If you had a pizza with crust thickness 'a' and radius 'z', what's the volume of the pizza?

In the attached figure, you can see a chessboard and two rooks placed on the chess board. What you have to find is the number of squares that do not contain the rooks. How many are there?

I have a five and a three-gallon jar. Assuming there is an infinite supply of water, how can I measure one-gallon water?

Find a 9-digit number, which you will gradually round off starting with units, then tenth, hundred etc., until you get to the last numeral, which you do not round off. The rounding alternates (up, down, up ...). After rounding off 8 times, the final number is 500000000. The original number is commensurable by 6 and 7, all the numbers from 1 to 9 are used, and after rounding four times the sum of the not rounded numerals equals 24.

I came first on earth but second on the heaven.
I also came twice in a week but found just once in a year.
I stay away from months but you can find me in February.

What am I?

What does this mathematical rebus means ?

A man dies of old age on his 25 birthday. How is this possible?

What has a bottom at the top?

On a magical land of Mexico , all the animal in the land are rational.

There are 10 tigers and one goat.
Tiger can eat goat but since it's a magical land , the tiger who eats the goat , turns into goat and then can be eaten by the remaining tiger(s).

If we leave them for some time then how many goat and tiger will be there , when we come back ?

A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:

There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?