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 Parallelogram: A=b*hArea 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
Practice Problem #2
How it is solved:
A=b*h
A=8*4.6
A=36.8 inches squared
Practice Problem #3
Real Life Tie-In
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
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
Practice Problem #2
How it is solved:
A=pi*r^2
A=pi*1.5 (half of diameter) ^2
A=7.068583471 yards squared
Practice Problem #3
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
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)
Practice Problem
1.cone
2.cylinder
3.sphere
4.rectangular pyramid
5.triangular prism
6.triangular pyramid
7.rectangular prism
8.hexagonal prism
Section 4: Surface Areas of Prisms and Cylinders
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
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
Real Life TIe-In
Other Formula for Surface Area of a Prism
Formula for Surface Area of a Cylinder
Section 5: Surface Areas of Pyramids and Cones
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
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
Section 6: Volume of Prisms and Cylinders
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
How it is solved:
V=B*h
V=(b*h)*h
V=(7*7)*7
V=343 centimeters cubed
Practice Problem #2
Other Formula for Finding Volume of a Prism
Formula for Finding Volume of a Cylinder
Section 7: Volumes of Pyramids and Cones
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
Practice Problem #2
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