How do I find the volume of a hemisphere?
Volume is ⅔πr³. Keep every measurement in the same unit before multiplying — mixing centimetres and metres is the most common source of an answer that is out by a factor of a thousand or a million. Volume units are cubic, so converting afterwards means cubing the conversion factor: 1 m³ is 1,000,000 cm³, not 100.
How do I find the surface area of a hemisphere?
Surface area is 3πr² (curved 2πr² plus the flat circular base πr²). Work out each face separately and add them, rather than trying to apply a single remembered formula — it is slower but far more reliable, and it makes it obvious when a face should be excluded because it is not exposed.
Why is a hemisphere's surface area 3πr² and not half a sphere's?
Because cutting a sphere in half creates a new flat face that did not exist before. Half the sphere's curved surface is 2πr², and the exposed circular base adds πr², giving 3πr² in total. If you only want the dome — a roof, say, with no floor — use 2πr². Which figure you need depends entirely on whether the flat face is part of what you are covering.
Where does the ⅔ in the volume come from?
A full sphere is (4/3)πr³, so half is (2/3)πr³. Archimedes proved the more elegant relationship: a sphere inscribed in a cylinder occupies exactly two-thirds of it. He considered this his finest result and asked for the diagram to be carved on his tomb.
What is the centre of mass of a solid hemisphere?
Three-eighths of the radius above the flat face, not halfway. That low centre of mass is why a solid hemisphere rights itself when tipped, and the principle behind self-righting lifeboats and weeble toys.
Why does my answer differ from the textbook's?
Usually one of three things. Rounding too early — carry full precision through the working and round only the final answer. Mixed units, which is the largest source of error. Or using a rounded value of π: use your calculator's π rather than 3.14, which introduces an error of about 0.05% that compounds when cubed.