Fundamental Group Two Tori

Question

Leg $H$ denote the topological space obtained by gluing two tori together along a copy of $D^{2}$ that embeds in each. Compute a presentation for $\pi_{1}$? I tried finding the generators of each and then gluing them together

Answer 1

The magic words are "van Kampen theorem." See the wikipedia article for Seifert-van Kampen theorem.

Answer 2

Hint: find a representation of $H$ as a cellular complex.