Surface Area & Volume

Vocabulary Polyhedron – a three-dimensional figure whose surfaces are polygons Face – each of the polygons of the polyhedron Edge – a segment that is formed by the intersection of two faces Vertex – a point where three or more edges intersect

Vocabulary Polyhedron – a 3-dimensional (3-D) solid with faces and edges

Prisms A prism is a polyhedron with exactly two congruent, parallel faces, called bases. Other faces are lateral faces (also congruent) An altitude of a prism is a perpendicular segment that joins the planes of the bases. The height h of the prism is the length of an altitude.

Prisms Prism – are named by the shape of their bases. This one is a _________________ prism.

Prisms Prism – are named by the shape of their bases. This one is a _________________ prism.

Prisms Prisms – are named by the shape of their bases. This one is a _________________ prism.

Prisms Have bases and faces that are polygons – therefore, these sides have all of the properties of polygons One of those properties is area.

Prisms Net – is what we would have if we took a polyhedron an unfolded it. If we made a cube into a net, here is what we would have:

Prisms If we made a triangular prism into a net, here is what we would have:

Surface Area of Prisms Surface area – is just that, the area of the surface of the entire polyhedron. How many surfaces does this have?

Surface Area of Prisms How might we go about finding the surface area of this square polyhedron?

Surface Area of Prisms Surface area = area of all of the lateral faces + area of the two bases

Surface Area of Prisms The lateral area of a prism is the sum of the areas of the lateral faces. The surface area is the sum of the lateral area and the area of the two bases.

Surface Area of Prisms The lateral area of a right prism is the product of the perimeter of the base and the height. LA = ph The surface area of a right prism is the sum of the lateral area and the areas of the two bases SA = LA + 2B

Surface Areas of Prisms Use a net to find the surface area. 8 ft 7 ft 6 ft

Surface Areas of Prisms Use a formula to find the lateral area and surface area. L.A.=ph=24*7=168 ft2 S.A.= L.A.+2 B= 216 ft2 8 ft 7 ft 6 ft

Homework Handout on Surface Area of Prisms

Surface Areas of Cylinders A cylinder has two congruent parallel bases. However, the bases of a cylinder are a circle. An altitude of a cylinder is a perpendicular segment that joins the planes of the bases. The height h of a cylinder is the length of an altitude.

Surface Areas of Cylinders Right cylinder Oblique cylinder

Surface Areas of Cylinders The lateral area of a right cylinder is the product of the circumference of the base and the height of the cylinder. LA = 2πrh or LA = πdh The surface area of a right cylinder is the sum of the lateral area and the areas of the two bases. SA = LA + 2πr2

Surface Areas of Cylinders The radius of the base of a cylinder is 16 in., and its height is 4 in. Find its surface area in terms of π. S.A.= 2πrh + 2πr2= 640 π

Homework Handout on surface area of cylinders

Volumes of Prisms and Cylinders Volume is the space that a figure occupies. It is measured in cubic units. Volume of a prism V = Bh B - area of the base For a rectangular prism, V = (l * w) * h

Volumes of Prisms and Cylinders Volume of a cylinder V = Bh B - area of the base In a cylinder, B= πr2, so V = πr2h.

Volumes of Prisms and Cylinders Find the volume of the figure to the nearest whole number. V = Bh V= (½ *3*5)*6 V= 45 in.3 6 in. 5 in. 3 in.

Volumes of Prisms and Cylinders Find the area of the figure to the nearest whole number. V = πr2h V = π(2)25= 63m3 5 m 2 m

Homework Handouts on the volume of prisms and cylinders

Surface Area Review: Prism – a 3-D polyhedron with 2 bases and a height that connects them Cylinder – a 3-D polyhedron with circles for bases Surface Area – the area of all of the sides of a polyhedron added together Net – a polyhedron unfolded into a flat, 2-dimensional figure

Surface Area of Pyramids A pyramid: Is a polyhedron Has only 1 base that is a polygon Has sides (lateral faces) that are triangles Has sides that meet at a vertex Has a height and a slant height We name pyramids by the shape of the base.

Surface Areas of Pyramids A regular pyramid is a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. The slant height l is the length of the altitude of a lateral face of the pyramid.

Surface Areas of Pyramids The lateral area of a regular pyramid is the sum of the areas of the congruent lateral faces. LA = ½ pl p – perimeter of the base l – slant height The surface area of a regular pyramid is the sum of the lateral area and the area of the base. SA = LA + B

Surface Areas of Pyramids Find the slant height of a square pyramid with base edges 12 cm and altitude 8 cm. First, draw and label the figure. l= 10 cm

Surface Areas of Pyramids Find the lateral area and surface area of the regular square pyramid below. LA = ½ pl = ½ (4*4 sides) 7 = 56 in.2 SA = LA + B = 56 + (4*4) = 72 in.2

Homework Handout on surface area of pyramids

Surface Areas of Cones A cone is like a pyramid, but its base is a circle. The slant height is the distance from the vertex to a point on the edge of the base.

Surface Areas of Cones The lateral area of a right cone is half the product of the circumference of the base and the slant height. LA = ½ (2πr)l = πrl The surface area of a right cone is the sum of the lateral area and the area of the base. SA = LA + πr2

Surface Areas of Cones Find the lateral and surface area of a cone with radius 8 cm and slant height 17 cm. Leave your answer in terms of π. LA = πrl LA= 136π cm2 SA = LA + πr2 SA= 136π + 64π= 200π cm2

Surface Area of Cones Find the Surface Area of a cone with a base with radius 10ft and a slant height of 12ft: LA = πrl LA= 120π ft2 SA = LA + πr2 SA= 120π + 100π= 220π ft2

Homework Handout on Surface Area of Cones

Volumes of Pyramids and Cones Volume of a pyramid V = 1/3 Bh In a cone, B= πr2, so V = 1/3 πr2h.

Volumes of Pyramids and Cones Find the volume of the figure. V = 1/3 Bh V= 1/3 (6*6) *5 V= 60 ft3

Volumes of Pyramids and Cones Find the volume of a cone with diameter 3 m and height 4 m. V = 1/3 πr2h V= 1/3 π(3/2)24 V= 3π m3

Volumes of Pyramids and Cones Find the volume of a square pyramid with base edges 24 in. long and slant height 15 in. V = 1/3 Bh First find h. 122+h2=152 h=9 V= 1/3 (24*24)*9= 1728 in.3

Homework Handouts on Volume of Pyramids and Cones

Surface Areas and Volumes of Spheres A sphere is the set of all points in space equidistant from a given point called the center.

Spheres Half of a sphere is called a hemisphere. How do you think the volume of a hemisphere would compare to that of a sphere?

Spheres What is the relationship between a sphere and a cylinder? Same radius; volume of a sphere is less

Surface Area of a Sphere A sphere is a 3-Dimensional object and a circle is only 2-D (flat). What do we think we can say about the surface areas of the two in comparison to each other?

Surface Area of a Sphere If the surface area of a sphere is more than the area of a circle, then the formula must give us a value greater than that of a circle. Area of a Circle = π r2 Surface area of a Sphere = 4(π r2)

Surface Areas and Volumes of Spheres Surface Area of a Sphere SA = 4πr2 Volume of a Sphere V = 4/3 πr3

Surface Areas and Volumes of Spheres Find the surface area and volume of a sphere whose diameter is 13 cm. r = 13/2 = 6.5 cm SA = 4πr2= 4 π (6.5)2 =530.9 cm2 V = 4/3 πr3 V= 4/3 π (6.5)3 V= 1150.3 cm3

Surface Area of a Sphere A sphere has a total surface area of 676 π in2. What are the radius, the diameter, and the circumference? SA = 4πr2 =676 r =7.33 in d = 14.66 in Circumference= πd =46.0 in

Surface Areas and Volumes of Spheres Find the surface area of a sphere whose circumference is 19π mm. Leave your answer in terms of cm2. Circumference= 19π = 2πr r = 9.5 mm SA = 4πr2 =361 π mm2 SA= 1134.1mm2 * (cm2) = 11.3 cm2 (102mm2)

Surface Areas and Volumes of Spheres Find the volume of a sphere with radius 5 m. V= 4/3 πr3= 523.6 m3

Surface Areas and Volumes of Spheres The volume of a sphere is 288π in.3 Find the surface area. Leave your answer in terms of π.

Homework Handouts on Surface Area and Volume of Spheres

Areas and Volumes of Similar Solids Similar solids – solids with the same shape and all corresponding dimensions proportional Similarity ratio - ratio of corresponding linear dimensions of two similar solids What types of solids will always be similar? Cubes and spheres

Areas and Volumes of Similar Solids If the similarity ratio of two similar solids is a:b, then: The ratio of their surface and/or lateral areas is a²:b². The ratio of their volumes is a³:b³.

Areas and Volumes of Similar Solids Find the similarity ratio of two similar cylinders with surface areas of 98π ft2 and 2π ft2. a²:b² 98π: 2π is equal to 49:1 a:b is equal to 7:1

Areas and Volumes of Similar Solids Two similar square pyramids have volumes of 48 cm3 and 162 cm3. The surface area of the larger pyramid is 135 cm2. Find the surface area of the smaller pyramid. The ratio of their volumes is a³:b³, or 162:48 Therefore, the ratio a:b is equal to 1.5 The ratio of a²:b² is equal to 2.25:1 SA (smaller)= 135/2.25 = 60 cm2

Homework Handout on Similarity

Volumes of Prisms and Cylinders A composite space figure is a three-dimensional figure that is the combination of two or more simpler figures.

Application Problems- Problem 1 A swimming pool is 24 ft long, 18 ft wide and 4.5 ft deep. The water resistant paint needed for the pool costs $1.50 per square foot. How much will it cost to paint the interior surfaces of the pool? How many gallons of water will be needed to fill the pool to the brim (top)? (7.481 gallons/ft3) How many gallons of water will be needed to fill the pool to 6 inches from the top? (7.481 gallons/ft3)

Problem 2 A moving company is trying to store boxes in a storage room with a length of 5 m, width of 3 m and height of 2 m. How many boxes can fit in this space if each is 50 cm long, 60 cm wide and 40 cm high?

Problem 3 Calculate the quantity of sheet metal in square meters that would be needed to make 10 cylindrical canisters with a diameter of 10 cm and a height of 20 cm.

Problem 4 Four cubes of ice with an edge 4 cm each are left to melt in a cylindrical glass with a radius of 6 cm. How high will the water rise when they have melted? (Assume that the total volume of the ice cubes will be equal to the total volume of water when the ice cubes melt.)

Problem 5 The dome of a cathedral has a hemispherical form with a diameter of 50 m. If the restoration costs $300 per m², what is the total cost of the restoration? SAhemisphere=1/2 * 4πr2= 2 π (25)2= 3927.0 m2 3927.0 m2 * $300/m2= $1,178,100.00

Problem 6 Calculate the area and volume of a sphere inscribed in a cylinder with a height of 2 m. Then calculate the volume of the void space. The void space is the volume inside the cylinder that is not taken up by the sphere. SAsphere= 4πr2= 4π(1)2= 12.6 m2 Volumesphere= 4/3 πr3 =4.2 m3 Void Space= Vcylinder-V sphere= πr2h - 4/3 πr3 6.28– 4.2= 2.1 m3

Problem 7 What is the capacity (volume) in gallons of the cylindrical tank pictured below with a hemispherical top and a conical bottom? Round your answer to the nearest gallon. (7.481 gallons/ ft3)

Volumehemisphere= ½ *4/3 πr3 =261.8 ft3 Volumecylinder= πr2 h=942.5ft3 Volumecone= 1/3 πr2h= 78.5 ft3 Total Volume= 1282.8 ft3 *7.481 gal/ft3 = 9,597 gallons

Review- formulas Solid Surface Area Volume Prism LA + 2B p*h + 2B B*h Cylinder 2 r*h + 2π r2 π r2*h Pyramid LA + B ½ p * l + B 1/3B*h Cone π r* l +π r2 1/3 π r2*h Sphere 4π r2 4/3 π r3 Hemisphere ½ (4π r2) ½ (4/3 π r3) π

Similar Solids Ratio Applies to… a:b Linear dimensions (height, radius, diameter, length, width, circumference, perimeter) a2:b2 Area (Surface Area, Lateral Area) a3:b3 Volume, weight