Chapter 10
Julia S
10.1
ParalellogramsA=bh Area= base x height A=area b=base h=height Example:
A=bh A=12x5 A=60 Real Life: If you want to find the area of your textbook the formula of A=bh would help you find it. | TrapazoidsA=1/2(b1+b2)h Area=1/2(base1+base2)xheight
Example: A=1/2(b1+b2)h A=1/2(2.5+6)5 A=1/2x8.5x5 A=21.25 Real Life: If for some reason you wanted to find the area of half of a stop sign you would use this formula to find the answer. You could also add the answer to its self to find the area of the whole sign | Vocabulary Base of parallelogram-length of any one of its sides Height of parallelogram-perpendicular distance between the base and the opposite side Base of trapezoid-its two parallel sides Height of trapezoid-perpendicular distance between the bases |
Paralellograms
A=bh
Area= base x height
A=area b=base h=height
Example:
A=bh
A=12x5
A=60
Real Life:
If you want to find the area of your textbook the formula of A=bh would help you find it.
Trapazoids
A=1/2(b1+b2)h
Area=1/2(base1+base2)xheight
Example:
A=1/2(b1+b2)h
A=1/2(2.5+6)5
A=1/2x8.5x5
A=21.25
Real Life:
If for some reason you wanted to find the area of half of a stop sign you would use this formula to find the answer. You could also add the answer to its self to find the area of the whole sign
10.2
A=πr^2
Area=pi x radius squared
π=pi r=radius ^2=squared
Example:
A=πr^2
A=π x 6^2
A=π x 36
A= 3.14 x 36
A= 113.04
Real Life:
If you are trying to find the area of a circular mirror to put in your mirror you would first find the radius of the mirror and square it to then multiple it by π to find the area of your mirror. Most time 3.14 can be used for π
Vocab
Area-the number of squared 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 of a circle- the distance between the center of any point on the circle
Diameter of a circle-the distance across the circle through the center
Circumference- the distance around the circle
Pi-the ratio of the circumference of a circle to its diameter
10.3
Vocab
Solid-a three-dimensional that encloses a part of space
Polyhedron-a solid that is enclosed by polygons
Face-the polygons that form a polyhedron
Prism-a solid, formed by polygons, that has two congruent bases lying in a parallel planes
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 lines
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 of a polyhedron-a line segment where two faces of the polyhedron meet
Vertex-the endpoint of the rays that form an angle
10.4
Prism S=2B+Ph Surface Area=2 x area of the base + perimeter of the base x height B=area of base P= perimeter of base Example: S=2B+Ph S= 2 x 48 + 32 x 6 S= 96 + 192 S=288 Real life: If you have a chocolate bar in this shape and you want to know the surface area of it you would find out its height, area of the base, and perimeter of the base to use it in the formula and find your answer to the surface area of your prism shaped chocolate bar. | Cylinder S=2B+Ch - S= 2πr^2+2πrh Surface Area= 2 x area of base + circumference x height Surface Area= 2 x pi x radius squared + 2 x pi x radius x height Example: S= 2 x π5^2 + 2π5 x 10 S= 157 + 314 S= 471 Real life: If you want to find the surface area of a can of soup you want to eat you would first need the radius and the height. You would plug those into either formula and solve to find the surface area of your soup can | Vocab Net- a two-dimensional pattern that forms a solid when it is folded Surface Area- the sum of the areas of the faces of the polyhedron |
Prism
Surface Area=2 x area of the base + perimeter of the base x height
B=area of base P= perimeter of base
Example:
S=2B+Ph
S= 2 x 48 + 32 x 6
S= 96 + 192
S=288
Real life:
If you have a chocolate bar in this shape and you want to know the surface area of it you would find out its height, area of the base, and perimeter of the base to use it in the formula and find your answer to the surface area of your prism shaped chocolate bar.
Cylinder
Surface Area= 2 x area of base + circumference x height
Surface Area= 2 x pi x radius squared + 2 x pi x radius x height
Example:
S= 2 x π5^2 + 2π5 x 10
S= 157 + 314
S= 471
Real life:
If you want to find the surface area of a can of soup you want to eat you would first need the radius and the height. You would plug those into either formula and solve to find the surface area of your soup can
10.5
PyramidS=B+1/2Pl Surface area=area of base +1/2 x perimeter of base x slant height l= slant height Example: S=B+1/2Pl S= 27.7 + 1/2 x 24 x 6 S= 99.7 Real Life: If you have a pyramid shaped roof and you are looking to find the surface area of that roof you would need to find the area of the base, perimeter of the base, and slant height. Then you plug those numbers into that formula and you have your answer to the surface area of your roof. | ConeS=πr^2+πrl Surface area= pi x radius squared + pi x radius x slant height Example: S=πr^2 + πrl S= π x 4^2 + π x 4 x 9 S= 163.4 Real Life: If you wanted to find the surface area of birthday hats that you are planning to use in your birthday party you would need to measure to find the radius and the slant height of your birthday hat. That is all that you would need to then plug the numbers into the formula and find the answers of the surface of the area of your hats. | Vocabulary Slant Height- (l) of a regular pyramid is the height of a lateral face, that is, any face that is not the base. |
Pyramid
S=B+1/2Pl
Surface area=area of base +1/2 x perimeter of base x slant height
l= slant height
Example:
S=B+1/2Pl
S= 27.7 + 1/2 x 24 x 6
S= 99.7
Real Life:
If you have a pyramid shaped roof and you are looking to find the surface area of that roof you would need to find the area of the base, perimeter of the base, and slant height. Then you plug those numbers into that formula and you have your answer to the surface area of your roof.
Cone
S=πr^2+πrl
Surface area= pi x radius squared + pi x radius x slant height
Example:
S=πr^2 + πrl
S= π x 4^2 + π x 4 x 9
S= 163.4
Real Life:
If you wanted to find the surface area of birthday hats that you are planning to use in your birthday party you would need to measure to find the radius and the slant height of your birthday hat. That is all that you would need to then plug the numbers into the formula and find the answers of the surface of the area of your hats.
10.6
Prisms V=Bh Volume= area of base x height Example: V=Bh V=(4 x 2 x 1/2) x 3 V= 4 x 3 V= 12 (note: a mini formula might have to be used to find the area of the base) Real Life: There are many reasons that you would want to find the volume of a prism. A prism can have many different shaped bases like the triangular one in the above example. Lets say that for this example you want to paint the walls and ceiling of a rectangular shaped room. You would then need to find the area of the base and multiple that by the height of the room. This answer will help you determine how much paint to buy. | CylindersV=Bh OR πr^2h Volume= area of base x height OR pi x radius squared x height Example: V=Bh V= 24 x 7 V=168 (note: a mini formula might have to be used to find the area of the base) Real life: If you are wondering how much soup is in a cylinder shaped can that you want to eat you would need to find the volume. The formula to do so is either V=Bh or V= πr^2h. So then for the first you would need the area of the base and the height. And for the second you would need the radius and the height. | Vocabulary Volume- measure of the amount of space it occupies, of a solid. |
Prisms
Volume= area of base x height
Example:
V=Bh
V=(4 x 2 x 1/2) x 3
V= 4 x 3
V= 12
(note: a mini formula might have to be used to find the area of the base)
Real Life:
There are many reasons that you would want to find the volume of a prism. A prism can have many different shaped bases like the triangular one in the above example. Lets say that for this example you want to paint the walls and ceiling of a rectangular shaped room. You would then need to find the area of the base and multiple that by the height of the room. This answer will help you determine how much paint to buy.
Cylinders
V=Bh OR πr^2h
Volume= area of base x height OR pi x radius squared x height
Example:
V=Bh
V= 24 x 7
V=168
(note: a mini formula might have to be used to find the area of the base)
Real life:
If you are wondering how much soup is in a cylinder shaped can that you want to eat you would need to find the volume. The formula to do so is either V=Bh or V= πr^2h. So then for the first you would need the area of the base and the height. And for the second you would need the radius and the height.
10.7
PyramidV=1/3Bh Volume= 1/3 x area of base x height (note: make sure that 1/3 is typed into the calculator first so it is not interpreted as x 1 / 3) Example: V=1/3Bh V=1/3 x 21 x 4 V=28 (note: a mini formula might have to be done to find the are of the base) Real life: If you have a mini sculpture of a pyramid and you want to know the volume of it to compare to that of the real pyramid, you would need to find the area of the base ad the height and multiply them by 1/3. This would then give you your answer to compare to the real pyramid. | ConeV=1/3Bh OR V=1/3πr^2h Volume= one third x area of base x height OR one third x pi radius squared x height Example: V=1/3Bh V=1/3 x 14 x 6 V=28 (note: you might have to do a mini formula to find the area of the base) Real life: If you want to know the maximum ice cream that can fill a waffle cone you would need to find the area of the base and height to multiple by one third for the first formula. For the second, you need to find the radius to plug into the formula. Now you know how much ice cream you can put in your cone. | SphereV=4/3πr^3 Volume= four thirds x pi x radius cubed Example: V=4/3πr^3 V=4/3 x π x 12^3 V=7234.56 Real life: If you want to know how much space is in the ball for your hamster you would first need to find the radius. Then, all you would need to do is find the radius and plug this into the formula and you have the volume of your hamster ball. |
Pyramid
V=1/3Bh
Volume= 1/3 x area of base x height
(note: make sure that 1/3 is typed into the calculator first so it is not interpreted as x 1 / 3)
Example:
V=1/3Bh
V=1/3 x 21 x 4
V=28
(note: a mini formula might have to be done to find the are of the base)
Real life:
If you have a mini sculpture of a pyramid and you want to know the volume of it to compare to that of the real pyramid, you would need to find the area of the base ad the height and multiply them by 1/3. This would then give you your answer to compare to the real pyramid.
Cone
V=1/3Bh OR V=1/3πr^2h
Volume= one third x area of base x height OR one third x pi radius squared x height
Example:
V=1/3Bh
V=1/3 x 14 x 6
V=28
(note: you might have to do a mini formula to find the area of the base)
Real life:
If you want to know the maximum ice cream that can fill a waffle cone you would need to find the area of the base and height to multiple by one third for the first formula. For the second, you need to find the radius to plug into the formula. Now you know how much ice cream you can put in your cone.
Sphere
V=4/3πr^3
Volume= four thirds x pi x radius cubed
Example:
V=4/3πr^3
V=4/3 x π x 12^3
V=7234.56
Real life:
If you want to know how much space is in the ball for your hamster you would first need to find the radius. Then, all you would need to do is find the radius and plug this into the formula and you have the volume of your hamster ball.
Vocabulary
Sphere-a solid formed by all points in a space that are the same distance form a fixed point called the center
(THERE IS OTHER VOCAB FOR THIS SECTION THAT CAN BE FOUND IN PREVIOUS SECTIONS)
Formulas
Area of Parallelograms- A=bh
Area of Trapezoids- A=1/2(b1+ b2)h
Area of Triangles- 1/2bh
Area of Circles- A=πr^2
Circumference of Circles- 2πr
Surface Area of Prisms- S=2B+Ph
Surface Area of Cylinders- S=2B+Ch
Surface Area of Pyramids- S=B+1/2Pl
Surface Area of Cones- S= πr^2+πrl
Volume of Prisms- V=Bh
Volume of Cylinders- V=Bh
Volume of Pyramids- V=1/3
Volume of Cones- V= 1/3Bh
Volume of Cones- V=1/3Bh
Volume of Spheres- V= 4/3πr^3
Reminders
When you are working with Surface Area or just Area the answer is unit squared ex. cm^2
When you are working with Volume the answer is units cubed ex. cm^3
The surface area and volume for a sphere are the same...with differently squared units
REMEMBER that there are always two bases(make sure you are looking at the shape in the correct way)
Make sure to be plugging in the right mini formulas when working with Area of the Base and Perimeter of the Base
Be aware of when it is asking to use the actual pi button or 3.14. Also be aware of what number it tells you to round to if needed.
Extra Formulas
Pentagon-A=1/4 x Squared root 5(5+2 squared root 5) a^2
hexagon-A=3 Squared root 3 / 2 x a^2