chapter 12 surface area and volume identify the parts of prisms, pyramids, cylinders and cones. find...
TRANSCRIPT
![Page 1: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/1.jpg)
Chapter 12Surface Area and Volume
• Identify the parts of prisms, pyramids, cylinders and cones.
• Find the lateral areas, total areas and volumes of right prisms, regular pyramids, cones and cylinders.
![Page 2: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/2.jpg)
12-1: Prisms
Objectives
• Learn and apply the area and volume formula for a prism.
![Page 3: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/3.jpg)
Anatomy of a Prism
BaseBase Edge
Lateral edge
Lateral Face
Altitude
![Page 4: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/4.jpg)
Oblique vs. Right
The lateral edges of a right prism are perpendicular to the base.
![Page 5: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/5.jpg)
The name of a prism comes from its base.
Names of Prisms
Rectangular Oblique Prism Triangular Right Prism
![Page 6: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/6.jpg)
Surface Area
The total surface area of a prism is the sum of all the lateral faces and the two bases.
![Page 7: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/7.jpg)
Lateral Area
The lateral area of a right prism is the product of the perimeter of the base and the height.
LA = p x h
![Page 8: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/8.jpg)
Volume
The volume of a right prism is the product of the area of the base and the height.
V = B x h
![Page 9: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/9.jpg)
Total Area
• TA=LA+2B
![Page 10: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/10.jpg)
White Board Practice
A right triangular prism has base edges of 5, 12, and 13. It has a volume of 450. Find the height of the prism.
![Page 11: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/11.jpg)
White Board Practice
The base of a triangular prism is an isosceles right triangle with legs of 3 cm. The height of the prism is 10 cm. Find the lateral area, total area, and volume of the prism
![Page 12: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/12.jpg)
12-2:Pyramids
Objectives
• Learn and apply the area and volume formula for a pyramid.
![Page 13: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/13.jpg)
Anatomy of a Pyramid
Base
Base Edge
Lateral edge
Lateral Face
Altitude
Slant Height
Vertex
![Page 14: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/14.jpg)
Rotate the pyramid…..this is hard
![Page 15: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/15.jpg)
Oblique vs. Right
The altitude of a right pyramid passes through the center of the base.
![Page 16: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/16.jpg)
The name of a pyramid comes from its base.
Names of Pyramids
Triangular Oblique Pyramid Square Right Pyramid
![Page 17: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/17.jpg)
Lateral Area
The lateral area of a right pyramid is one half the product of the perimeter of the base and the slant height.
p
l
1
2LA pl
![Page 18: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/18.jpg)
Total AreaThe total surface area of a pyramid is the sum of all the lateral
faces and the base or it is the total are plus one base area
TA = LA + B
![Page 19: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/19.jpg)
Volume
The volume of a right pyramid is one third the product of the area of the base and the height.
h
B
1
3V Bh
![Page 20: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/20.jpg)
White Board Practice
• A regular square pyramid had base edge 6m and lateral edge 5m.
a) Find the length of a slant height
b) Find the Lateral Area
c) Find the Base Area
d) Find the Total Area
e) Find the Length of the altitude
f) Find the Volume
![Page 21: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/21.jpg)
12-3: Cylinders and Cones
Objectives
• Find the area and volume for a cylinder and a cone.
![Page 22: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/22.jpg)
Cylinder
A cylinder is a prism with a circular base.
r
h
Base
![Page 23: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/23.jpg)
LA and TA
The lateral area of a cylinder is the product of the perimeter(?) of the base and the height.
r
h
2TA LA B
LA = ph
LA = (2Пr) h
TA = LA + 2 Пr2
![Page 24: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/24.jpg)
Volume
The volume of a cylinder is the product of the area of the base and the height.
V Bh
r
h
![Page 25: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/25.jpg)
Cone
A cone is a pyramid with a circular base.
r
h
Base
![Page 26: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/26.jpg)
LA and TA
The lateral area of a cone is one half the product of the circumference of the base and the slant height.
TA LA B r
h
LA = ½ pl
LA = ½ (2Пr) h
LA = Пrh
![Page 27: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/27.jpg)
Volume
The volume of a cone is one third the product of the area of the base and the height.
1
3V Bh
r
hV = 1/3 Пr2h
![Page 28: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/28.jpg)
White Board Practice
• Find the lateral area, total area and volume of a cone with height and radius 3 cm.cm33
![Page 29: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/29.jpg)
White Board Practice
• Find the lateral area, total area and volume of a cone with height and radius 3 cm.
LA = 18 cm2
TA = 27 cm2
V = 9 cm3
cm33
3
![Page 30: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/30.jpg)
12-4: Spheres
Objectives
• Determine the area and volume of a sphere.
![Page 31: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/31.jpg)
Sphere
r
A sphere is the locus (set) of points in space equidistant from a given point.
![Page 32: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/32.jpg)
Volume of a Sphere
r
V = 4/3 Пr3
![Page 33: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/33.jpg)
Area of a Sphere
The area of a sphere is four times the area of the circle with the same radius.
r
24A r
![Page 34: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/34.jpg)
White Board Practice
• If the surface area of a sphere is 16, find the diameter and the volume
![Page 35: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/35.jpg)
White Board Practice
• If the surface area of a sphere is 16, find the diameter and the volume
d = 4
V = 3
32
![Page 36: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/36.jpg)
White Board Practice
• Find the area of the circle formed when a plane passes 9 cm from the center of a sphere with a radius of 15 cm.
![Page 37: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/37.jpg)
White Board Practice
• Find the area of the circle formed when a plane passes 9 cm from the center of a sphere with a radius of 15 cm.
144
![Page 38: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/38.jpg)
White Board Practice
• Find the area of the circle formed when a plane passes 9 cm from the center of a sphere with a radius of 15 cm.
15 9
![Page 39: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/39.jpg)
White Board Practice
• Betty made two wax candles, one in the shape of a sphere with radius 5cm and another in the shape of a cylinder with radius 5cm and height 6 cm. Which candle required more wax?
![Page 40: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/40.jpg)
White Board Practice
• Betty made two wax candles, one in the shape of a sphere with radius 5cm and another in the shape of a cylinder with radius 5cm and height 6 cm. Which candle required more wax?
Volume of Sphere = 166 2/3 cm3
Volume of Cylinder = 150 cm3
![Page 41: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/41.jpg)
12-5: Areas and Volumes of Similar Solids
Objectives
• Determine the ratios of the areas and volumes of solids.
![Page 42: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/42.jpg)
Similar SolidsFor two solids to be similar, all angles must be congruent and
all corresponding measurements must be proportional.
rh rh
![Page 43: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/43.jpg)
Remember
• Scale Factor a:b
• Ratio of perimeters a:b
• Ratio of areas a2:b2
![Page 44: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/44.jpg)
Theorem 12-11If the scale factor of two similar solids is a:b, then
• the ratio of their perimeters is a:b
• the ratio of their areas is a2:b2
• the ratio of their volumes is a3:b3
rr
rr
![Page 45: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/45.jpg)
• Scale Factor a:b
• Ratio of (anything that is not area of volume) a:b– Radii, diameter, heights, circumference,
perimeter, etc….
• Ratio of areas a2:b2
• Ratio of volumes a3:b3
![Page 46: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/46.jpg)
Ex 1
• Two regular pyramids have equilateral triangular bases with sides 4 and 6. Their heights are 6 and 9 respectively. Are the two pyramids similar?
![Page 47: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/47.jpg)
Ex 2
• Two similar cones have bases with area ratios of 4:9. Find the ratios of the following:
a) Radii
b) Heights
c) Total areas
d) Volumes
![Page 48: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/48.jpg)
Ex 3
• The volumes of two spheres have a ratio of 27:64. Find the area of the larger sphere if the area of the smaller sphere is 18.
![Page 49: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/49.jpg)
Ex 4
• The radii of two similar cylinders are 2 and 5. Find the ratios of their volumes and of their lateral areas.
![Page 50: Chapter 12 Surface Area and Volume Identify the parts of prisms, pyramids, cylinders and cones. Find the lateral areas, total areas and volumes of right](https://reader035.vdocuments.site/reader035/viewer/2022070408/56649e665503460f94b60f6b/html5/thumbnails/50.jpg)
Ex 5
• The volumes of two similar rectangular solids are 125 and 64. Find the ratio of their base perimeters.