As we faced many problem to work in social media about size of profile picture to use. Here is the dimension that analysis by renowned email marketing tools provider constant contact. Also I add some value to enrich the list –
Facebook
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Cover photo 815×315
Profile photo 180×180 (display area 160×160)
Fan page profile photo 200×200 (display area 176×176)
Tab 111×74
Link Image 1200×627
Image 1200×1200
Highlighted/milestone image 1200×717
Twitter
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Header 1500×500
Profile photo 400×400
Image display size 880×440 (recommended)
Google+
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Profile photo 250×250
Cover photo 2120×1192
Shared image 800×600
LinkedIn
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Profile photo 200×200
Cover photo 646×220
Pinterest
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Profile photo 600×600
Pins 600xINFINITE
Board thumbnail 222×150
Instagram
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Profile photo 161×161
Image viewed on desktop in lightbox as 612×612
Image feed 510×510
YouTube
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Profile photo 800×800
Channel art 2560×1224
Custom video thumbnail 1280×720
* all dimensions listed in pixels
Hope it may help you to engineering the web for your client.