Volume of Cylinders, Cones, and Spheres Quiz

Volume of Cylinders

• Volume formula for a cylinder: ( V = πr^2h )
• Example with radius 1 inch and height 1 inch: Volume is approximately 3.14 cubic inches.
• If the height is twice the radius, volume can be expressed as: ( V = 2πx^3 ).

Volume of Cones

• Volume formula for a cone: ( V = \frac{1}{3}πr^2h )
• When the base diameter and height equal ( x ) units, volume is represented as: ( \frac{1}{2}πx^3 ).
• For a cone with height as twice its radius, volume expression is: ( \frac{2}{3}πx^3 ).
• Example with diameter 4 and height 5 yields a volume of 12.56 cubic units.

Volume of Spheres

• Volume formula for a sphere: ( V = \frac{4}{3}πr^3 )
• With a diameter of 4 centimeters, volume is represented as ( \frac{64}{3}π , cm^3 ).

Ratios of Areas

• For a cylinder inscribed in a square prism, the area ratio of the circle to the square is expressed as ( \frac{π}{4} ).
• Volume of the prism using area ratio: ( \frac{π}{4}(4r^2)(h) ).
• For a cone inside a square pyramid, similarly, the area ratio of the circle to square is ( \frac{π}{4} ).

Oblique Cylinders and Cones

• Oblique cylinder example: A cylinder with a diameter of 14 units and volume 1,176π cubic units has a height of 24 units.
• Volume of an oblique cone, where height equals base diameter, is given as 18π cubic units; the radius calculated is 3 units.