Chapter 10

Mason Schulz

Section 1

Areas of Parallelograms and Trapezoids

Section 1 Areas of Parallelograms and Trapezoids

Area of a Parallelogram

The area of a parallelogram is the product of its base and height.

A=b*h

Area of a Trapazoid

The area of a trapezoid is one half the product of the bases and height.

A=.5*(b1+b2)*h

How to Change a Parallelogram into a Rectangle

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
Algebra - Area of a Trapezoid

Example 1

If a trapezoid has a base 4in long, another base 6in long, and a height 3in long what is its area?

  • 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

If a parallelogram has a base 10ft long and a height 3in long what is its area?


  • Step one: Write the formula for area of parallelogram
  • Step two: Substitute base and height
  • Step three: Multiply
A=b*h


A=10*3



Answer=30 feet

Real World Example

All around are parallelograms and trapezoids. For example a window. When installing a window, builders use the formula of parallelograms. Also when building houses, if the roof is in the shape of a trapezoid they may need to know the area to figure out how much wood or shingles they would need.

Section 2

Areas of Circles

Area of a Circle

The area of a circle is the product of pi and its radius squared.


A=pi*r^2

Vocab

Area-the amount of surface the figure covers

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

Area of a Circle - MathHelp.com - Math Help

Real World Example

If you had a circular drive way and needed to fill in the center with grass, you would need to find the area of the circle in order to know how many rolls of grass to buy.

Example

Find the area of a circle with a radius of 2in


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

Three-Dimensional Figures

Vocab

Solid- a three-dimensional figure that encloses a part if space.

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)

Prism

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.

Classifying Solids

Real World Example

Everyday objects such as boxes, books, and even pencils can be classified. If you needed to solve for the volume of an object you would need to know its shape. This is because depending on the objects shape, the formulas would change. You may ask "Why would I need to solve for the volume of an object". For example if you needed to know how many books you could put in your backpack, you would need to know how to classify solids.

Example

Classify the Solid. Tell whether it is a polyhedron. Also tell how many faces, vertices, and edges it has.


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

Surface Areas of Prisms and Cylinders

Vocab

Net- two-dimensional pattern that forms a solid when it is folded.

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=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.

Surface Area of a 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

Example

Draw a net of a triangular prism


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

Big image

Real World Example

If you needed to paint a room, you would need to find the surface area of the spot you are painting. When you figure out the area you are painting, you can go to the store. Next you can go to the store and on the can of paint it will tell you how much paint is needed per area. This will not only save money, but also time.

Section 5

Surface Areas of Pyramids and Cones

Vocabulary

slant height-l of a rectangular pyramid is the height of a lateral face, or any face that is not the base

height of pyramid- the perpendicular distance between the vertex and the base

Other Notes

The net for a rectangular pyramid has a rectangular polygon as the base and congruent isosceles triangles on each side of the base.


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

S= pi*r^2+pi*r*l


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

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.

Example 1

Find the surface area of the cone. Round to the nearest tenth.

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

Find the surface area of a Triangular Pyramid. Round to the nearest tenth. B=24 m Three edges= 3m Slant height= 8m


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

Surface Area of a Right Cone

Real World Example

Many things including ice cream cones and pyramids themselves relate to this lesson. In fact if you needed to you could find the lateral surface area.

Section 6

Volume of Prisms

Vocab

volume of a solid-a measure of the amount of space a solid occupies.


Other Notes

Volume is measured in cubic units.

Volume of a Prism

The volume of a prism is the product of the area of the base B and the height h.

V=B*h

Example

Find the volume of a prism with the length of twelve, width of eight, and a height of two.

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

Volumes of Pyramids and Cones

Vocab

Pyramid-a polyhedron, with one base and other faces are triangles

Cone-a solid with one circular base

Volume-a measure of the amount of space a solid occupies

Volume of a Pyramid

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

Example

Find the volume of the rectangular pyramid with a height of 15 and the area of the base as 30.


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

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

Real World Example

Volume of an Ice Cream Cone

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