The best known packings of equilateral triangles into an equilateral triangle are illustrated above for the first few cases (Friedman).

The best known packings of equilateral triangles into a circle are illustrated above for the first few cases (Friedman).

The best known packings of equilateral triangles into a square are illustrated above for the first few cases (Friedman).

Stewart (1998, 1999) considered the problem of finding the largest convex area that can be nontrivially tiled with equilateral triangles whose sides are integers for a given number of triangles and which have no overall common divisor. There is no upper limit if an arbitrary number of triangles are used. The following table gives the best known packings for small numbers of triangles.

Friedman, E. "Circles in Triangles." http://www.stetson.edu/~efriedma/cirintri/.Friedman, E. "Squares in Triangles." http://www.stetson.edu/~efriedma/squintri/.Friedman,
E. "Triangles in Triangles." http://www.stetson.edu/~efriedma/triintri/.Graham,
R. L. and Lubachevsky, B. D. "Dense Packings of Equal Disks in an
Equilateral Triangle: From 22 to 34 and Beyond." Electronic J. Combinatorics2,
No. 1, F1, 1-39, 1995. http://www.combinatorics.org/Volume_2/Abstracts/v2i1f1.html.Stewart,
I. "Squaring the Square." Sci. Amer.277, 94-96, July 1997.Stewart,
I. "Mathematical Recreations: Monks, Blobs and Common Knowledge. Feedback."
Sci. Amer.279, 97, Aug. 1998.Stewart, I. "Mathematical
Recreations: The Synchronicity of Firefly Flashing. Feedback." Sci. Amer.280,
106, Mar. 1999.