Which of the following statements are incorrect?
(A) Volume of a cone \(=\displaystyle\frac{1}{3}\pi r^3h\)
(B) Volume of a cone \(=\displaystyle\frac{1}{3}\pi r^2h\)
(C) Volume of a hemisphere \(=\displaystyle\frac{2}{3}\pi r^2\)
(D) Volume of a hemisphere \(=\displaystyle\frac{2}{3}\pi r^3\)
(E) Volume of a cylinder \(=\displaystyle\frac{1}{3}\pi r^3h\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (E) only
The standard volumes are: cone \(\frac{1}{3}\pi r^2h\), hemisphere \(\frac{2}{3}\pi r^3\) (half of the sphere’s \(\frac{4}{3}\pi r^3\)) and cylinder \(\pi r^2h\). A volume is a product of three lengths, which is a quick way to spot a wrong formula.
(A) Volume of a cone \(=\frac{1}{3}\pi r^3h\) Incorrect. The radius is squared, not cubed; \(r^3h\) has four lengths.
(B) Volume of a cone \(=\frac{1}{3}\pi r^2h\) Correct. A cone holds one-third of the cylinder on the same base and height.
(C) Volume of a hemisphere \(=\frac{2}{3}\pi r^2\) Incorrect. \(r^2\) gives an area; the volume needs \(r^3\).
(D) Volume of a hemisphere \(=\frac{2}{3}\pi r^3\) Correct. It is half of the sphere’s volume.
(E) Volume of a cylinder \(=\frac{1}{3}\pi r^3h\) Incorrect. The volume of a cylinder is \(\pi r^2h\), with no \(\frac{1}{3}\) and the radius squared.