# Chapter 10

### Julia D.

## Chapter 10 Formulas

-Area of a Trapezoid: A=1/2(b1+b2)h

-Area of a Circle: A=pi*r^2

-Area of a Triangle: A=1/2*b*h

-Surface Area of a Prism: S=2B+Ph

-Surface Area of a Pyramid: S=B+1/2*P*l

-Surface Area of a Cylinder: S=2(pi*r^2)+(2*pi*r*)h or S=2B+Ch

-Surface Area of a Sphere: S=4*pi*r^2

-Surface Area of a Cone: S=(pi*r^2)+(pi*r*l)

-Volume of a Cylinder: V=pi*r^2*h

-Volume of a Sphere: V=4/3*pi*r^3

-Volume of a Prism: V=Bh

-Volume of a Pyramid: V=1/3*B*h

-Volume of a Cone: V=1/3*B*h

-Perimeter (circumference) of a Circle: C=2*pi*r

-Slant Height: l=square root of (r^2+h^2)

-E+2=F+V (find out if edges, vertices, and faces are correct on a 3-dimensional object)

**Notes:**

-capital letters in formulas are usually associated with area

## Section 1: Areas of Parallelograms and Trapezoids

**Formulas:**

Area of a Rectangle: A=b*h

Area of a Trapezoid: A=1/2(b1+b2)(h)

-b equals base

-h equals height

-b1 equals first base of trapezoid

-b2 equals second base of trapezoid

Vocabulary:

-base of a parallelogram: The length of any one of a parallelogram's sides.

-height of a parallelogram: The perpendicular distance between the base and the opposite side.

-bases of a trapezoid: A trapezoid's two parallel sides.

-height of a trapezoid: The perpendicular distance between the bases.

**Notes:**

-parallelogram has same equation for area as rectangle

-the base of a shape is always there but the height doesn't have to be (it can be a hidden

line)

-base is connected to height by a right angle (base is perpendicular to the height)

-trapezoid's bases are always parallel

-with area, units must be squared! ex. in^2

## Practice Problem #1 Answer: 39 inches squared | ## Practice Problem #2 Answer: 36.8 inches squared How it is solved: A=b*h A=8*4.6 A=36.8 inches squared | ## Practice Problem #3 Answer: 10 inches squared |

## Real Life Tie-In You are building a garden the shape of a parallelogram and need to know much area you have to plant flowers. (It is 24 feet long and 9 feet wide.) |

1. start with any parallelogram

2. cut the parallelogram to form a right triangle and a trapezoid

3. move the triangle to form a rectangle

## Section 2: Areas of Circles

**Formulas:**

Area of a Circle: A=pi*r^2 (r squared)

-A equals area

-pi= 3.14 or button on calculator

-r=radius

**Vocabulary:**

-area: The number of square units covered by a figure.

-circle: The set of 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 and any point on the circle.

-diameter: The distance across the circle through the center.

-circumference: The distance around a circle.

-pi: The ratio of the circumference of a circle to its diameter.

**Notes:**

-r^2 (r squared) does not equal the diameter

-if the area is given for a problem, and you want to find the radius (r), divide both sides by pi and then find the square root of both sides to get r by itself

-radius is half of diameter

-circle is not a polygon

-with area, units must be squared! ex. in^2

## Practice Problem #1 Answer: 1520.530844 feet squared | ## Practice Problem #2 Answer: 7.068583471 yards squared How it is solved: A=pi*r^2 A=pi*1.5 (half of diameter) ^2 A=7.068583471 yards squared | ## Practice Problem #3 Answer: 78.539881634 units squared |

## Section 3: Three-Dimensional Figures

**Formulas:**

-E+2=F+V (find out if edges, vertices, and faces are correct on a 3-dimensional object)

-E= edges

-F= faces

-V= vertices

**Vocabulary:**

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

Polyhedron: A solid that is enclosed by polygons.

Prism: A polyhedron, has two congruent .bases that lie in parallel planes. The other faces are rectangles.

Pyramid: Is a polyhedron, has one base, and 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 space that are the same distance from a fixed point called the center.

Edge: The segments where faces of a polyhedron meet.

Vertex: A point where three or more edges meet. (Plural of vertex is vertices.)

**Notes:**

-solid shape (prism, pyramid) is defined by its base

ex: rectangular prism, rectangular pyramid

-lateral area is the area of the sides (everything but the base/bases)

## Section 4: Surface Areas of Prisms and Cylinders

**Formulas:**

Surface Area of Prisms/Cylinders: S=2B+Ph

-S=surface area

-B=area of the base

-P=perimeter of the base

-h=height

**Vocabulary:**

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

Surface Area: The sum of a polyhedron's areas of its faces.

**Notes:**

-with surface area, units must be squared! ex. in^2

## Practice Problem #1 Answer: 122 feet squared How it is solved: S= 2B+Ph S= 2(b*h)+(l+w+l+w)h S= 2(4*7)+(7+4+7+4)3 S= 56+(7+4+7+4)3 S= 56+66 S=122 feet squared | ## Practice Problem #2 Answer: 6.283185307 centimeters squared |

## Section 5: Surface Areas of Pyramids and Cones

**Formulas:**

Surface Area of a Pyramid: S=B+1/2Pl

Surface Area of a Cone: S=pi*r^2+pi*r*l

-S= surface area

-B= area of the base

-P= perimeter of the base

-l= slant height

-r= radius

-pi= button on calculator/3.14

-r^2= radius squared

**Vocabulary:**

Slant Height: The height of a lateral face, that is, any face not the base.

**Notes:**

-with surface area, units must be squared! ex. in^2

## Practice Problem #1 Answer: 2225 feet squared How it is solved: S=B+1/2*P*l S=(b*h)+1/2*(l+w+l+w)*l S=(25*25)+1/2*(25+25+25+25)*l S=625+1/2*100*32 S=625+1600 S=2225 feet squared | ## Practice Problem #2 Answer: 301.5928947 inches squared |

## Section 6: Volume of Prisms and Cylinders

**Formulas:**

Volume of a Prism/Cylinder: V=B*h

-V= volume

-B= area of the base

-h= height

**Vocabulary:**

Volume: A measure of the amount of space a solid occupies.

**Notes:**

-with volume, unit must be cubed! ex. in^3

## Practice Problem #1 Answer: 343 centimeters cubed How it is solved: V=B*h V=(b*h)*h V=(7*7)*7 V=343 centimeters cubed | ## Practice Problem #2 Answer: 56.54866776 centimeters cubed |

## Section 7: Volumes of Pyramids and Cones

**Formulas:**

Volume of a Pyramid/Cone: V=1/3*B*h

-V= volume

-B= area of the base

-h= height

**Vocabulary:**

Pyramid: A solid, formed by polygons, that has one base.

Cone: A solid with one circular base.

Volume: The amount of space the solid occupies.

**Notes:**

-with volume, unit must be cubed! ex. in^3

## Practice Problem #1 Answer: 167.5516082 millimeters cubed | ## Practice Problem #2 Answer: 128 units cubed How it is solved: V=1/3*B*h V=1/3*(b*h)*h V=1/3*(12*4)*8 V=128 units cubed |

## Real Life Tie-In You are making water cups the shape of cones, and you need to know how much water would fit in the cup. (ex. it is 4 inches tall and the base's diameter is 2 and a half inches) |