Surface area of Spheres
Finding the surface area of a sphere
In lesson 10.7 a circle was described as a locus of points in a plane that are a given distance from a point. A sphere is the locus of points space that are a given distance from a point.
Finding the surface area of a sphere
A diameter is a chord that contains the center. As with all circles the terms radius and diameter also represent distances and the diameter is twice the radius
Theorem 12.11: Surface Area of s sphere
The surface area of a sphere with readius r is S=4πr2
Thursday, May 29, 2008
Secants and tangents
Lesson 8-5
Central Angle
Central Angle Theorem
The measure of a center angle is equal to the measure of the intercepted arc.
Inscribed Angle
Intercepted Arc
Secants, Tangents, and Angle Measures
Secant – A line that intersects a circle in two points
Interior Angle Theorem
Exterior Angles
Exterior Angle Theorem
Example: Exterior Angle Theorem
Central Angle
Central Angle Theorem
The measure of a center angle is equal to the measure of the intercepted arc.
Inscribed Angle
Intercepted Arc
Secants, Tangents, and Angle Measures
Secant – A line that intersects a circle in two points
Interior Angle Theorem
Exterior Angles
Exterior Angle Theorem
Example: Exterior Angle Theorem
May 16thh
•May 16th
•Finding the surface area of a pyramid.
•A pyramid is a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex. The intersection of two lateral faces is a lateral edge. The intersection of the base and a lateral face is a base edge. The altitude or height of a pyramid is the perpendicular distance between the base and the vertex.
•More on Pyramids
A regular pyramid has a regular polygon for a base and its height meet the base at its center. The slant height of a regular pyramid is the altitude of any lateral face. A non-regular pyramid does not have a slant height.
•Ex 1. Finding the Area of Lateral Face.
•Architecture. The lateral faces of the Pyramid Arena in Memphis, Tennessee are covered with steel panels. Use the diagram of the arena to find the area of each lateral face of this regular pyramid.
•Finding the surface area of a pyramid.
•A pyramid is a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex. The intersection of two lateral faces is a lateral edge. The intersection of the base and a lateral face is a base edge. The altitude or height of a pyramid is the perpendicular distance between the base and the vertex.
•More on Pyramids
A regular pyramid has a regular polygon for a base and its height meet the base at its center. The slant height of a regular pyramid is the altitude of any lateral face. A non-regular pyramid does not have a slant height.
•Ex 1. Finding the Area of Lateral Face.
•Architecture. The lateral faces of the Pyramid Arena in Memphis, Tennessee are covered with steel panels. Use the diagram of the arena to find the area of each lateral face of this regular pyramid.
Equations of a Circle
Objectives
Develop and use the equation of a circle.
Adjust the equation for a circle to move the center in a coordinate plane.
The Equation of a Circle
Given the definition of a circle: The set of all points in a plane that are equidistant from a given point called the center, we know that we are dealing with points in the coordinate plane that are a given distance, r, from a single point, the center, that we can represent with a set of coordinates that we’ll call (h, k) so that we can use them anywhere in the plane.
The Equation of a Circle, Cont.
Since we are working with Distance, the Distance Formula will be used.
The Equation of a Circle, Cont.
An equation for a circle with center at (h, k), and a radius of r units is:
Key Skills
Key Skills
Develop and use the equation of a circle.
Adjust the equation for a circle to move the center in a coordinate plane.
The Equation of a Circle
Given the definition of a circle: The set of all points in a plane that are equidistant from a given point called the center, we know that we are dealing with points in the coordinate plane that are a given distance, r, from a single point, the center, that we can represent with a set of coordinates that we’ll call (h, k) so that we can use them anywhere in the plane.
The Equation of a Circle, Cont.
Since we are working with Distance, the Distance Formula will be used.
The Equation of a Circle, Cont.
An equation for a circle with center at (h, k), and a radius of r units is:
Key Skills
Key Skills
May 5th
•Math Notes
•May 5th, 2008
•Area Investigation
•what is the area of a triangle?
•1/2 bh.
•What if the triangle is an equilateral triangle then what is the formula
•Area Of an Equilateral Triangle
•The area of an equalateral is one fourth the square of the length of the side times the square root of 3.
•Area Investagation
•In a regular polygon, how many triangles can be formed with a vertex at the center and one side along the side of the polygon?
•Definitions
•Apothem- the distance from the center to any side of the polygon.
•Area Investigation
•calculate the area of a regular pentagon, in terms of the side length, s, and the apothem.
•
•in terms of s and a what is the area of a heptagon?
•
•in terms of s and a, what is th earea of an ngon?
•Tetrahedron
•A regular four sided 3-D shape.
•4 faces
•6 vertices
•Tetrahedron net
•Tetrahedron model
•Spin a tetrahedron
•Area Investigation
•Calculate the area of a regular pentagon, in terms of the side, s, and the apothem
•
•The area of each triangle is 1/2 sa
•
•There are 5 triangles, so the total area is 1/2 sa (5)
•
•In terms of s and a what is the area of a heptagon?
•
•In terms of s and a what is the area of a n-gon?
•Cube
•6 faces
•8 vertices
•12 edges
•Cube net
•Cube model
•Spin a cube
•Octahedron
•8 faces
•6 vertices
•12 edges
•Octahedron net
•Octahedron model
•Spin a octahedron
•May 5th, 2008
•Area Investigation
•what is the area of a triangle?
•1/2 bh.
•What if the triangle is an equilateral triangle then what is the formula
•Area Of an Equilateral Triangle
•The area of an equalateral is one fourth the square of the length of the side times the square root of 3.
•Area Investagation
•In a regular polygon, how many triangles can be formed with a vertex at the center and one side along the side of the polygon?
•Definitions
•Apothem- the distance from the center to any side of the polygon.
•Area Investigation
•calculate the area of a regular pentagon, in terms of the side length, s, and the apothem.
•
•in terms of s and a what is the area of a heptagon?
•
•in terms of s and a, what is th earea of an ngon?
•Tetrahedron
•A regular four sided 3-D shape.
•4 faces
•6 vertices
•Tetrahedron net
•Tetrahedron model
•Spin a tetrahedron
•Area Investigation
•Calculate the area of a regular pentagon, in terms of the side, s, and the apothem
•
•The area of each triangle is 1/2 sa
•
•There are 5 triangles, so the total area is 1/2 sa (5)
•
•In terms of s and a what is the area of a heptagon?
•
•In terms of s and a what is the area of a n-gon?
•Cube
•6 faces
•8 vertices
•12 edges
•Cube net
•Cube model
•Spin a cube
•Octahedron
•8 faces
•6 vertices
•12 edges
•Octahedron net
•Octahedron model
•Spin a octahedron
12-1 & 12-2
•12-1 & 12-2: Space Figures, Nets, & Diagrams.
•Objective: To recognize nets and to make isometric drawings of space figures.
•Polyhedron
•A three dimensional figure whose surfaces are polygons.
- Faces: the polygons making up the sides.
-Edges: A segment that is formed by the intersection of two faces.
-Vertex: a point where the three or more edges intersect.
•Euler’s Formula
F+V=E+2
5+6=9+2
11=11
•Space figures and Nets
Net: A two dimensional pattern that you can fold to form a three dimensional figure.
•Cross-Sections
•Cross Section: the intersection of a solid and a plane.
•Objective: To recognize nets and to make isometric drawings of space figures.
•Polyhedron
•A three dimensional figure whose surfaces are polygons.
- Faces: the polygons making up the sides.
-Edges: A segment that is formed by the intersection of two faces.
-Vertex: a point where the three or more edges intersect.
•Euler’s Formula
F+V=E+2
5+6=9+2
11=11
•Space figures and Nets
Net: A two dimensional pattern that you can fold to form a three dimensional figure.
•Cross-Sections
•Cross Section: the intersection of a solid and a plane.
Monday, May 12, 2008
notes for the 12th
12.2 Surface Area of Prisms & Cylinders
Prism
A polyhedron with two congruent faces (bases) that lie in parallel planes.
- The lateral faces are all parallelogram.
- The height is the perpendicular distance between the bases.
Surface Area: The sum of the area of its faces
Lateral Area: The sum of the area of the lateral faces.
Surface Area of a Right Prism
The surface area, S, of a right prism is:
S=2B+Ph
B= area of one base.
P= perimeter of one base.
H=height of the prism.
2B represents the “base area”
Ph represents the “lateral area”
Find lateral area & surface area
Prism
A polyhedron with two congruent faces (bases) that lie in parallel planes.
- The lateral faces are all parallelogram.
- The height is the perpendicular distance between the bases.
Surface Area: The sum of the area of its faces
Lateral Area: The sum of the area of the lateral faces.
Surface Area of a Right Prism
The surface area, S, of a right prism is:
S=2B+Ph
B= area of one base.
P= perimeter of one base.
H=height of the prism.
2B represents the “base area”
Ph represents the “lateral area”
Find lateral area & surface area
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