Feb 17 Week
Missed 4 days last week due to weather
M Warm Up
1. Find the area of a square with 5” to a side
2. Find the area of a triangle with a base of 7
feet and height of 5 feet
3. Find the area of a rectangle with base of 3cm
and height of 9 cm
4. Find the area of a trapezoid with the large
base of 9 m, small base of 5 m, and height of
3m
5. Find the area of a circle with a radius of 4
yards
Areas
• Triangle
– A = ½ bh
• Square
– A = s2
• Rectangle
– A = bh
• Trapezoid
– A = 1/2(b1 + b2)h
• Circle
– A = r2
Volumes – Two Bases
•
•
•
•
Base: different from “base” in area
Base in Volume = “what the shape sits on”
The bases in two base solids are parallel
The height is the perpendicular distance
between the two bases
• The “sides” are quadrilaterals, in our case,
rectangles, unless we are talking about a
cylinder
Volumes – Two Bases
• Two base solids are called Prisms (or cylinders)
• Names are based on shape of base:
– Right means the sides are perpendicular to the
bases, the sides are rectangles
– Oblique means the sides are not perpendicular to
the bases, the shape “leans” ( volume stays the
same: Cavalieri’s principle)
– Rectangular, Triangular, etc., is the shape of the
base
– Cylinder – the base is a circle
• V = B times h
Volumes – One Base
• The end opposite the base in one base solids
is a point
• The height is the perpendicular distance
between the base and the point
• The “sides” are normally triangles, unless we
are talking about a cone.
Volumes – One Base
• One base solids are called Pyramids or Cones
• Names are based on shape of base, as for two
bases:
– Right means the sides are congruent, or the point is
directly over the center of the circle base (cone)
– Oblique means the sides are not congruent, the shape
“leans”
– Rectangular, Triangular, etc., is the shape of the base
– Cone – the base is a circle
• V = 1/3 B times h
Answers to Handout
1. 15 yd cubed
2. 10 mi cubed
3. 15 yd cubed
4. 3.1 km cubed
5. 37.7 in cubed
6. 8 m cubed
7. 22.5 yd cubed
8. 0.7 in cubed
9. 1026 cm cubed
10. 680 cm cubed
11. 1200 m cubed
12. 1,325.3 m cubed
13. 13,823 mi cubed
14. 1693.3 yd cubed
15. 836 in cubed
16. 1734 mi cubed
17. 197 cm cubed
18. 2094 cm cubed
19. 718.4 yd cubed
20. 840 m cubed
T Warm Up
1. Find the volume and surface area of a sphere
with a circumference of 12”.
2. Find the volume and surface area of the above
sphere if the radius is tripled.
3. What is the scale factor between the two
spheres?
4. Write a proportion to show the ratio of the radii
to the ratio of the surface areas.
5. Write a proportion to show the ratio of the radii
to the ratio of the volumes
6. What conclusions can we draw to relate scale
factor to area and volume of spheres?
Today’s Standards
• MCC9-12.G.GMD.3 Use volume formulas for
cylinders, pyramids, cones, and spheres to
solve problems.
Wed Warm Up
• If a sphere has a volume of 1,000 cubic inches,
what would the volume be if the radius were
made 5 times larger?
• If a sphere has an area of 1,000 square inches,
what would the area be if the radius were
made 5 times smaller?
1. Find the area and volume of a sphere with a
radius of 3 inches
2. Find the area and volume of a sphere with a
radius 5 times larger that the above
3. Write a proportion to show the ratio of the radii
to the ratio of the surface areas.
4. Write a proportion to show the ratio of the radii
to the ratio of the volumes
5. What conclusions can we draw to relate scale
factor to area and volume of spheres?
• If a sphere has a volume of 8,000 cubic inches,
what would the volume be if the radius were
made 4 times larger?
• If a sphere has an area of 8,000 square inches,
what would the area be if the radius were
made 4 times smaller?
Volume of Composite Shapes
from Student CCGPS Frameworks page 38
• Approximate the Volume of the Backpack that
is 17in x 12in x 4in.
Vol = Prism + ½ cylinder
Vol = Bh + ½ Bh
Vol = bh*h + ½r2h
Vol = 12*4*(17-12/2) +
½ *62*4
Vol = 528 + 72
Vol  754.2 in3
Volume of Composite Shapes
from Student CCGPS Frameworks page 38
• Find the Volume of the Grain Silo shown below
that has a diameter of 20ft and a height of 50ft
Vol = Bh + ½ Volsphere
Vol = (r2)h + ½ (4/3 r3)
Vol = *102*40 +
½ *4/3* *103
Vol = 4666 2/3  14,661 ft3
The diameter of a baseball is about 1.4 in.
How much leather is needed to cover the
baseball?
How much rubber is needed to fill it?
Find the Volume
Find the Volume
Convert:
• 165
• 230
• 7pi/9 radians
• 4pi/3 radians
• What is volume of a sphere that has a
circumference of 32
• What is the volume of a cylinder that can just
hold the above sphere?
• What is the ratio of the volume of the sphere
to the volume of the cylinder?
• Prove it with variables, not numbers
• I have posted the graphic organizers for angles
and segments in circles on my wiki, along with
some helpful video links
• I have a link to Ms. Brassard’s wiki, which has
more instructional videos for your use
• You may log onto and use
carnegielearning.com for review
• You may use USATestPrep
• What is surface area of a sphere that has a
circumference of 32?
• What is the surface area of a cylinder
(including the “ends”) that can just hold the
above sphere?
• What is the ratio of the surface area of the
sphere to the surface area of the cylinder?
• Prove it with variables, not numbers
• In his work On the Sphere and Cylinder,
Archimedes proved that the ratio of the volume
of a sphere to the volume of the cylinder that
contains it is 2:3. In that same work he also
proved that the ratio of the surface area of a
sphere to the surface area of the cylinder that
contains it, together with its circular ends, is also
2:3. Because expressions for the volume and
surface area of a cylinder were known before his
time, Archimedes’’ results established the first
exact expressions for the volume and surface
area of a sphere.
• From
http://www.math.nyu.edu/~%20crorres/Archime
des/Tomb/Cicero.html
• Assume a cylindrical watering can hold
100cm3 of water.
• How much water can it hold if the height is
tripled and the radius stays the same?
• How much water can it hold if the radius is
made 1/5 the original and the height stays
the same?
• How much water can it if both the radius and
height were made 4 times larger?
• Go to Geo Sketch 5 Various Problems
• Go to Geo Sketch 5 Angle Problems
• Go to Geo Sketch 5 arc length and sector area
find
area
between
triangle
and
circle
r = 7'
angle =23 deg
• Find the volume of a pyramid with a regular
octagonal base that would be inscribed in a 12
meter diameter circle and 14 meters high.
2x
24x
110
20y
C
4x
10y
12x