Chapter 12 Areas and Volumes of Solids (page 474) Essential Question How can you calculate the area and volume of any solid?

Volume Problem This problem was actually used in a movie. How can you get exactly 4 gallons of water, if you only have a 5 gallon container, a 3 gallon container, and an unlimited supply of water? What was the movie? What was the situation in the movie? Check out this site for this problem.

Lesson 12-1 Prisms (page 475) Essential Question How can you calculate the area and volume a prism?

Prisms

BASES: lie in parallel planes and are congruent polygons. Prisms are named for their bases, ie. triangular prism rectangular prism pentagonal prism hexagonal prism octagonal prism etc.

… Prisms ALTITUDES: a segment joining the two base planes and perpendicular to both. HEIGHT: the length of an altitude (h). LATERAL FACES: the faces that are not bases. Lateral faces are parallelograms that intersect each other in parallel segments which are called lateral edges.

lateral face lateral edge altitude base base edge basebase edge

RIGHT PRISM: a prism in which all the lateral faces are rectangles. In a right prism, each lateral edge is an altitude. This is the prism we will study.

OBLIQUE PRISM: a prism in which the lateral faces are not rectangles.

… Prisms LATERAL AREA: ( L.A. ) of a prism is the sum of the areas of its lateral faces.

Nets Imagine that you cut some edges of a right hexagonal prism and unfolded it. The two-dimensional representation of all of the faces is called a NET.

The lateral area of a right prism equals the perimeter of a base times the height of the prism. Theorem 12-1 L.A. = p  h

The lateral area of a right prism equals the perimeter of a base times the height of the prism. Theorem 12-1 L.A. = p  h

TOTAL AREA: (T.A.) of a prism is the sum of the areas of all of its faces. T.A. = L.A. + 2B B = base area

Take Note: This textbook uses TOTAL AREA (TA), but others use SURFACE AREA (SA).

CUBE: a rectangular solid with square faces. T.A. = 6e 2 e e e

Cavalieri’s Principle If two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

CUBE: a rectangular solid with square faces. T.A. = 6e 2 V = e 3 e e e

The volume of a right prism equals the area of a base times the height of the prism. Theorem 12-2 V = B  h

Example: Draw a triangular prism with a right triangle for its base and its legs 6 cm and 8 cm, and its height 12 cm. Then find its lateral area, total area, & volume.

6 cm 8 cm h = 12 cm

6 cm 8 cm h = 12 cm 10 cm

6 cm 8 cm h = 12 cm 10 cm

6 cm 8 cm h = 12 cm 10 cm

6 cm 8 cm h = 12 cm 10 cm

Assignment Written Exercises on pages 478 REQUIRED: 1, 9, 15, 17, 19 BONUS: #30 on page 479 How can you calculate the area and volume of a prism?