8G Slides Chapter 6
5.1.2 -Volume of Cylinders
Cylinders
Volume of a Cylinder
• Volume = π • radius2 • height
• v = πr2h
• Remember that volume is written in
Example – Find Volume
Example – Find Volume
• Find the volume of a cylinder with a radius of 6 inches and a height of 10 inches.
• Volume = πr2h or π62 • 10 = 360π
Example – Find Radius
• A cylinder has a volume of 726π cm3
Example – Find Radius
• A cylinder has a volume of 726π cm3
and a height of 6cm. Find the radius. • Volume = πr2h
726π = πr2 6
121 = r2
Example – Find Height
Example – Find Height
• A cylinder has a radius of 4cm and a volume of 88π cm3. Find the height.
• Volume = πr2h
88π = π42 • h
Classwork
1) What is the volume of a cylinder with a height of 8cm and a radius of 4cm?
2) What is the height of a cylinder with a volume of 275π cm3 and a radius of 5cm?
3) What is the diameter of a cylinder with a volume of 510π cm3 and a height of
Classwork
1) What is the volume of a cylinder with a height of 8cm and a radius of 4cm?
128π cm3 or 401.9 cm3
2) What is the height of a cylinder with a volume of 275π cm3 and a radius of 5cm?
11cm
3) What is the diameter of a cylinder with a volume of 510π cm3 and a height of 17cm?
Volume of Cones
Key Vocabulary
• A cone is a three dimensional shape
that tapers from a flat round surface to a single point.
Cylinders and Cones
Cylinder and Cones
• If the volume of a cylinder = πr2h can
Cylinder and Cones
• If the volume of a cylinder = πr2h can
we derive the volume of a cone? • Volume of a Cone =
p
r
2h
Cylinder v Cone
• https://www.youtube.com/watch?v=0
Examples
• Find the volume of a cone with a radius of 2cm and a height of 6cm.
Examples
• Find the volume of a cone with a radius of 2cm and a height of 6cm.
V = πr2h/3, so V = π(22)(6)/3
V = 8π cm3
• Find the radius of a cone whose volume is 18π cm3 and whose height is 6cm.
V = πr2h/3, so 18π = πr2(6)/3
Classwork
• Find the volume of a cone with a radius of 8cm and a height of 12cm.
Classwork
• Find the volume of a cone with a radius of 8cm and a height of 12cm.
V = 256πcm3
• Find the height of a cone with a volume of 42cm3 and a diameter of 6cm.
Volume of Spheres
Key Vocabulary
Spheres/Cones/Cylinders
• If a sphere, a cone and a cylinder have the same radius and height, which
Spheres/Cones/Cylinders
• The cylinder has the greatest volume, the cone the least volume and the
Cylinder v Sphere
• https://www.youtube.com/watch?v=h
Volume of a Sphere
• Volume = 4/3πr3
Volume of a Sphere
• Volume = 4/3πr3
• Where does the r3 come from?
Examples
• Find the volume of a sphere with a radius of 4cm.
Examples
• Find the volume of a sphere with a radius of 4cm.
V = 4/3πr3, so V = 4/3π(43)
V = 85.3π cm3
• Find the radius of a sphere with a volume of 36π cm3.
V = 4/3πr3, so 36π = 4/3πr3
Classwork
• Find the volume of spheres with radii of 3cm, 6cm, and 12cm.
• What is the pattern?
• Find the volume of cones with a height of 10cm and radii of 3cm, 6cm, and 12cm.
• What is the pattern?
6.1.1 - 6.1.2 Lines of
Symmetry and Reflection
• Key Skills:– WWBAT define and identify symmetric figures and lines of symmetry.
Background Vocabulary
Key Vocabulary
Key Vocabulary
Key Vocabulary
Transformation I
Line of Reflection
• The line of reflection is the line
exactly in between the two figures.
Classwork
• Page 262 #1-6
6.2.1 - Rotation Symmetry
Key Vocabulary
Transformation II
Counter Examples
Rotation Symmetry?
Key Vocabulary
• Angle of Rotation is the smallest angle you need to turn the figure for it to
match with original.
Find the Angles of Rotation
Find the Angle of Rotation
Rotation Direction
• Rotations can be clockwise or counterclockwise.
Clockwise Rotation
Clockwise Rotation
Clockwise Rotation
Counterclockwise Rotation
Counterclockwise Rotation
Counterclockwise Rotation
Classwork
6.3.1 - Translations
What did we do here?
#1
Key Vocabulary
Transformation III
Key Vocabulary
Key Vocabulary
• Vector is the distance and direction of a translation.
• A vector MUST have BOTH distance AND direction.
Translation Example
#1
#2 Vector:
A Horizontal Example
#1 #2
Vector: Rule: (x, y) to
A Diagonal Example
#1
#2 Vector:
Classwork
• Pages 287-288 #9-12
6.3.1 - Combined Transformation
What Happened Here?
#1
What Happened Here?
#1
What Happened Here?
#1
Classwork
6.3.4 - Dilation
Dilation Example
A
B C
A’
Counter Example
Key Vocabulary
Transformation IV
•A Dilation is a new figure that is similar, but not necessarily congruent, to the
Key Vocabulary
• Similar figures have the exact same shape, but are not necessarily the same size.
• Similar figures will have proportional
What did we do here?
Key Vocabulary
• Scale Factor is the ratio between
corresponding side lengths of similar figures.
• For example, if one side of a figure is twice as long as its dilation, then
every side must be twice as long. • Scale factors 0 < x < 1 will make a
What is the Scale Factor?
What is the Scale Factor?
Placement
• Note that a dilation created on a grid MUST be placed on EXACTLY the
correct coordinates.
• Multiply the corners coordinates of the original figure by the scale factor to
Classwork
6.4.4 - Creating Dilations
• Key Skill: WWBAT create dilations on a coordinate plane and by using the
Using the Cartesian Plane
Draw a
Using the Cartesian Plane
Each point in the diamond is multiplied by 1/2 to get the points for
the dilation Point (6,6) becomes (3,3), point
Projection Method
Projection Method
Projection Method
• Step 1: pick a point inside the figure. Center is best.
Projection Method
• Step 2: Measure the distance from the point to each corner.
Projection Method
• Step 3: Multiply the measurement by the scale factor and draw new lines that length.
Projection Method
• Step 4: Connect the new dots to form the dilation.
Classwork
6.3.3 - Tessellations
Symmetry around us
• http://blog.ted.com/2009/10/29/sym
Key Vocabulary
Classwork
• Tessellation project on pages 290-293. • Work in your seating pairs.
• Materials:
– 1 index card (you may have another if you don’t like your shape)
– 3 pieces of construction paper (3 different colors) – Scissors
– Glue Sticks (will need to be shared)
