Feb 19, 2015 - Perimeter = Sum of all sides; 2(L+W) or 2L+2W or .... Find the volume of the rectangular prism shown below. V = Bh ..... for the follow...
Recap on Unit 7 The test from Unit 7 included the following: • Perimeter = Sum of all sides; 2(L+W) or 2L+2W or L+L+W+W • Circumference = πd or 2πr • Area of a Rectangle = b x h • Area of a Triangle =
• Area of a Circle = π r²
1 bh 2
or
𝑏ℎ 2
Unit 7
vs.
Unit 8
1. Volume of a Triangular Prism V = Bh 1
1. Area of a Rectangle or Square A = bh
V = 2bh x h 2. Volume of a Rectangular Prism V = Bh V = bh x h 3. Volume of a Triangular Pyramid V= V=
2. Area of a Triangle
A=
1 bh 2
1 Bh 3 1 1 ( bh) 3 2
xh
4. Volume of a Rectangular Pyramid 1 3 1 3
V = Bh V = (bh) x h
What is
?
Volume is the amount of space occupied by a threedimensional object; and it’s measured in cubic units. Ex. cm³, ft³, in³
Face, Edge, and Vertex • Face: A face is any of the individual surfaces of a solid object (Think Two-Dimensional). • Edge: An edge is a line segment that joins two vertices.
• Vertices: A vertex (plural: vertices) is a point where two or more straight lines (Edges) meet.
Cube
Faces of a Cube Faces are also known as Surfaces
• A cube has 6 faces. • All the faces of a cube are squares.
Face
Vertices of a Cube Vertices are also known as Corners vertex
• A cube has 8 vertices.
Edges of a Cube Edges are also known as Straight Lines Edge
• A cube has 12 edges.
Cuboid
How many faces does a cuboid have?
Faces of a Cuboid Faces are also known as Surfaces
• A cuboid has 6 faces. • A cuboid can have 6 faces which are rectangles.
• A cuboid can have 4 faces which are rectangles and 2 faces which are squares.
How many vertices does a cuboid have?
Vertices of a Cuboid Vertices are also known as Corners
• A cuboid has 8 vertices.
How many edges does a cuboid have?
Edges of a Cuboid Edges are also known as Straight Lines
• A cuboid has 12 edges.
A Cube has 6 square faces.
The length of each square face is equal.
height2
height1 base
Volume of CUBE = Bh = b X h1 X h2 = units³
V = Bh Where B = Area
A rectangular prism that has 6 congruent faces is called a cube.
A rectangular prism that has 6 congruent faces is called a cube.
A rectangular prism that has 6 congruent faces is called a cube.
A rectangular prism that has 6 congruent faces is called a cube.
A rectangular prism that has 6 congruent faces is called a cube.
A rectangular prism that has 6 congruent faces is called a cube.
The solid shown below is made up of twenty-seven 1-cm cubes. Find the volume of the solid.
V = Bh Where B = Area
Find the Volume: bh x h 3 cm
3 cm
3 cm
Volume of cube = 3 x 3 x 3 cm³ = 27 cm³
The volume of the cube is 27 cm³
Example 1 Find the volume of the cube.
5 cm
V = Bh 5 cm
Where B = Area
5 cm
Volume of cube = b x h x h =5 x 5 x 5 = 125cm³
Example 2
10 cm
The box shown on the left is a present given to class the Spirit Squad for winning the Cheer Competition. Find the volume of the V = present if it is a cube.
Volume of present = b x h x h = 10 x 10 x 10 = 1000 cm³
Bh
CUBOID (Rectangular Prism) A cuboid also has 6 faces but NOT all the faces are equal height2
base
height1
V = Bh Where B = Area
Volume of CUBOID = base X height1 X height2 = b X h X h
Example 3 Find the volume of the rectangular prism shown below. 4cm
V = Bh
4cm
10cm
Volume = b x h x h = 10 x 4 x 4 = 160 cm³
Example 4 V = Bh 12.5 m
5m 7m
Volume of a Rectangular Prism =b x h x h = 7 x 5 x 12.5 = 437.5 m³
Practice Try these questions yourself. Make sure you solve each question in your journal.
Question 1 Find the volume of the following solid.
V = Bh
Volume of cube = b x h x
h
=
2
2cm
2cm
x
2
2
2cm
=
8
cm³
x
Question 2
V = Bh
Volume of cube = b x h x h =
6
= 216
x
6
x
Find the volume of the cube shown below.
6
cm³ 6 cm
Question 3 V = Bh
Volume of a = 4 Rectangular Prism
x
3 =
x 10.5 126
10.5m
m³
3m 4m
Question 4
6.5 cm
V = Bh
11cm 7.5 cm
Volume of = 11 x 7.5 a Rectangular Prism = 536.25 cm³
x
6.5
Bell Ringer
2/20/15
Area of a Rectangle =
Volume of a Rectangle =
A = bh
V = Bh 5 cm
5 cm
5 cm 12 cm
A = 12 x 5 A = 60 cm²
12 cm
V = 12 x 5 x 5 V = 300 cm³
Area of Triangles What is the formula? Area =
1 b 2
x h or
Area =
h b
𝑏ℎ 2
Area of Triangles
Example: Find the area of the triangle
1 A= bh 2 8 in
1 A = 8(7) 2 7 in
A = 28 in²
Area of Triangles
Example: Find the area of the triangle
1 A= bh 2 11 in
1 A = 11(3) 2 3 in
A = 16.5 in²
Unit 7 Area of a Triangle = 𝑏ℎ 2
or 1 bh 2
Unit 8 Volume of a Triangular Prism = V = Bh (STAAR Reference Sheet)
or V=
𝑏ℎ xh 2
or V=
1 bh x h 2
Volume of a Triangular Prism
Volume of a
h b
To find the volume of the triangular prism, we must first find the area of the triangular base (shaded in yellow).
Volume of a • To find the area of the Base…
Area (triangle) = b x h 2 h b
This gives us the Area of the Base (B).
Volume of a • Now to find the volume…
B
h
We must then multiply the area of the base (B) by the height (h) of the prism. This will give us the Volume of the Prism.
Volume of a
Volume (triangular prism) B
h
V =
B x h
Volume of a Volume = V = B x h V = (8 x 4) x 12 2 V = 16 x 12 V =
192 cm3
Practice Try these questions yourself. Make sure you solve each question in your journal.
Volume of a Volume = V = B x h
V = (6 x 4) x 12 2 V = 12 x 12 V =
144 cm3
Volume of a Volume = V = B x h V = (7 x 3) 2 V = 10.5 V =
x 10
x 10
105 m3
Volume of a Volume = V = B x h V = (8 x 10) x 60 2 V = 40 x 60 V =
2,400 cm3
Volume of a Volume = V = B x h
9.5 in
17 in
67 in
V = (17 x 9.5) x 67 2 V = 80.75 x 67 V = 5,410.25 in3
Bell Ringer Area of a Triangle:
2/26/15 Volume of a Triangular Prism: Round to the nearest hundredth.
A=
𝑏ℎ 2
or
1
bh
2
V = Bh
3 ft 8 in
3.5 ft
3.5 ft
12.8 in
A = 12.8 x 8 ÷ 2 A = 51.2 in²
V=
3.5 x 3.5 2
V = 300 cm³
x2
Vs. Prisms Have: • 2 Congruent Bases • Rectangular Side Faces
Pyramids Have: • 1 Base • Triangular Side Faces That Meet at One Vertex
Volume of a
Pyramid
Base
Faces: 5 total faces • The 4 Side Faces are Triangles • The Base is a Square It has 5 Vertices (corner points) It has 8 Edges
Net 1 3
Volume = Bh
B = Base area B = b xh
Height
1 3
Volume = BH 1 3
V = (8 x 7) x 6 1 3
V = (56) x 6 1 3
V = (336) V=
336 3
= 112in³
1 3
Volume = BH 1 3
V = (24 x 8) x 10 1 3
V = (192) x 10 1 3
V = (1,920) V=
1,920 3
= 640in³
1 3
Volume = Bh 1 3
V = (10 x 8) x 6 1 3
V = (80) x 6 1 3
V = (480) V=
480 3
= 160in³
Practice Try these questions yourself. Make sure you solve each question in your journal.
1 3
Volume = Bh 1 3
V = (6 x 6) x 15 1 3
V = (36) x 15 1 3
V = (540)
V=
540 3
= 180m³
1 3
Volume = Bh 1 3
V = (8 x 8) x 9 1 3
V = (64) x 9 1 3
V = (576)
V=
576 3
= 192cm³
Bell Ringer
2/27/15
Volume of a Rectangular Pyramid: 1
V = Bh 3
1
B = bh so V = bhh 3
6.8 ft
4.5 ft 4.5 ft 1
V = 4.5 x 4.5 x 6.8 3
V = 45.9 ft³
Pyramid
Base
Faces: 4 total faces • The 3 Side Faces are Triangles • The Base is also a Triangle It has 4 Vertices (corner points) It has 6 Edges
Net 1
Volume = 3 Bh B = Base area 1 𝑏ℎ B = bh or 2
2
height
1 Volume = Bh 3 1 8𝑥6 V= ( ) x 12 3 2 1 3
V = (24) x 12 1 3
V = (288) V=
288 3
= 96cm³
1 Volume = Bh 3 1 8 𝑥 12 V= ( ) x 16 3 2 1 3
V = (48) x 17 1 3
V = (816) V=
816 3
= 272cm³
1 Volume = Bh 3 1 14 𝑥 8 V= ( ) x 18 3 2 1 3
V = (56) x 18 1 3
V = (1,008) V=
1,008 3
= 336cm³
18
Practice Try these questions yourself. Make sure you solve each question in your journal.
1 Volume = Bh 3 1 4 𝑥 10 V= ( ) x 12 3 2 1 3
V = (20) x 12
4
1 3
V = (240) V=
240 3
= 80cm³
10
1 Volume = Bh 3 1 8.5 𝑥 16.5 V= ( ) 3 2 1 3
V = (70.125) x 12 1 3
V = (841.5)
V=
841.5 3
m
x 12
= 280.5 m³
8.5 m
16.5 m
1 Volume = Bh 3 1 16 𝑥 8 V= ( ) x 24 3 2 1 3
24.6 cm
V = (64) x 24
16 cm
1 3
V = (1,536)
V=
1,536 3
= 512cm³
8 cm
Bell Ringer
3/2/15
Volume of a Triangular Pyramid: 1
V = Bh 3
1 1
B = bh so V = ( bh) x h 3 2
6.8 ft 4.5 ft
4.5 ft 1
4.5 x 4.5
3
2
V= x
x 6.8
V = 4.5 x 4.5 x 6.8 x 0.5 ÷ 3 V = 22.95 ft³
What is
surface area?
The number of square units needed to cover all of the surfaces (bases and lateral faces) In order to find the total surface area all you have to do is find the area of each shape and add them together.
base
Lat. Face
Lat F.
Base Lat. Face Lat. Face
Base Lat. Face
Lat. Face
The answer is always in square units. Ex. units²
Remember: In order to find the total surface area all you have to do is find the area of each shape and add them together. Find the total surface area. 9 yd
6.4 yd
4.6 yd
6.4 yd
Base
TSA = 2(9 x 6.4) + 2(4.6 x 6.4) + 2(9 x 4.6) TSA = 115.2 + 58.88 + 82.8 TSA = 256.88 yd²
Base
Remember: In order to find the total surface area all you have to do is find the area of each shape and add them together. Find the total surface area. TSA =
7.6 𝑥 5.2 2
+
6.4 𝑥 5.2 3( 2 )
TSA = 19.76 + 49.92
7.6 in.
TSA = 19.76 + 3(16.64)
TSA = 69.68 in² 5.2 in.
Find the
for the following shapes. 9 yd
9 in.
6 yd
6 in.
5 yd
5 yd
Base
Base
T.S.A. = ___________
T.S.A. = ___________
What is
surface area?
The number of square units needed to cover the lateral view (area excluding the base(s) of a three-dimensional figure)
Find the triangular pyramids lateral surface area.
Remember: The Lateral Surface Area is everything except the base(s). LSA =
9.3 𝑥 7 2( ) 2
+
9.3 𝑥 4 2( ) 2
LSA = 65.1 + 37.2 LSA = 102.3 ft²
4 ft
7 ft
9.3 ft
Find the rectangular pyramids lateral surface area. (Paint the 4 walls in your bedroom, but not the floor or the ceiling) 7 yd
