CHAPTER 12 SOLID

GEOMETRY

12.1 SOLIDS A Polyhedron- (polyhedra or

polyhedrons) Is formed by 4 or more polygons (faces)

that intersect only at the edges. Encloses a region in space. Includes prisms and pyramids. Does not include cylinders, cones, and

spheres.

EULER’S FORMULA/ THEOREM V vertices E edges F faces

For any polyhedron, V - E + F = 2

REGULAR POLYHEDRON Regular polyhedron- all faces are

congruent, regular polygons Platonic solids are regular polyhedrons. Tetrahedron (4 faces)- 3 triangles Cube (6 faces)- 3 squares Octahedron (8 faces)- 4 triangles Dodecahedron (12 faces)- 3 pentagons Icosahedron (20 faces)- 5 triangles

Convex- any 2 points on the surface can be connected by a segment that lies entirely inside or on the polyhedron.

Cross section- intersection of a plane and a solid.

IS IT A POLYHEDRON?

.

Find the number of faces, vertices, and edges of the regular octahedron.

Check your answer using Euler’s Theorem.

Describe the shape formed by the intersection of the plane and the polygon.

12.2 SURFACE AREA OF PRISMS Lateral faces, edges, bases, lateral area

and surface area

Right prism- lateral edges are perpendicular to both bases

Oblique prism- lateral edges are not perpendicular to the bases

SA = 2B + Ph

NETS

12.2 SURFACE AREA OF CYLINDERS height, base areas, lateral area, surface

area

Right cylinder- the side of the cylinder is perpendicular to the bases.

Oblique cylinder- the side of the cylinder is not perpendicular to the bases

SA = 2B + Ch or SA = 2∏r2 + 2∏rh

EXAMPLES

12.3 SURFACE AREA OF PYRAMIDS Regular Pyramid- a regular polygon for

the base and the segment from the common vertex to the base is perpendicular

Lateral faces- congruent isosceles triangles

Slant height, l- the altitude of the lateral face (triangle)

LA = Pl/2 SA = Pl/2 + B

SURFACE AREA OF CONES Cone- formed by a circular base and a

curved surface that connects the base to a vertex

The radius of the base is the radius of the cone.

The height of the cone, h, is the perpendicular distance from the vertex to the base.

The slant height is the distance from the vertex to a point on the circle of the base.

LA = Cl/2 SA = Cl/2 + B or = ∏r2 + ∏rl

A regular square pyramid has a height of 15 centimeters and a base edge length of 16 centimeters. Find the area of each lateral face of the pyramid

Find the surface area of the regular hexagonal pyramid.

Find the surface area of the right cone shown.

Find the surface area of the cone.

12.4 VOLUME POSTULATES The volume of a cube is the cube of the

length of its sides. V = s3

If 2 polyhedra are congruent, then they have the same volume.

The volume of a solid is the sum of the volumes of all its non-overlapping parts.

CAVALIERI’S PRINCIPLE If 2 solids have the same height and the

same cross-sectional area at every level, then they have the same volume.

VOLUME OF A PRISM V = Bh

Find the volume of a right hexagonal prism with a height of 7cm and side length (of the hexagon) equal to 12cm.

VOLUME OF A CYLINDER V = Bh = ∏r2h

Fi nd the volume of a right cylinder with a height of 10ft and a radius of 40ft.

Find the volume of the oblique cylinder.

12.5 VOLUME OF PYRAMIDS AND CONES Pyramids: V = (1/3) Bh

Cones: V = (1/3)Bh or (1/3) ∏r2 h

EXAMPLES Find the volume of the hexagonal

pyramid.

EXAMPLE Two cones sharing a common base have

a radius of 10mm. One of the cones is 16mm high and the other is 18mm. Find the volume.

16mm18mm

r = 10mm

12.6 SPHERES Definitions- center, radius, chord,

diameter,

Great circle- the intersection formed by a plane that intersects the sphere through its center

Hemisphere- two halves of the sphere

SURFACE AREA OF A SHPERE SA = 4πr2

Find the surface area of the sphere.

VOLUME OF A SPHERE V = 4πr3/3

Find the volume of a beach ball with a diameter of 15in.

EXAMPLE Find the surface area of a globe that has

a circumference of 18π inches.

EXAMPLE: FIND THE VOLUME OF THE COMPOSITE