I really enjoyed my trip up to Entoto Hills/Mountains. I took about a half-day (a guided tour with a car) to go up to Entoto hill during a stay in Addis. Don't be fooled by the word "hills" into thinking of something small, these hills/mountains are quite large, takes about 45 minutes to drive to the top.
The countryside you will see is stunning and worth the time and effort. The area is in a thick, cool, well watered, conifer forest. You will have great views of Addis Ababa. Called the "Lung of Addis Ababa", you will not be disappointed in the fresh air! You will also see the hundreds of peasants who make less than a $1 per day transporting wood down into the the city.
The Menelek II Museum and Palace is worth a look. If only Haile Selassie had stayed in the humble set of buildings (instead of squandering the country's wealth on huge ornate palaces in the city while his people starved), perhaps things would have turned out differently for Ethiopia. (There is a small circular church next to the palace that has great atmosphere.)
The other highlight of the area is the quite large, multilevel and altogether spectacular octagonal "Entoto Maryam Church" which hosted Menelik's coronation. Don't miss it. There is a small rock-hewn church here you should peek into (though it is in terrible condition) as well as a small museum with artifacts of Menelik's life.
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