1. 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

2. 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.

3. Thm 12.9Thm 12.9 – Volume of a Pyramid– Volume of a Pyramid
• Works for right & oblique pyramidsWorks for right & oblique pyramids
because of Cavalieri’s Principle.because of Cavalieri’s Principle.
• A pyramid takes up 1/3 the spaceA pyramid takes up 1/3 the space
of a prism with the same base andof a prism with the same base and
height.height.
SO,SO,
V = 1/3 BhV = 1/3 Bh

4. Ex: What is the volume of theEx: 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
VV ≈ 17.32 cm≈ 17.32 cm33
2 cm2 cm
5cm5cm
1 cm1 cm
√√3cm3cm

5. 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 aA cone takes up 1/3 the space of a
cylinder with the same base andcylinder with the same base and
height.height.
SO,SO,
V = 1/3 BhV = 1/3 Bh
OROR
V = 1/3V = 1/3 ππrr22
hh

6. ExEx: What is the volume of the cone?: What is the volume of the cone?
V = 1/3V = 1/3 ππrr22
hh
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
10.2m10.2m

7. Ex. : Using the Volume of a Cone

8. Ex. 4: Finding the volume of a solid
Nautical prisms. A nautical prism is a
solid piece of glass as shown. Find its
volume.

9. 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

11. 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
).

12. Ex. : Using the volume of a cone
First, find the approximate volume of
the funnel.

13. 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: