Chapter 10
Mason Schulz
Section 1
Section 1 Areas of Parallelograms and Trapezoids
Area of a Parallelogram
A=b*h
Area of a Trapazoid
A=.5*(b1+b2)*h
How to Change a Parallelogram into a Rectangle
First
Second
Vocab
- Base of a Parallelogram-length of any one of its sides
- Height of a Parallelogram-perpendicular distance from a base to the opposite side
- Base of a Trapezoid-a trapezoids two parallel sides
- Height of a Trapezoid-perpendicular distance from a base to the opposite side
Example 1
- Step one: Write the formula for a trapezoid.
- Step two: Substitute Base one, base two, and height.
- Step three: Multiply to solve for area
A=.5*(b1+b2)*h
A=.5*(4+6)*3
Answer= 15 inches
Example 2
- Step one: Write the formula for area of parallelogram
- Step two: Substitute base and height
- Step three: Multiply
A=10*3
Answer=30 feet
Real World Example
Section 2
Area of a Circle
A=pi*r^2
Vocab
Circle-set of all points in a plane that are the same distance from the same point called the center
Radius-distance from the center to any point on the circle
Circumference-the distance around a circle
Diameter-distance across the circle through the center, or twice the radius
Pi-the quotient of a circles diameter and circumference 3.14159...... which this constant is represented by the Greek letter pi
Real World Example
Example
Step One: Write the Area formula
Step two: Substitute 3.14 for pi and substitute 2 for radius.
Step three: Multiply
A=pi*r^2
A=3.14*2^2
Answer=12.56
Section 3
Vocab
Polyhedron- solid that is enclosed by polygons. This also only has flat surfaces.
Face-the polygons that form a polyhedron
Edge-the segments where faces of a polyhedron meet
Vertex-point where three or more edges meet
Classifying Solids (Vocab Continued)
Polyhedron with two congruent bases that lie in parallel planes. The other faces are rectangles. A cube is a prism with six faces.
Pyramid
Polyhedron with one base. The other faces are triangles.
Cylinder
Solid with two congruent circular bases that lie in parallel planes.
Cone
Solid with one circular base.
Sphere
a solid formed by all point in space that are the same distance from a fixed point called the center.
Real World Example
Example
Step one: Ask yourself is the solid enclosed by polygons.
Step two: If it is it will be a polyhedron.
Step three: Use definitions of prism, pyramid, cylinder, cone, and sphere to classify the sold.
Step four: count all of the vertices, faces, and edges.
Answer: This is a rectangular prism which is a polyhedron. It has 6 faces, 12 edges, 8 vertices.
Section 4
Vocab
Surface area- the sum of the areas of its faces
(Picture)-the picture to the right is a net of a rectangular prism
Using a Formula to Find Surface Area of a Prism
S=surface area
B=area of base (may change depending on shape of base)
P=bases perimeter
h=height
In other words the surface area of a prism is the sum of twice the area of a base B and the product of the base's perimeter and the height h.
Surface Area of a Cylinder
Or the surface area of a cylinder is the sum of twice the area of a base B and the product of the base's circumference C and the height.
C=bases circumference
B=area of base
Example
Step one: Draw one base with a rectangle adjacent to each other.
Step two: Draw the other base adjacent to one of the rectangles.
Answer: Picture below
Real World Example
Section 5
Vocabulary
height of pyramid- the perpendicular distance between the vertex and the base
Other Notes
You can use the pythagorean therm to find the slant height of a cone.
You can use net of a pyramid to find the surface area of the pyramid.
Surface area=area of base+number of triangles*area of each triangle.
You can use the net of a cone to find its surface area. The curved surface of a cone is a section of a circle with radius l, the slant height of a cone. The area of the curved, lateral surface, called the lateral surface area of the cone is, A=pi*r*l, where r is the radius of the base of the cone.
Surface Area of a Cone
The surface area of a cone is the sum of the area of the circular base with the radius r and the product of pi, the radius r of the base, the slant height l.
Surface area of a Pyramid
The surface area of a rectangular pyramid is the sum of the area of the base B and one half the product of the vase perimeter P and the slant height l.
Example 1
It has a 2m radius, 6m height.
First: Write out the formula for surface area of a cone.
Next: Substitute values into the formula
Then: Multiply Note: you need to use the pathegrean therm to find slant height.
S=pi*r^2+pi*rl
S=3.14*2^2+3.14*2*square root 40
S= 52.3m
Examle 2
Step one: Find the perimeter of the base.
Step two: Substitute into the formula for surface area.
Step three: Solve
P=3+3+3=9
S=B+1/2Pl
S=24+1/2*9*8
S=60m
Real World Example
Section 6
Vocab
Other Notes
Volume is measured in cubic units.
Volume of a Prism
V=B*h
Example
In inches.
First: Because the base is a rectangle, use length times width to find the area of the base.
Second: Substitute values into the formula.\
Third: Multiply
V=Bh
=lwh
=12(8)(2)
=192
The volume is 192 cubic inches
Section 7
Vocab
Cone-a solid with one circular base
Volume-a measure of the amount of space a solid occupies
Volume of a Pyramid
V=1/3*B*h
Example
First: Write formula for volume of a pyramid.
Second: Substitute 30^2 for B and 15 for h.
Third: Multiply
V=1/3*B*h
V=1/3*30^2*15
Answer: 4500 cubic feet
Volume of a Cone
V=1/3*B*h=1/3*pi*r^2*h
Real World Example
All Formulas
Area of Parallelogram
The area of a parallelogram is the product of its base and height.A=b*h
Area of Trapezoid
The area of a trapezoid is one half the product of the bases and height.
A=.5*(b1+b2)*h
Area of Circle
The area of a circle is the product of pi and its radius squared.
A=pi*r^2
Area of Prism
S=2B+Ph
S=surface area
B=area of base (may change depending on shape of base)
P=bases perimeter
h=height
In other words the surface area of a prism is the sum of twice the area of a base B and the product of the base's perimeter and the height h.
Area of Cylinder
S=2B+Ch=2*pi*r^2+2*pi*r*h
Or the surface area of a cylinder is the sum of twice the area of a base B and the product of the base's circumference C and the height.
C=bases circumference
B=area of base
S= pi*r^2+pi*r*l
Area of Cone
The surface area of a cone is the sum of the area of the circular base with the radius r and the product of pi, the radius r of the base, the slant height l.
Area of Pyramid
S=B+1/2*Pl
The surface area of a rectangular pyramid is the sum of the area of the base B and one
half the product of the vase perimeter P and the slant height l.
Prism Volume
The volume of a prism is the product of the area of the base B and the height h.
V=B*h
Pyramid Volume
The volume of a pyramid is one third the product of the area of the base B and the height h.
V=1/3*B*h
Cone Volume
The volume of a cone V is one third the product of the area of the Base B and the height h.
V=1/3*B*h=1/3*pi*r^2*h
Rectangle:
Area = Length X Width
A = lw
Perimeter = 2 X Lengths + 2 X Widths
P = 2l + 2w
Parallelogram
Area = Base X Height
a = bh
Triangle
Area = 1/2 of the base X the height
a = 1/2 bh
Perimeter = a + b + c
(add the length of the three sides)
Trapezoid
Perimeter = area + b1 + b2 + c
P = a + b1 + b2 + c
The distance around the circle is a circumference. The distance across the circle is the diameter (d). The radius (r) is the distance from the center to a point on the circle. (Pi = 3.14)
d = 2r
c = pd = 2 pr
A = pr2
(p=3.14)
Sphere
The volume V of a sphere is four thirds the product of pi and the cube of the radius r.
V=4/3*pi*r^3
S=4*pi*r^2