Etymology
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beetle (n.2)
"heavy wooden mallet used to drive wedges, pack earth, etc.," Old English bietl "mallet, hammer," from Proto-Germanic *bautilo-z, from *bautan "to beat," from PIE root *bhau- "to strike."
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beetle (n.1)
insect of the order Coleoptera, Old English bitela "beetle," apparently originally meaning "little biter, biting insect," from bitel "biting," from Proto-Germanic *bitan, from PIE root *bheid- "to split," with derivatives in Germanic referring to biting.

By normal evolution it would be *bittle, but it seems to have been influenced by beetle (n.2). Sometimes applied to soft insects, as black beetle, an old name for the cockroach. As a nickname for the original Volkswagen car, 1946, translating German Käfer.
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beetle (v.)
"project, overhang," apparently a Shakespearean back-formation (in "Hamlet," 1602) from bitelbrouwed "grim-browed, sullen" (mid-14c.), from bitel "sharp-edged, sharp" (c. 1200), probably a compound from Old English *bitol "biting, sharp" (related to bite (v.)), + brow, which in Middle English meant "eyebrow," not "forehead." Meaning "to overhang dangerously" (of cliffs, etc.) is from c. 1600. Related: Beetled; beetling.
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sledgehammer (n.)
late 15c., from sledge (n.1) + hammer (n.). As a verb, from 1834. Old English had slegebytel "hammer," from beetle (n.2).
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battledore (n.)
mid-15c., "bat-like implement used in washing clothes," of unknown origin, perhaps from Old Provençal batedor, Spanish batidor "beater, bat," from batir "to beat;" perhaps blended with Middle English betel "hammer, mallet" (see beetle (n.2)). As a type of racket used in a game, from 1590s, from similarity of shape.
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*bhau- 
Proto-Indo-European root meaning "to strike."

It forms all or part of: abut; baste (v.3) "beat with a stick, thrash;" battledore; beat; beetle (n.2) "heavy wooden mallet;" botony; boutonniere; butt (n.1) "thick end;" butt (v.) "strike with the head;" buttocks; button; buttress; confute; halibut; rebut; refute; sackbut; turbot.

It is the hypothetical source of/evidence for its existence is provided by: Latin *futare "to beat" (in compounds); Old English beadu "battle," beatan "to beat," bytl "hammer, mallet."
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*bheid- 
Proto-Indo-European root meaning "to split," with derivatives in Germanic "referring to biting (hence also to eating and to hunting) and woodworking" [Watkins].

It forms all or part of: abet; bait (n.) "food used to attract prey;" bait (v.) "to torment, persecute;" bateau; beetle (n.1) "type of insect; bit (n.1) "small piece;" bite; bitter; bitter end; boat; boatswain; -fid; fissile; fission; fissure; giblets; pita; pizza; vent (n.).

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit bhinadmi "I cleave," Latin Latin findere "to split, cleave, separate, divide," Old High German bizzan "to bite," Old English bita "a piece bitten off, morsel," Old Norse beita "to hunt with dogs," beita "pasture, food."
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weevil (n.)

Old English wifel "small beetle," from Proto-Germanic *webilaz (source also of Old Saxon wibil, Old High German wibil, German Wiebel "beetle, chafer," Old Norse tordyfill "dung beetle"), cognate with Lithuanian vabalas "beetle," from PIE root *(h)uebh- "to weave," also "to move quickly" (see weave (v.)). The sense gradually narrowed by 15c. to a particular kind of beetle that, in larval or adult stages, bores into plants, often destroying them.

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cockchafer (n.)

popular name of a common European beetle, the May-beetle, 1690s, from cock (n.1), in reference to its size, + chafer "beetle."

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