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Is there a way to curve a rectangle outwards (give arc shape effect, see attached image)? Or do I have do it manually using the pen tool by adding anchors to the rectangle line? I have also tried to do it with arc Up and arc down horizontal (make with warp) tool. But its a complex process. Is there any simpler and better way to do it?

enter image description here

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You could use an "Inflate" warp on the rectangle...

enter image description here

Either using an effect (Effect → Warp → Inflate...), or using an envelope distortion (Object → Envelope Distort → Make With Warp... then "Inflate" from the Style dropdown).

You can "apply" the distortion by going to "Object → Expand/Expand Appearance" if needs be.

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You can also use the curvature tool to curve specific lines of an object.

If you want to do that for a rectangle, click on the rectangle, then click on a line, click the curvature tool and drag the line.

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