Very educational.

Troy Sartain on March 19, 2019:

How about a similar article for ellipses? Just a thought. Obviously, another level of complexity, even if not rotated.

Eugene Brennan (author) from Ireland on March 19, 2019:

Thanks Troy, I'll keep it in mind. Parabolas will probably come first though.

Mazin G A on April 01, 2019:

Hi,

How can I calculate the angle at the center of an arc knowing radius and center, start, and end points? I know how to do that if I have the length of the arc, but in my case I don't have it.

Eugene Brennan (author) from Ireland on April 05, 2019:

If you mean you know the coordinates of the start and end points of the chord, you can work out the length of the chord using Pythagoras's theorem. Then use the equation for length of a chord (2Rsin(θ/2) to find θ.

Lakshay on September 19, 2019:

Good efforts

darrell on April 06, 2020:

how do i calculate the length of a segment of a circle

Eugene Brennan (author) from Ireland on April 07, 2020:

If you mean the chord length, it's 2Rsin(θ/2).

See the derivation above.

Austen SMITH on April 28, 2020:

Hi I have a simple but frustrating problem- I want to build a regular curved wall a set distance from a straight wall - the centre of the circle /arc of the wall falls within the building.

I need to work out distance from the straight wall to measure, at regular intervals, to create the perfect curve starting and ending on the chord (2nd) forming the distance from the straight wall.(1st chord)

Hope you can help.

Eugene Brennan (author) from Ireland on April 30, 2020:

Hi Austen, I spent hours trying to figure this out using angles, but it turned out that since the chord length is known between two ends of the curved wall (is this correct?), it can easily be worked out using Pythagoras's Theorem. I've drawn it up as an example at the bottom of the article, hope it helps.

Suggestion, you could put the values into a spreadsheet to do the calculations.

Austen on May 31, 2020:

Many thanks for solving the curved wall issue.

I’ racked my brains back 45 plus used the existing formulae on your website to crack it- but as always when someone who “knows” tells you “how” - it becomes so clear you wonder how you couldn't see it before.

Thanks for relighting the knowledge thirst.

A

Eugene Brennan (author) from Ireland on May 31, 2020:

Thanks Austen.

I worked out D in the diagram above knowing R and L/2. In reality that's probably not necessary because you may already know the distance from the centre of the arc to the inside of the wall. Adding this to S gives you D.