# Surface Area and Volume of Solids - Andrews University

SURFACE AREA AND VOLUME OF SOLIDS Geometry Chapter 12 This Slideshow was developed to accompany the textbook Larson Geometry By Larson, R., Boswell, L., Kanold, T. D., & Stiff, L. 2011 Holt McDougal Some examples and diagrams are taken from the textbook.

Slides created by Richard Wright, Andrews Academy [email protected] 12.1 EXPLORE SOLIDS Polyhedron Solid with polygonal sides Flat sides Face

Side Edge Line segment Vertex Corner 12.1 EXPLORE SOLIDS Prism Polyhedron with two congruent

surfaces on parallel planes (the 2 ends (bases) are the same) Named by bases (i.e. rectangular prism, triangular prism) Cylinder Solid with congruent circular bases on parallel planes (not a polyhedron)

12.1 EXPLORE SOLIDS Pyramid polyhedron with all but one face intersecting in one point Cone circular base with the other surface meeting in a point (kind of like a pyramid) Sphere all the points that are a given distance from the

center 12.1 EXPLORE SOLIDS Eulers Theorem The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by Convex Any two points can be connected with a segment completely inside the polyhedron Concave

Not convex Has a cave 12.1 EXPLORE SOLIDS Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges and describe as convex or concave 12.1 EXPLORE SOLIDS

Regular polyhedron Polyhedron with congruent regular polygonal faces Only 5 types (Platonic solids) Tetrahedron 4 faces (triangular pyramid) Hexahedron 6 faces (cube) Octahedron 8 faces (2 square pyramids put together)

Dodecahedron 12 faces (made with pentagons) Icosahedron 20 faces (made with triangles) 12.1 EXPLORE SOLIDS Cross Section Imagine slicing a very thin slice of the solid

The cross section is the 2-D shape of the thin slice 12.1 EXPLORE SOLIDS Find the number of faces, vertices, and edges of a regular dodecahedron. Check with Eulers Theorem. Describe the cross section.

798 #2-40 even, 44-60 even = 29 ANSWERS AND QUIZ 12.1 Answers 12.1 Homework Quiz 12.2 SURFACE AREA OF PRISMS AND CYLINDERS Surface area = sum of the areas of each surface of the solid In order to calculate surface area it is sometimes easier

to draw all the surfaces 12.2 SURFACE AREA OF PRISMS AND CYLINDERS Nets Imagine cutting the three dimensional figure along the edges and folding it out. Start by drawing one surface, then visualize unfolding the

solid. To find the surface area, add up the area of each of the surfaces of the net. 12.2 SURFACE AREA OF PRISMS AND CYLINDERS Parts of a right prism Bases parallel congruent surfaces

(the ends) Lateral faces the other faces (they are parallelograms) Lateral edges intersections of the lateral faces (they are parallel) Altitude segment perpendicular planes containing the two bases with an endpoint on each plane Height length of the altitude

Base Lateral Face Lateral Edge Altitude

12.2 SURFACE AREA OF PRISMS AND CYLINDERS Right prism Prism where the lateral edges are altitudes Oblique prism Prism that isnt a right prism

12.2 SURFACE AREA OF PRISMS AND CYLINDERS Lateral Area (L) of Prisms Area of the Lateral Faces L = Ph L = Lateral Area P = Perimeter of base h = Height

12.2 SURFACE AREA OF PRISMS AND CYLINDERS Base Area (B) In a prism, both bases are congruent, so you only need to find the area of one base and multiply by two Surface Area of a Right Prism Where S = surface area, B = base area, P =

perimeter of base, h = height of prism 12.2 SURFACE AREA OF PRISMS AND CYLINDERS Draw a net for a triangular prism. Find the lateral area and surface area of a right rectangular prism with height 7 inches, length 3 inches, and width 4 inches.

12.2 SURFACE AREA OF PRISMS AND CYLINDERS Cylinders are the same as prisms except the bases are circles Lateral Area = L = 2rhrh Surface Area of a Right Cylinder Where S = surface area, r = radius of base, h = height of prism

12.2 SURFACE AREA OF PRISMS AND CYLINDERS The surface area of a right cylinder is 100 cm2. If the height is 5 cm, find the radius of the base. Example: Draw a net for the cylinder and find its surface area. 2 5

806 #2-28 even, 31-37 all = 21 ANSWERS AND QUIZ 12.2 Answers 12.2 Homework Quiz 12.3 SURFACE AREA OF PYRAMIDS AND CONES Pyramids All faces except one intersect at one

point called vertex The base is the face that does not intersect at the vertex Lateral faces faces that meet in the vertex Lateral edges edges that meet in the vertex Altitude segment that goes from the vertex and is perpendicular to the base

12.3 SURFACE AREA OF PYRAMIDS AND CONES Regular pyramid base is a regular polygon and the vertex is directly above the center of the base In a regular pyramid, all the lateral faces are congruent isosceles triangles The height of each lateral face is called the slant height () Lateral Area L = P Surface Area of a Regular Pyramid

Where B = base area, P = base perimeter, = slant height 12.3 SURFACE AREA OF PYRAMIDS AND CONES Find the surface area of the regular pentagonal pyramid. 12.3 SURFACE AREA OF PYRAMIDS AND CONES Cones

Cones are just like pyramids except the base is a circle Lateral Area = rhr Surface Area of a Right Cone Where r = base radius, = slant height 12.3 SURFACE AREA OF PYRAMIDS AND CONES Example: The So-Good Ice Cream Company makes Cluster Cones. For packaging, they must cover

each cone with paper. If the diameter of the top of each cone is 6 cm and its slant height is 15 cm, what is the area of the paper necessary to cover one cone? 814 #2-32 even, 35-39 all = 21 Extra Credit 817 #2, 6 = +2 ANSWERS AND QUIZ

12.3 Answers 12.3 Homework Quiz 12.4 VOLUME OF PRISMS AND CYLINDERS

Create a right prism using geometry cubes Count the lengths of the sides Count the number of cubes. Remember this to verify the formulas we are learning today. 12.4 VOLUME OF PRISMS AND CYLINDERS Volume of a Prism Where B = base area, h = height of prism

Volume of a Cylinder Where r = radius, h = height of cylinder 12.4 VOLUME OF PRISMS AND CYLINDERS Find the volume of the figure 12.4 VOLUME OF PRISMS AND CYLINDERS

Find the volume. 12.4 VOLUME OF PRISMS AND CYLINDERS There are 150 1-inch washers in a box. When the washers are stacked, they measure 9 inches in height. If the inside hole

of each washer has a diameter of inch, find the volume of metal in one washer. 12.4 VOLUME OF PRISMS AND CYLINDERS Cavalieris Principle If two solids have the same height and the same crosssectional area at every level, then they have the same volume.

Find the volume. 822 #2-40 even = 20 ANSWERS AND QUIZ 12. 4 Answers 12.4 Homework Quiz 12.5 VOLUME OF PYRAMIDS AND CONES

How much ice cream will fill an ice cream cone? How could you find out without filling it with ice cream? What will you measure? 12.5 VOLUME OF PYRAMIDS AND CONES Volume of a Pyramid Where B = base area, h = height of pyramid Volume of a Cone

Where r = radius, h = height of cone 12.5 VOLUME OF PYRAMIDS AND CONES Find the volume. 832 #2-30 even, 34, 36, 40, 44-52 even = 23 Extra Credit 836 #2, 4 = +2 ANSWERS AND QUIZ

12.5 Answers 12.5 Homework Quiz 12.6 SURFACE AREA AND VOLUME OF SPHERES Terms Sphere all points equidistant from center Radius segment from center to surface

Chord segment that connects two points on the sphere Diameter chord contains the center of the sphere Tangent line that intersects the sphere in exactly one place 12.6 SURFACE AREA AND VOLUME OF SPHERES

Intersections of plane and sphere Point plane tangent to sphere Circle plane not tangent to sphere Great Circle plane goes through center of sphere (like equator) Shortest distance between two points on sphere Cuts sphere into two hemispheres

12.6 SURFACE AREA AND VOLUME OF SPHERES Surface Area of a Sphere Where r = radius If you cut 4 circles into 8ths you can put them together to make a sphere Volume of a Sphere Where r = radius

12.6 SURFACE AREA AND VOLUME OF SPHERES Find the volume of the empty space in a box containing three golf balls. The diameter of each is about 1.5 inches. The box is 4.5 inches by 1.5 inches by 1.5 inches.

842 #2-36 even, 40-44 even = 21 ANSWERS AND QUIZ 12.6 Answers 12.6 Homework Quiz 12.7 EXPLORE SIMILAR SOLIDS Russian Matryoshka dolls

nest inside each other. Each doll is the same shape, only smaller. The dolls are similar solids. 12.7 EXPLORE SIMILAR SOLIDS Similar Solids Solids with same shape but not necessarily the

same size The lengths of sides are proportional The ratios of lengths is called the scale factor 12.7 EXPLORE SIMILAR SOLIDS Congruent Solids Similar solids with scale factor of 1:1

Following four conditions must be true Corresponding angles are congruent Corresponding edges are congruent Areas of corresponding faces are equal The volumes are equal 12.7 EXPLORE SIMILAR SOLIDS Determine if the following pair of shapes are similar, congruent or neither.

Cone A: r = 4.3, h = 12, slant height = 14.3 Cone B: r = 8.6, h = 25, slant height = 26.4 Ratios: , . Not proportional so neither Right Cylinder A: r = 5.5, height = 7.3 Right Cylinder B: r = 5.5, height = 7.3 1:1 ratio so congruent. 12.7 EXPLORE SIMILAR SOLIDS Similar Solids Theorem

If 2 solids are similar with a scale factor of a:b, then the areas have a ratio of a2:b2 and the volumes have a ratio of a3:b3 12.7 EXPLORE SIMILAR SOLIDS Cube C has a surface area of 216 square units and Cube D has a surface area of 600 square units. Find the scale factor of C to D. Find the edge length of C. Use the scale factor to find the volume of D.

850 #2-26 even, 30-48 even = 23 Extra Credit 854 #2, 4 = +2 ANSWERS AND QUIZ 12.7 Answers 12.7 Homework Quiz 12.REVIEW

861 #1-17 all = 17

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