# Chapter 10

### Mason Schulz

## Section 1

**Areas of Parallelograms and Trapezoids**

## Section 1 Areas of Parallelograms and Trapezoids

## Area of a Trapazoid

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

## How to Change a Parallelogram into a Rectangle

## First Start with any parallelogram. | ## Second Cut off one of the right triangles from the parallelogram. This will form a partial trapezoid. Add the right triangle to the other side of parallelogram. |

## 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

**Areas of Circles**

## 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

**Three-Dimensional Figures**

## 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)

**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.

## 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

**Surface Areas of Prisms and Cylinders**

## 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

## Real World Example

## Section 5

**Surface Areas of Pyramids and Cones**

## Vocabulary

*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

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

## Section 6

**Volume of Prisms**

## 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

**Volumes of Pyramids and Cones**

## 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**

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