12.5 Volume of Pyramids & 12.5 Volume of Pyramids & ConesCones

p. 752p. 752

NCSCOS: 1.02, 2.03, 2.04NCSCOS: 1.02, 2.03, 2.04

Finding Volumes of Pyramids and Cones In Lesson 12.4, you learned

that the volume of a prism is equal to Bh, where B is the area of the base, and h is the height. From the figure at the left, it is clear that the volume of the pyramid with the same base area B and the same height h must be less than the volume of the prism. The volume of the pyramid is one third the volume of the prism.

Thm 12.9Thm 12.9 – Volume of a Pyramid – Volume of a Pyramid

• Works for right & oblique pyramids Works for right & oblique pyramids because of Cavalieri’s Principle.because of Cavalieri’s Principle.

• A pyramid takes up 1/3 the space A pyramid takes up 1/3 the space of a prism with the same base and of a prism with the same base and

height.height.

SO,SO,

V = 1/3 BhV = 1/3 Bh

Ex: What is the volume of the Ex: What is the volume of the oblique pyramid?oblique pyramid?

V = 1/3 BhV = 1/3 Bh

B = ½ PaB = ½ Pa

B = ½ (12)(B = ½ (12)(√3√3))

B = 6B = 6√√33

V = 1/3 (6V = 1/3 (6√√3)(5)3)(5)

V =V = 1010√√33

V V ≈ 17.32 cm≈ 17.32 cm33

2 cm2 cm

5 c

m5

cm

1 cm1 cm

√√3

cm

3 c

m

Thm 12.10Thm 12.10 – Volume of a Cone – Volume of a Cone

• Works for right & oblique conesWorks for right & oblique cones

• A cone takes up 1/3 the space of a A cone takes up 1/3 the space of a cylinder with the same base and cylinder with the same base and

height.height.

SO,SO,

V = 1/3 BhV = 1/3 Bh

OROR

V = 1/3 V = 1/3 ππrr22hh

ExEx: What is the volume of the cone?: What is the volume of the cone?

V = 1/3 V = 1/3 ππrr22hh

V = 1/3V = 1/3ππ(6.7(6.722)(10.2))(10.2)

V = 479.49 mV = 479.49 m33

6.7 m6.7 m

10.2

m1

0.2 m

Ex. : Using the Volume of a Cone

Ex. 4: Finding the volume of a solid

Nautical prisms. A nautical prism is a solid piece of glass as shown. Find its volume.

Example continued

To find the volume of the entire solid, add the volumes of the prism and the pyramid. The bases of the prism and the pyramid are regular hexagons, made up of six equilateral triangles. To find the area of each base, B, multiply the area of one of the equilateral triangles by 5 or

Ex. : Using the volume of a cone

Automobiles. If oil is being poured into the funnel at a rate of 147 milliliters per second and flows out of the funnel at a rate of 42 milliliters per second, estimate the time it will take for the funnel to overflow. (1 mL = 1 cm3).

Ex. : Using the volume of a cone

First, find the approximate volume of the funnel.

Ex. : Using the volume of a cone

The rate of accumulation of oil in the funnel is 147 – 42 = 105 mL/s. To find the time it will take for the oil to fill the funnel, divide the volume of the funnel by the rate of accumulation of oil in the funnel as follows: