Volume and Dilation of Figures

The video tutorial explains how to calculate the volume of different geometric figures, including prisms, cylinders, pyramids, and oblique prisms. It uses Cavalieri's principle to demonstrate that figures with the same height and cross-sectional area have the same volume. The tutorial also covers how a dilation of the base affects the volume, specifically for figure E, which has a base scaled by a factor of 1.5. The instructor provides step-by-step calculations for each figure, emphasizing the relationship between base area, height, and volume.

Which figure does not have a square base?

• Figure A

• Figure B

• Figure C

• Figure D

What is the volume of figure A if its base area is b and height is h?

• b + h

• b - h

• b * h

• b / h

Figure B is a cylinder. What principle explains its volume being the same as a rectangular prism with the same base area and height?

• Pythagorean Theorem

• Cavalieri's Principle

• Archimedes' Principle

• Pascal's Law

What is the formula for the volume of a pyramid like figure C?

• Base area + height

• 1/3 * base area * height

• 1/2 * base area * height

• Base area * height

How is the volume of figure C calculated if its base area and height are known?

• Base area / height

• 1/3 * base area * height

• Base area - height

• Base area + height

What type of prism is figure D?

• Right prism

• Oblique prism

• Triangular prism

• Hexagonal prism

Which principle is used to calculate the volume of figure D?

• Cavalieri's Principle

• Pythagorean Theorem

• Archimedes' Principle

• Newton's Laws

How is the base of figure E different from figure A?

• It is a reduction by a scale factor of 0.5

• It is a triangle

• It is a circle

• It is a dilation by a scale factor of 1.5

What is the scale factor used to dilate the base of figure E?

• 2.0

• 1.2

• 1.5

• 2.5

To find the volume of figure E, which calculation is necessary?

• Add 2.25 to the volume of figure A

• Multiply 2.25 by the volume of figure A

• Multiply 1.5 by the volume of figure A

• Divide the volume of figure A by 2.25