was a British operation to get desperately needed supplies to the island of Malta in August 1942, during the Second World War. Malta was the base from which surface ships, submarines and aircraft attacked Axis convoys carrying essential supplies to the Italian and German armies in North Africa. In 1941–42, Malta was effectively under siege, blockaded by Axis air and naval forces. To sustain Malta, the United Kingdom had to get convoys through at all costs. Despite serious losses, just enough supplies were delivered for Malta to survive, although it ceased to be an effective offensive base for much of 1942. The most crucial supply was fuel delivered by the Ohio
SS , an American-built tanker with a British crew. The operation started on 9 August 1942, when the convoy sailed through the Strait of Gibraltar.
The convoy is also known as the "Battle of Mid-August" in Italy and as the Konvoj ta' Santa Marija
in Malta. The arrival of the last ships of the convoy on 15 August 1942, coincided with the Feast of the Assumption (Santa Marija
) and the name Santa Marija Convoy
or Sta Marija Convoy
is still used. That day's public holiday and celebrations, in part, celebrate the arrival of the convoy. The attempt to run fifty ships past bombers, E-boats, minefields, and submarines has gone down in military history as one of the most important British strategic victories of the Second World War. However, it was at a cost of more than 400 lives, with only five of the original fourteen merchant ships reaching the Grand Harbour.
In 1942, the British Empire was waging a land war against Italian forces in North Africa and their allies, Rommel's Afrika Korps. Malta was a critical component to this campaign, as the island could be used as a base to interdict efforts to resupply Axis forces operating in North Africa. During this stage of the war, Malta was critically short on munitions, food, and fuel for both military operations and civil use. Attempts to run the blockade and resupply Malta proved to be costly and often ended in failure; the two-pronged effort to resupply Malta during June 1942—known as Operations Harpoon
(from Gibraltar) and Vigorous
(from Alexandria, Egypt)—were unsuccessful, only two merchantmen from Harpoon
and none from Vigorous
reached Malta, while many others (including the only tanker included in Harpoon
) were sunk, and the escorting warships also suffered heavy casualties. Military planners knew Malta would be forced to surrender if fuel, food, and ammunition did not get through before the end of August. The local air commander had warned the planners that there remained only a few weeks' supply of aviation fuel. The Admiralty drew up plans for another convoy to sail at the earliest practical date, mid-August.
The Admiralty started to plan Operation Pedestal
in the early weeks of July 1942, and it was soon recognized that this was to be the main effort to relieve Malta, and would employ the largest escort force yet deployed for a single convoy, including two battleships and three aircraft carriers. At the same time, there would be several minor but important operations carried out under the cover of Pedestal
: Operation Berserk
would be an aircraft carrier exercise to improve coordination between the various carriers in the convoy, Operation Bellows
would be a reinforcement of Malta's air defences by Spitfires flown from an aircraft carrier, and Operation Ascendant
would be an effort to return the two surviving merchantmen from the Harpoon
convoy from Malta to Gibraltar. The main participants during these discussions were Rear Admiral A. L. St. G. Lyster, CB, CVO, DSO
, Rear-Admiral H. M. Burrough, CB, DSO
, Vice-Admiral E. N. Syfret, CB
and the Naval Staff.
The supplies were to be carried by fourteen merchant ships, the most important being Ohio
, the only large, fast tanker available, an American-built ship under the British flag and with a British crew.[a] As partial insurance against Ohio
s loss, the others would carry some fuel supplies in drums. The convoy was to be protected by two large forces of warships, one as distant cover (Force Z), and the other as close support all the way to Malta (Force X). Between the two forces there were two battleships, three aircraft carriers, seven cruisers and thirty-two destroyers. Once they reached the Sicilian channel, Force Z
(the battleships, the aircraft carriers, and three cruisers) was to return to Gibraltar, leaving the convoy to continue to Malta escorted by the remaining four cruisers and destroyer flotilla of Force X
The Italian Regia Marina was hampered by a lack of fuel oil, which compelled it to keep its largest vessels in port. When the British convoy was detected, Axis commanders decided to attack with German and Italian aircraft based in Sardinia and send ten submarines, E-Boats and MAS motor torpedo boats into the Sicilian Channel. An Italian cruiser division was to deliver the final attack, which required oil to be transferred from inactive battleships to the cruisers.
The overall operational commander, Vice-Admiral E. N. Syfret, transferred to Nelson
HMS on 27 July when Nelson
returned to Scapa Flow from Freetown, West Africa. Syfret convened a conference on 29 July, for Flag and Commanding Officers of the naval forces for Pedestal
currently assembled at Scapa, to consider the orders for the operation. On 31 July, Nelson
and destroyers sailed from Scapa to rendezvous with Eagle
from Gibraltar and Indomitable
HMS and Phoebe
, from Freetown, for Operation Berserk
successfully exercised fighter direction and co-operation between the three carriers, in preparation for the impending convoy.
The convoy, named with a bogus "WS"[b] prefix, escorted by Nigeria
HMS , Kenya
and destroyers sailed from the Clyde overnight on 2 August and joined the other escorts during the following morning. Just prior to sailing, but after the "normal" convoy conference, Rear-Admiral Burrough met with the Convoy Commodore, A.G. Venables, and the Masters of the individual merchant ships on board his flagship, HMS Nigeria
, and the whole plan was explained in detail. A similar meeting was held with radio operators of the merchantmen to explain fleet communications and procedures. Personal messages signed by the First Lord of the Admiralty wishing the Masters "God Speed
" and contained in envelopes marked "Not to be opened until 0800 hours August 10
" were handed to the ships' masters.
Shortly before the departure from Scapa, the Admiralty decided that Furious
HMS should carry out Operation Bellows
to reinforce Malta with Spitfires at the same time as Pedestal
. This made alterations to the Operation Orders necessary, which were then distributed to all concerned. Technical difficulties[c] delayed the carrier's departure but, with Manchester
HMS , she joined Nelson
and the convoy three days before the start of the operation. The passage of the convoy from the UK to the rendezvous with the aircraft carriers west of the Straits was successful, though there were many alarms over U-Boat contacts en route and a Coastal Command Sunderland flying boat was shot down by friendly fire. The convoy was exercised in anti-aircraft gunnery, in emergency turns and in changing from one cruising disposition to another, using both signal flags and short range W/T. The risk to security in breaking W/T silence was accepted and as a result of these exercises the convoy attained an efficiency in manoeuvring "comparable to that of a fleet unit."
The force's aircraft performed dummy air attacks in the afternoon of 8 August, followed by a fly past. These were done to exercise the radar reporting and fighter direction organisation and to give ships' gun crews an opportunity to recognise the markings of friendly aircraft. The resulting volume of radio traffic must have been very apparent to hostile listening stations, but this risk to security was worth the benefit gained from the rehearsals.[d]
The passage of the Straits and 10 August were uneventful. Fishing boats and one merchant vessel were passed at close quarters, but due to a moonless night and indifferent visibility, it was thought improbable that the force had been sighted from the shore. Reports received later, however, showed that the enemy was fully aware of the convoy's passage of the Straits.
The convoy completed refuelling by dawn on 11 August, from the tankers Dingledale
and Brown Ranger
. Previous Malta convoys had refuelled at Malta but now the island had no oil to spare. The refuelling of three cruisers and twenty-six destroyers at sea, under enemy observation and in U-boat infested waters, was an anxious one; failure could have seriously upset the whole operation.
The main coup for the Axis during the day happened early in the morning. The aircraft carrier Eagle
HMS was hit by four torpedoes from U-73
[e] and sank seventy nautical miles south of Cape Salinas. Most of the crew survived (160 lost out of 927), rescued by her escorts. The sinking of Eagle
deprived the force of a quarter of its fighter strength.
was being torpedoed, Furious
successfully finished Operation Bellows
, flying off thirty-seven much needed Spitfires to Malta. The flying distance between Furious
and Malta was 555 nmi (639 mi; 1,028 km) to 584 nmi (672 mi; 1,082 km). Her part of the mission complete, Furious
returned to Gibraltar with her escorts. A destroyer escorting Furious
HMS , rammed and sank the Italian submarine Dagabur
s bow was seriously damaged, but she reached Gibraltar for repairs.
A large number of sightings of torpedoes, air and U-Boat attacks were reported throughout the whole convoy. The Axis made every form of air attack, including minelaying ahead of the fleet. Up to 19:00 on 11 August, there were four heavy air attacks on the fleet. Despite the weight of these attacks, the only casualties were one merchant ship with her speed reduced by a near miss, a destroyer torpedoed but afloat (though later scuttled) and Indomitable
HMS put out of flying action but capable of steaming at 28 kn (32 mph; 52 km/h). At 2000, an Italian air attack (using S.84, CR.42, C.202, and Re.2001 fighters and bombers) against the aircraft carriers caused minor damage to the flight deck of Victorious
As the convoy moved eastwards towards the enemy's bases in Sardinia and Sicily, it was expected that the U-Boat and air threat would increase and a U-Boat concentration was expected near the Galite Islands. Anti-submarine measures were stepped up to meet this threat. Later in the day, Ithuriel
HMS rammed and sank the Italian submarine Cobalto
.[f] This tactic was discouraged by the Admiralty, due to the damage that often resulted. In Ithuriel
s case, she badly damaged herself, put her Asdic gear out of action, and was missing from the escort screen during the next air attack. She also lost two of her crew who boarded and attempted to keep the sinking submarine afloat. Only two Italian seamen were lost, the rest becoming prisoners of war. The carriers launched Fairey Fulmars and Hawker Sea Hurricanes as air cover. The first air attack by nineteen Ju 88s was met by both anti-aircraft fire and the fighters. Four bombers were claimed for the loss of a fighter and two more were shot down by the escorting ships.
The Italian cruiser division, consisting of three heavy cruisers (Gorizia
, and Trieste
), three light cruisers (Eugenio di Savoia
, Raimondo Montecuccoli
, and Muzio Attendolo
) and seventeen destroyers sailed to meet the British convoy.
Throughout the day, the force was under continual observation by Italian aircraft and later in the day there were three more heavy air attacks. Force X, after parting company, was attacked at dusk by bombers and torpedo-bombers. During daylight, British fighters were active; they provided early warning of raids and destroyed enemy aircraft, although frequently outnumbered. The convoy's anti-aircraft guns also proved successful as deterrents.
Submarines, however, remained a potent threat. The Italian submarine Brin
was driven off by destroyers. A Sunderland flying boat attacked another submarine, Giada
, which was waiting for the convoy off Algiers, damaging it and a subsequent air attack by another flying boat caused more damage. Before heading for shelter to the Spanish port of Valencia, where she remained until the 14th, Giada
shot down the Sunderland with her own guns. To prevent any further submarine attack the destroyers dropped depth charges every 10 minutes, between 14:00 and 19:00.
At noon, another air attack developed, by Italian and German bombers and Italian fighters. The Axis planes did not attack in continuous waves as planned, and the convoy enjoyed brief respites. Even so, the attack wrecked the flight deck of Indomitable
, leaving Victorious
as the only working carrier. Indomitable
s aircraft had to be landed on Victorious
and several fighters had to be ditched overboard to make space for further landings.
It had been intended that Force Z should return to Gibraltar when the Skerki Channel was reached at 1915. Indomitable
was on fire forward and aft, so Syfret brought the manoeuvre forward by twenty minutes, and at 18:55, Force Z was ordered to return to Gibraltar, leaving Force X
to continue to Malta. This separation went unnoticed by the Axis and was not discovered by them until about 20:30. In view of the weight of the Axis air attack between 18:30 and 18:50, it seemed improbable that a further significant air attack on Force X could be possible before dark, and having reached the Skerki Bank, it was hoped that the submarine threat would subside.
The main threat to the convoy appeared to be E Boat attacks during the night and by aircraft the following morning. Thus, the successful submarine attack by the Italian submarine Axum
at 20:00, when Nigeria
HMS , Cairo
HMS and Ohio
were torpedoed was unexpected and its effect far reaching. The timing was critical, for the convoy was at that moment changing its setup from four to two columns. For this manoeuvre the cruisers were needed as column leaders. The torpedoing of HMS Nigeria
, the diversion of HMS Ashanti
, and the detachment of four Hunt-class destroyers to stand by the damaged cruisers, temporarily deprived Force X of its Commander, deprived two columns of their leaders, lost the convoy nearly half its escort, and the entire force its two Fighter Direction ships. On hearing that Nigeria
had been torpedoed, Syfret ordered Charybdis
HMS , Eskimo
to reinforce Force X. From about 20:35–21:00, the convoy was subjected to a severe dusk air attack by dive-bombers and torpedo-bombers.
laid a smokescreen to cover the light western horizon, but this did not prevent the attack from being effective. The merchantman Empire Hope
was bombed and sunk. A separate torpedo-bomber attack was thought to have sunk Deucalion
at 21:30 near the Cani Rocks when under escort of Bramham
HMS . The commanding officer, Kenya
HMS described the state of the convoy after these attacks as 'chaotic', and in fact there are different versions of what really happened during these confusing hours. However, though the convoy was in a confused state, all of the ships in the convoy were steaming in their correct direction. Most of the convoy got safely as far as Kelibia[g], though some were damaged, by early morning. The destroyer Foresight
HMS was sunk in the same attack. Roughly around 21:00, the Italian submarine Alagi
reported that it had sunk the merchant ship Empire Hope
and damaged the cruiser Kenya
; in fact two freighters sank at this moment, while Kenya
turned to avoid a torpedo. Another submarine, Bronzo
, stated that it had sunk the merchant ship Deucalion
. It is possible that the unknown second freighter sunk by the Italian submarine Alagi
or the damage to Kenya
was not directed to either ship but to the freighter Brisbane Star
MV , that supposedly had its bows torn off during this encounter.
and the other damaged ships turned back to Gibraltar with HMS Wilton
HMS as escorts. Other submarines, Emo
, and Dandolo
among them, were driven off by depth charges.
Passing through minefields between Africa and Sicily around midnight, the convoy met eight Italian and seven German torpedo-boats which made fifteen attacks. The long line of merchant ships and the reduced number of escort ships provided easy opportunities for attacks by the torpedo-boats which were lying in wait off Kelibia. Here three of the merchant ships which failed to reach Malta were torpedoed. Of these, Wairangi
was hit in the engine room and was permanently disabled, while the American Almeria Lykes
was hit at the joint of No. 1 bulkhead hold and could not continue steaming to Malta. At 01:00, the cruiser Manchester
HMS was torpedoed by two Italian E-Boats (MS-16
), leaving her dead in the water and listing. She restored power and some of her crew (156 men) were transferred to HMS Pathfinder
in the ship's boats and Carley floats, but she was later scuttled off Cape Bon by order of her Commanding Officer. She was the largest ship sunk by motor torpedo boats during World War II.[h] Several hundred of her complement landed in Vichy-controlled Tunisia and were interned. According to most sources, a dozen of her crew died from the torpedo explosions, while at least another one-hundred and fifty went missing at sea during her evacuation. The merchantman Santa Elisa
was hit by torpedo, the entire ship catching fire and being abandoned by the crew. Glenorchy
was torpedoed and on fire although most of her crew survived her eventual sinking; another merchantman, Rochester Castle
, was torpedoed but kept going. Fighters from Malta were fired upon by the convoy in the confusion because communications between the convoy and the RAF were still out.
Twelve Junkers 88s attacked, hitting Waimarama
. The aviation fuel on deck burst into flame and the ship exploded and sank, with 80 of the 107 crew killed. The destroyer Ledbury
HMS passed through the flames to save what few survivors there were. The wreckage of Waimarama
started fires on the merchantman Melbourne Star
, and several of her crew abandoned ship prematurely, some of whom were later rescued by Ledbury
. Soon afterwards, sixty Stukas
attacked, this time concentrating on the damaged Ohio
. Near-misses buckled the tanker's hull plates and the forward tank filled with water. A downed Ju 88 crashed onto the deck of Ohio
. Another plane, an Italian dive bomber, bounced off the water, and crashed onto Ohio
. Although the tanker had avoided mines, torpedoes and circling torpedoes, she was eventually straddled by two bombs and with her boilers ruptured, was left dead in the water at 10:50.
was disabled by three near-miss bombs, and the engine room was flooded. The high octane fuel caught fire and the merchantman was abandoned. Twelve Italian torpedo bombers attacked, and the Port Chalmers
caught an aerial torpedo in its paravane.[i] A bomb nearby set Kenya
s forward engine room on fire, but the fire was quickly put out. Fighters from Malta provided some air cover to the blitzed convoy. Rochester Castle
, Port Chalmers
and Melbourne Star
steamed on to meet escort from Malta, reaching Grand Harbour in Valletta at 18:00. Another aerial attack on Ohio
tried to tow Ohio
, but the tanker was listing to its side and snapped the tow line. During another attack, Ohio
eventually broke its back when a bomb hit the same area as a previous torpedo hit. The crew, led by Master D. W. Mason, abandoned the ship, while Italian torpedo bombers were about to dive in for yet another attack.
Field Marshal Albert Kesselring, commander of the German Air Command based in Sicily, denied air coverage to the Italian cruiser division, having little regard for the fighting capability of the Regia Marina
, and preferred to use his aircraft for direct attacks on the British convoy. Without air cover and given the closeness of the air base at Malta, the Supermarina
High Command) withdrew its cruisers to Messina. They passed through the area patrolled by the British submarines Safari
HMS and Unbroken
and were attacked. Unbroken
torpedoed the Bolzano
, which was hit in her oil tank and ran aground; Attendolo
lost 60 ft (18 m) of her bow. Neither ship returned to action during the war.
arrived in Malta, its bow torn off by a torpedo, but it successfully discharged its supplies in the harbour. Ohio
was surrounded by a flotilla of ships, aiming to nurse the stricken tanker to Grand Harbour. Several American volunteers, themselves survivors from the sunken American freighter Santa Eliza
, manned anti-aircraft guns on Ohio
during the tow. The weight of the tanker kept breaking the tow lines, while constant air attacks were made by twenty bombers. An attack destroyed the rudder, making a hole in her stern. The decks of the ship were now awash. Finally, the tanker was successfully towed while it was supported by the two destroyers Ledbury
on each side, with a minesweeper Rye
HMS to act as a stabilizer at the stern. The remainder of the convoy was either sunk or falling back on Gibraltar. Several further air attacks disrupted the towing formation, until it was re-established with Bramham
s port side for the remainder of the journey.
was towed into Grand Harbour by the two destroyers and a set of tugs at 09:30 to cheering crowds and a band playing Rule Britannia.
In his book, First Light, Geoffrey Wellum, who flew a Spitfire from HMS Furious
as part of Operation Bellows
, reports that the crowd fell silent as the ships entered harbour as a mark of respect for the crew who lost their lives aboard. The tanker discharged its cargo into two tankers and settled on the bottom just as the last of the fuel it transported left her holds.[j]
The arrival of the four merchant ships, and the survival of the tanker Ohio
ensured the arrival of enough materials to maintain the island, but it did not mean its siege was at an end. The ultimate result of Operation Pedestal was that it ensured that Malta stayed in the war. For the high price of nine merchantmen sunk, one aircraft carrier (Eagle
), two cruisers (Manchester
), and a destroyer (Foresight
) sunk, the Royal Navy and the Merchant Navy had saved Malta, as roughly 32,000 short tons (29,000 t) of general cargo had reached the Grand Harbour, together with petrol, oil fuel, kerosene and diesel fuel, enough to give the island about ten weeks more life beyond the existing stocks of only a few weeks. Royal Navy gunners and Fleet Air Arm fighters shot down 42 of the approximately 330 attacking Axis aircraft.
Operation Pedestal was a tactical disaster, of a magnitude comparable to the German attack on Convoy PQ-17. The defeat, however, was turned into a strategic victory in that it served as a great uplift to the besieged island's morale and it delivered thousands of tons of needed food stores and eliminated the possibility of surrender due to famine. Indeed, for several months after this convoy, Malta was still dependent on essential stores and stocks being delivered by fast minelayers, like Manxman
HMS , and of mine-laying submarines. From the moment the shield of Spitfires patrolled over the unloading battered ships, it became obvious that ships could now arrive and be protected, meaning that more ships would come in due course, thus sustaining the will to endure.
German reports on 17 August stated that all the tankers in the recent Mediterranean convoy were sunk and not one of the transports reached their assumed destination in Egypt. A revived Malta led to a shift in the North African balance immediately preceding the Second Battle of El Alamein. In August 1942, with Malta still besieged, 35% of Axis convoys to North Africa did not get through. In September, with Malta resupplied, Allied forces sank 100,000 long tons (100,000 t) of Axis shipping, including 24,000 tons of fuel destined for Rommel, leaving him desperately short of supplies during his assault at El Alamein on 23 October 1942. Hence, it was no longer a question of "How many days to Cairo?" for the Axis armies, but of whether Rommel could hold the Allied attack back when it would materialise. Submarines and torpedo-carrying Bristol Beauforts escorted by their variants the Bristol Beaufighters, regularly attacked Axis supply ships known to the Allies through Ultra intercepts received from Bletchley Park. This interdiction of sea-lanes from Malta, significantly contributed to the Axis’ worsening state of supply and lead to their eventual inability to compete with the British build-up for an offensive towards the end of the year.[k]
For the Axis powers in general, and for the Italians in particular, the inability of the fleet to fully display its power and especially the inability of Axis air forces to provide for cover clearly demonstrated that the tide in this campaign had turned. Ultimately, Malta was still alive while any hope to maintain North Africa was quickly fading away, with the now looming possibility of having the Allies opening up a Third Front on the Italian mainland.
Operation Pedestal was the subject of a 1953 black and white British film, Malta Story, which interspersed archive footage of the SS Ohio
with scripted studio scenes.
In recognition of their fortitude during the siege and air attacks during all of the Mediterranean campaign, Malta was awarded the George Cross in the months immediately preceding this operation. Vice-Admiral Syfret was knighted (KCB) for his "bravery and dauntless resolution in fighting an important Convoy through to Malta in the face of relentless attacks by day and night from enemy submarines, aircraft, and surface forces."
The Master of the tanker Ohio
, Dudley William Mason was awarded the George Cross for showing "skill and courage of the highest order and it was due to his determination that, in spite of the most persistent enemy opposition, the vessel, with her valuable cargo, eventually reached Malta and was safely berthed." Several other officers, crew members and commanders of both the Royal and Merchant Navies, including the commander of Ledbury
HMS , Roger Hill, received military awards ranging from the Distinguished Service Order and Conspicuous Gallantry Medal down to Mentions in Despatches for the bravery and intrepidity shown in ferrying the merchantmen to Malta.
The Merchant Marine Distinguished Service Medal was awarded to Frederick August Larsen, Jr., Junior Third Officer and to Francis A. Dales, Cadet-Midshipman, U.S. Merchant Marine Academy, for "Heroism beyond the call of duty."
Allied ships that took part in this operation included:
Axis ships that took part in this operation included:
Media related to Operation Pedestal at Wikimedia Commons
At the time, the Ohio
was also the largest tanker ship afloat.
W.S convoys were normally those from U.K. to Suez via the Cape of Good Hope.
This was connected with the aircraft's propellers and the aircraft carrier's flying deck which was not level, but sloped upwards to a point amidships.
At 13:00, when HMS Indomitable
joined the force, it was believed to be the first time that five British aircraft carriers had operated together at sea.
Captained by Kapitänleutnant
The rescued crew members (3 officers and 38 crew) confirmed her destruction.
Some 20 mi (32 km) south of Cape Bon in Tunisia.
There is some disagreement about Manchester's
fatalities among the sources: The following websites mention 150 "lost":
Submerged floats meant to catch mines.
Water had been pumped into the tanker as the fuels were extracted to minimize the chance of a structural failure.
In December 1942 four convoys sailed into the island without loss, and during this same month some 200,000 tons of stores of all kinds were brought ashore.
In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid
in American English and as a trapezium
in English outside North America. The parallel sides are called the bases
of the trapezoid and the other two sides are called the legs
or the lateral sides (if they are not parallel; otherwise there are two pairs of bases). A scalene trapezoid
is a trapezoid with no sides of equal measure, in contrast to the special cases below. A trapezoid with vertices ABCD is denoted ABCD
There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some define a trapezoid as a quadrilateral having exactly
one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Others define a trapezoid as a quadrilateral with at least
one pair of parallel sides (the inclusive definition), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral ill-defined. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals.
The term trapezium
has been in use in English since 1570, from Late Latin trapezium
, from Greek τραπέζιον (trapézion
), literally "a little table", a diminutive of τράπεζα (trápeza
), "a table", itself from τετράς (tetrás
), "four" + πέζα (péza
), "a foot, an edge". The first recorded use of the Greek word translated trapezoid
, "table-like") was by Marinus Proclus (412 to 485 AD) in his Commentary on the first book of Euclid's Elements.
This article uses the term trapezoid
in the sense that is current in the United States and Canada. In all other languages using a word derived from the Greek for this figure, the form closest to trapezium
(e.g. French trapèze
, Italian trapezio
, Spanish trapecio
, German Trapez
, Russian трапеция
) is used.]
In an isosceles trapezoid, the legs (AD
in the figure above) have the same length, and the base angles have the same measure. In a right trapezoid
(also called right-angled trapezoid), two adjacent angles are right angles. A tangential trapezoid is a trapezoid that has an incircle.
Under the inclusive definition, all parallelograms (including rhombuses, rectangles and squares) are trapezoids.
Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:
Additionally, the following properties are equivalent, and each implies that opposite sides a
(also called the median or midline) of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases. Its length m
is equal to the average of the lengths of the bases a
of the trapezoid,
The midsegment of a trapezoid is one of the two bimedians (the other bimedian divides the trapezoid into equal areas).
(or altitude) is the perpendicular distance between the bases. In the case that the two bases have different lengths (a
), the height of a trapezoid h
can be determined by the length of its four sides using the formula
are the lengths of the legs. This formula also gives a way of determining when a trapezoid of consecutive sides a
, and d
exists. There is such a trapezoid with bases a
if and only if
The area K
of a trapezoid is given by
are the lengths of the parallel sides, and h
is the height (the perpendicular distance between these sides.) In 499 AD Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya
(section 2.8). This yields as a special case the well-known formula for the area of a triangle, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.
Therefore the area of a trapezoid is equal to the length of this midsegment multiplied by the height
From the formula for the height, it can be concluded that the area can be expressed in terms of the four sides as
When one of the parallel sides has shrunk to a point (say a
= 0), this formula reduces to Heron's formula for the area of a triangle.
Another equivalent formula for the area, which more closely resembles Heron's formula, is
is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula, but it differs from it, in that a trapezoid might not be cyclic (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral).
From Bretschneider's formula, it follows that
The line that joins the midpoints of the parallel sides, bisects the area.
The lengths of the diagonals are
are the bases, c
are the other two sides, and a
If the trapezoid is divided into four triangles by its diagonals AC
(as shown on the right), intersecting at O
, then the area of is equal to that of , and the product of the areas of and is equal to that of and . The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.
Let the trapezoid have vertices A
, and D
in sequence and have parallel sides AB
. Let E
be the intersection of the diagonals, and let F
be on side DA
be on side BC
such that FEG
is parallel to AB
. Then FG
is the harmonic mean of AB
The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.
The center of area (center of mass for a uniform lamina) lies along the line joining the midpoints of the parallel sides, at a perpendicular distance x
from the longer side b
If the angle bisectors to angles A
intersect at P
, and the angle bisectors to angles C
intersect at Q
The term trapezium is sometimes defined in the USA as a quadrilateral with no parallel sides, though this shape is more usually called an irregular quadrilateral. The term trapezoid was once defined as a quadrilateral without any parallel sides in Britain and elsewhere, but this does not reflect current usage. (The Oxford English Dictionary says "Often called by English writers in the 19th century".)
According to the Oxford English Dictionary
, the sense of a figure with no sides parallel is the meaning for which Proclus introduced the term "trapezoid". This is retained in the French trapézoïde
, German Trapezoid
, and in other languages. A trapezium in Proclus' sense is a quadrilateral having one pair of its opposite sides parallel. This was the specific sense in England in 17th and 18th centuries, and again the prevalent one in recent use. A trapezium as any quadrilateral more general than a parallelogram is the sense of the term in Euclid. The sense of a trapezium as an irregular quadrilateral having no sides parallel was sometimes used in England from c. 1800 to c. 1875, but is now obsolete. This sense is the one that is sometimes quoted in the US, but in practice quadrilateral is used rather than trapezium.
In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering towards the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually isosceles trapezoids.