Principle of Parallel Slices in the Plane: If planar (2-D) figures of equal altitude have identical horizontal cross-sectional lengths at each height, then the figures have the same area.

Aim #9: How do we use Cavalieri's Principle to find the volume of prisms and cylinders?Do Now: The bases of triangular prism T and rectangular prism R lie in the same plane. A plane parallel to the bases and also a distance 3 from the bottom base intersects both solids and creates cross-sections T' and R'.

CC Geometry H

a) Find Area(T').

b) Find Area(R').

c) Find Vol(T).

d) Find Vol(R).e) If a height other than 3 were chosen for both cross-sections, would the cross sectional areas of the solids change? Would the volumes change?

h

1) ΔABC and ΔA'B'C' both have area 15 u2. The distance between DE and CC' is 3. a) Find CB.

b) What is the height of ΔA'B'C'? ____ (since AB' ll CC')

c) Find A'B'. ____

e) What is the ratio of the corresponding heights for each pair of triangles?_____

d) Why is ΔABC ~ ΔDEC, and ΔA'B'C' ~ ΔD'E'C'? ____

f) Find the lengths of DE and D'E', two corresponding horizontal distances.

Cavalieri's Principle: Given two solids that are included between two parallel planes, if every plane parallel to the two planes intersects both solids in cross-sections of equal area, then the volumes of the two solids are equal.

Cross-Sections and Equal Volumes

The two solids below meet Cavalieri's criteria: Both solids have the same altitude and have cross-sections of equal area at every height. Therefore, their volumes are equal.

2. Are the volumes of a right cylinder and an oblique cylinder the same? Why?

3. Morgan tells you that Cavalieri‛s principle cannot apply to the general cylinders shown below because their bases are different. Do you agree or disagree? Explain.

4. A triangular prism has an isosceles right triangular base as shown below . A square prism also shown below has volume equal to that of the triangular prism. What are the dimensions of the square base?

5. The diagram below shows two figures. Figure A is a right triangular prism and figure B is an oblique triangular prism. The base of figure A has a height of 5 and a length of 8 and the height of prism A is 12. The base of figure B has a height of 8 and a length of 5 and the height of prism B is 12. Use Cavalieri's Principle to explain why the volumes of these two triangular prisms are equal.  

1212

6. Which solid has greater volume? Show all work.- a square prism with base edges 8√2 and height 6 OR- a triangular prism with base an equilateral triangle with each side 12√3 and the same height as the square prism.

7. An oblique cylinder has height 4 cm. and volume 48π cm3. Find the radius of the circular base, in simplest radical form.

Let's Sum it Up!

• Principle of parallel slices in the plane: If two planar figures of equal altitude have identical horizontal cross-sectional lengths at each height, then the regions of the figures have the same area.

• Cavalieri's Principle: Given two solids that are included between two parallel planes, if every plane parallel to the two planes intersects both solids in cross-sections of equal area, then the volumes of the two solids are equal.

Name________________________ CC Geometry H Date _________________ HW #9

3. A triangular prism with a right triangular base and a right circular cylinder are equal in volume. If the base edges of the prism are 12 in and 16 in., and both figures have height 18 in., find the diameter of the base of the cylinder, in terms of π.

12 in.

16 in.18 in.

18 in.

1. An oblique cylinder has height 5 and volume 45π. Find the radius of the circular base.

2. Use Cavalieri's Principle to explain why a circular cylinder with base of radius 5 and a height 10 has the same volume as a square prism whose base is a square with edge length 5 and whose height is also 10.

4. Do you think the right square pyramid and the oblique trapezoidal pyramid are equal in volume, based on Cavalieri's Principle? Explain.

5m3m

9.5m2 9.5m2

5m3m

10cm

8cm

2. A right circular cylinder fits tightly inside a right regular hexagonal prism so that each base edge of the prism is tangent to the base of the cylinder. If the radius of the hexagon base is 8, and the height of the cylinder/prism is 10 cm, find the volume of the space outside the cylinder but inside the prism, to the nearest tenth.

Review:1. An oblique cylinder has a base with radius 8 in. A hole in the cylinder is the shape of a right cylinder with a radius of 5 in. The cylinders' height is 20 in. Find the volume of the remaining solid (inside the larger cylinder but outside the smaller cylinder), in terms of π and to the nearest tenth of a cubic inch.

h = 20 in.

8 in.

5 in.

3. An oblique prism has a rectangular base that is 16 in. x 9 in. A hole in the prism is also the shape of an oblique prism with a rectangular base that is 3 in. wide and 6 in. long, and the prism‛s height is 9 in. (as shown in the diagram). Find the volume of the oblique prism with the hole removed.

9 in.

3 in.6 in.

16 in.

9 in.