Lesson 11.2 Prisms pp. 464-470.

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Presentation transcript:

Lesson 11.2 Prisms pp. 464-470

Objectives: 1. To state Cavalieri’s principle and use it to prove theorems. 2. To derive a formula for the volume of a prism. 3. To find volumes of prisms using the formula.

Postulate 11.5 Cavalieri’s Principle. For any two solids, if all planes parallel to a fixed plane form sections having equal area, then the solids have the same volume.

Theorem 11.2 A cross section of a prism is congruent to the base of the prism.

Theorem 11.2 refers to cross-sections that are parallel to the bases. A cross section is the intersection of a three-dimensional figure and a plane that passes through the figure that is perpendicular to the altitude. Theorem 11.2 refers to cross-sections that are parallel to the bases.

Theorem 11.3 The volume of a prism is the product of the height and the area of the base: V = BH.

EXAMPLE Find the volume of the following regular prism. 8 14 3

EXAMPLE Find the volume of the following regular prism. 1. Find the apothem 42 + a2 = 82 a2 = 64 – 16 = 48 a 4 8 a = 4 3

EXAMPLE Find the volume of the following regular prism. 2. Find the base area B = ½ap a 4 8 B = ½(4 3)(48) B = 96 3 sq. units

EXAMPLE Find the volume of the following regular prism. 8 14 3 3. Find the volume V = BH V = (96 3)(4 3) V = 4032 cu. units

Square: A = s2 Area Formulas Rectangle: A = bh Parallelogram: A = bh Triangle: A = ½bh Trapezoid: A = ½h(b1 + b2) Rhombus: A = ½d1d2 Equilateral triangle: A = s2 Regular polygon: A = ½ap Circle: A = r2 4 3

Practice: Find the volume. 10 cm 12 cm 6 cm

Practice: Find the volume. 6″ 4″ 8″ 9 5 ″

Definition The semiperimeter of a triangle is one-half of the perimeter of a triangle: s = 2 c b a +

Heron’s Formula s(s – a)(s – b)(s – c) A = If ABC has sides of lengths a, b, and c and semiperimeter s, then the area of the triangle is s(s – a)(s – b)(s – c) A =

Practice: Find the volume. The bases are regular hexagons. 24 mm 6 mm

6 mm

6 mm 6 mm 6 mm

Find the apothem. 32 + a2 = 62 9 + a2 = 36 a2 = 27 a ≈ 5.2 6 mm a 3 mm

B = 1/2ap = 1/2a(36) = 1/2(5.2)(36) = 93.6 6 mm a 3 mm

Practice: Find the volume. The bases are regular hexagons. 24 mm 6 mm V = BH = 93.6(24) = 2246.4 mm3

Homework pp. 467-470

Find the volume of each prism. 1. ►A. Exercises Find the volume of each prism. 1. 3 18

Find the volume of each prism. 7. ►A. Exercises Find the volume of each prism. 7. 10 10.4 41

Find the volume of each prism. 9. ►A. Exercises Find the volume of each prism. 9. 10 12 8

■ Cumulative Review 22. Perimeter of a square State a formula for each. 22. Perimeter of a square

■ Cumulative Review State a formula for each. 23. Area of a square

■ Cumulative Review 24. Surface area of a cube State a formula for each. 24. Surface area of a cube

■ Cumulative Review State a formula for each. 25. Volume of a cube

■ Cumulative Review 26. Surface area of a circular cylinder State a formula for each. 26. Surface area of a circular cylinder