Geometric Probability using Area and Volume

A circle is inscribed in a square. If the side length of the square is 4 units, what is the probability that a randomly chosen point inside the square also lies inside the circle?

A cube has a side length of 2 units. A sphere is inscribed within the cube. What is the probability that a randomly chosen point inside the cube also lies inside the sphere?

A rectangle has dimensions 4 units by 3 units. A circle with a radius of 1 unit is placed inside the rectangle. What is the probability that a randomly chosen point inside the rectangle also lies inside the circle?

A cylinder has a height of 6 units and a base radius of 2 units. What is the probability that a randomly chosen point inside the cylinder also lies within a sphere of radius 2 units centered at the base of the cylinder?

A square has a side length of 8 units. A smaller square with a side length of 4 units is placed inside it, sharing the same center. What is the probability that a randomly chosen point inside the larger square also lies inside the smaller square?

A cone has a base radius of 3 units and a height of 6 units. What is the probability that a randomly chosen point inside the cone also lies within a cylinder of the same base radius and height?

A rectangular prism has dimensions 4 units by 3 units by 2 units. What is the probability that a randomly chosen point inside the prism also lies within a sphere of radius 2 units centered at one of the corners of the prism?

A triangle has vertices at (0,0), (4,0), and (0,4). What is the probability that a randomly chosen point inside the triangle also lies within a circle of radius 2 units centered at the origin?

A sphere has a radius of 3 units. A cube is inscribed within the sphere. What is the probability that a randomly chosen point inside the sphere also lies inside the cube?

A hexagon is inscribed in a circle with a radius of 2 units. What is the probability that a randomly chosen point inside the circle also lies inside the hexagon?