# Chapter 10

## Formulas

Areas of squares, parallelograms, and rectangles~A=b*h

Area of a trapezoid~ A=1/2(b1+b2)*h

Area of circles~ A=πrr

Surface area of prisms~ S=2lw+Ph

Surface area of cylinders~ S=2πrr+2πrh

Surface area of pyramids~ S=B+πrl

Surface area of cones~ S=πrr+πrl

Surface area of spheres~ S=4πrr

Volume of prisms~ V=Bh

Volume of cylinders~ V=πrrh

Volume of pyramids~ V=1/3*Bh

Volume of cones~ V=1/3*πrrh

Volume of spheres~ V=4/3*πrrr

## Section 1

Areas of Parallelograms and Trapezoids

KEY VOCABULARY

base of a parallelogram------ the length of any side of the parallelogram can be used at the base
height of a parallelogram--- the perpendicular distance between the side whose length is the base and the opposite side
bases of a trapezoid ---------- the lengths of the parallel sides of the trapezoid
height of a trapezoid --------- the perpendicular distance between the bases of a trapezoid
FORMULAS

area of a parallelogram~ A=bh
area of a trapizoid~ A=1/2(b1+b2)

Real Life Scenario

You are trying to find the area of a window to replace but it is a parallelogram. The height of the window is 12 inches and the base is 15 inches. What is the area of the window.

A=b*h

A=15*12

A=180

The area of the window is 180 inches squared

Practice Problems

1. Find the area of a trapezoid when Base 1=4in Base 2=5in and the height=8in

2. Find the area of a parallelogram with the base=6in and the height=7in

## Section 2

AREAS OF CIRCLES
KEY VOCABULARY

Area ------------------ the number of square units covered by a figure

Circle------------------ the set of all points in a plane that are the same distance, called the radius, from a fixed point, called the center

Radius --------------- the distance between the center of a circle and any point

Diameter------------ the distance across the circle through the center

Circumference --- the distance around the circle

Pi (π)--------------- the ratio of the circumference of a circle to its diameter

Formula

A=πrr
REAL LIFE SCENARIO

You need to find an area to build a silo. The silo's diameter is 10 inches. What is the area? Round to the nearest tenth.

A=π*r*r

A=π*5*5

A=π*25

A=78.53981634.....

The area is about 78.5 inches squared

PRACTICE PROBLEMS

3. Find the area of a circle where the radius is 6 feet. Round to the nearest tenth.

4. Find the area of a circle where the radius is 3 yards. Round to the nearest tenth.

## Section 3

THREE-DIMENSIONAL FIGURES
KEY VOCABULARY

Solid ------------- A three-dimensional figure that encloses a part of space

Polyhedron --- a solid that is enclosed by polygons

Face ------------- A polygon that is a side of the polyhedron

Prism ----------- A solid, formed by polygons, that has two congruent bases lying in parallel lines

Pyramid ------- A solid, formed by polygons, that has one base. The base can be any polygon, and the other faces are triangles

Cylinder ------- A solid with two congruent circular bases that lie in parallel planes

Cone ----------- A solid with one circular base

Sphere -------- A solid formed by all points in a space that are the same distance from a fixed point called the center

Edge ---------- A line segment where two faces of the polyhedron meet

Vertex --------- A point at which three or more edges of a polyhedron meet

## Section 4

SURFACE AREAS OF PRISMS AND CYLINDERS
KEY VOCABULARY

Net ---------------- A two-dimensional representation of a solid. This pattern forms a solid when it is folded.

Surface area --- The sum of the areas of the faces of the polyhedron

FORMULAS

Surface area of prism~ S=2B+Ph

Surface area of a cylinder~ S= 2πrr+2πrh

B=base

P=perimeter

REAL LIFE SCENARIO

You have been chosen to design a new cereal box. The company wants you to keep the same shape and size. But first you need to find the surface area. Here is a few dimensions that you need to know: B=160cm squared h=30cm P=56cm

S=2B+Ph

S= 2*160+56*30

S=320+1680

S=2000

The surface area is 2000 cementers squared

PRACTICE PROBLEMS

5. Find the surface area of a cylinder with the height being 8 inches and the radius 2 inches. Round to the nearest tenths.

6. Find the surface area of a prism with the base being 25 inches squared and the height being 7 inches

## Section 5

SURFACE AREAS OF PYRAMIDS AND CONES
KEY VOCABULARY

Slant height ----- the height of any face that is not the base of a regular pyramid

FORMULAS

Surface area of a pyramid~ S=B+1/2*Pl

Surface area of a cone~ S=πrr+πrl

B= base

P= perimeter

l= slant height

REAL LIFE SCENARIO

You are a construction worker and you need to paint a traffic cone orange. You only need to paint the lateral surface area or everything but the base. The radius is 5 inches and the height is 10 inches. What is the lateral surface area of the cone? round to the nearest tenth.

S=πrr+πrl

S=πrl

S=π*5*10

S=157.0796327

The surface area of the cone is about 157.1 inches squared.

PRACTICE PROBLEMS

7. Find the surface area of a pyramid with the base being 16 inches squared the perimeter being 4 inches and the slant height being 7 inches

8. Find the surface area of a cone where the radius is 8 cm and the slant height is 20 inches. Round to the nearest tenths.

## Chapter 6

VOLUMES OF PRISMS AND CYLINDERS
KEY VOCABULARY

Volume ----- The amount of space the solid occupies

FORMULAS

Volume of prisms~ V=Bh

Volume of cylinders~ V=2πrr+2πrh

REAL LIFE SCENARIO

You are trying to find the volume of two recycling bins. You want the one that holds the most. One is a cylinder and one is a rectangular prism. The cylinder's dimensions are the radius is 5 inches and the height is 30 inches. The rectangular prism's dimensions are the base is 40 inches squared and the height is 25 inches. Which recycling bin has the largest volume.

Cylinder-

V=2πrr+2πrh

V=2π5*5+2π5*30

V=50π+300π

V=1099.557429

Rectangular Prism-

V=Bh

V=40*25

V=1000

The cylinder recycling bin has a larger volume.

PRACTICE PROBLEMS

9. Find the volume of a prism with the base being 75 cm squared and the height being 20 cm

10. Find the volume of a cylinder with the radius being 7in and the height being 9 in. Round to the nearest tenth.

## Section 7

VOLUMES OF PYRAMIDS AND CONES
KEY VOCABULARY

Pyramid--- A solid formed by polygons that has one base. The base can be any polygon but the rest of the faces must be triangles

Cone------- A solid with one circular base

Volume--- The amount of space the solid occupies.

FORMULAS

Volume of a pyramid~ V=1/3*Bh

Volume of a cone~ V= 1/3*πrrh

REAL LIFE SCENARIO

You work at an ice cream parlor and you need to find the the volume of an ice cream cone. The dimensions are the radius is 2 inches and the height is 8 inches. What is the volume of the ice cream cone? Round to the nearest tenth.

V=1/3*πrrh

V=1/3*π*2*2*8

V=1/3*π*32

V=33.51032164

The volume of the ice cream cone is about 33.5 inches cubed.

PRACTICE PROBLEMS

11. Find the volume of a pyramid with the base being 25 inches cubed and the height being 5 inches.

12. Find the volume of a cone with the radius being 4 inches and the height being 7 inches. Round to the the nearest tenth.

## William Jones

William Jones was the inventor of pi. He invented pi in 1707- 1783. Jones was a math teacher when he discovered pi. He later discovered it was irrational.
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