Geometry Chapter 10 NC

A net is a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure. To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face.

A cross section is the intersection of a three-dimensional figure and a plane.

A cone is formed by a circular base and a curved surface that connects the base to a vertex.

A cube is a prism with six square faces. Other prisms and pyramids are named for the shape of their bases.

A cylinder is formed by two parallel congruent circular bases and a curved surface that connects the bases.

A pyramid is formed by a polygonal base and triangular faces that meet at a common vertex.

Volume of Cones: The volume of a cone with base area B, Radius R, and Height H is V=1/3Bh or V=1/3πr2h.

A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle dividers a sphere into two hemispheres. The figure shows a hemisphere and a cylinder with a cone removed from its interior. The cross sections have the same area at every level, so the volumes are equal by Cavalleri's Principle. You will prove that the cross sections have equal areas in exercise 39.

There are many ways to represent a three-dimensional object. An orthographic drawing shows six different views of an object: top, bottom, front, back, left side, and right side.

Isometric drawing is a way to show three sides of a figure from a corner view. You can use isometric dot paper to make an isometric drawing. This paper has diagonal rows of dots that are equally spaced in a repeating triangular pattern.

The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis.

Lateral Area and Surface Area of Right Cylinders: The later area of a right cylinder with radius r and height h is L=2πrh. The surface area of a right cylinder with lateral area L and base area B is S=L+2B, or S=2πRh + 2πr^2

The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism. The base of the rectangle is equal to the perimeter of the base of the prism.

Lateral Area and Surface Area of Right Prisms: The lateral area of a right prism with base perimeter P and height h is L=Ph. The surface area of a right prism with lateral prism with lateral area L and base area B is S=L+2B or S=Ph +2B. The surface area of a cube with edge length s is s=6s^2

In a perspective drawing, non-vertical parallel lines are drawn so that they meet at a point called a vanishing point. Vanishing points are located on a horizontal line called the horizon. A one-point perspective drawing contains one vanishing point. A two-point perspective drawing contains two vanishing points.

Prisms and cylinders have 2 congruent parallel bases. A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has least one non rectangular lateral face.

Volume of a Sphere: The volume of a sphere with radius r is V=4/3πr3.

Surface Area of a Sphere: The surface area of a sphere with radius r is S=4πr2.

An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude.

Surface area is the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces.

The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.

The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid.

Lateral and Surface Area of a Regular Pyramid: The lateral area of a regular pyramid with perimeter P and slant height L is L=1/2Pl The surface area of a regular pyramid with lateral area L and base area B is S=L+B or S=1/2Pl+B

The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the verge and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base.

Lateral and Surface Area of a Right Cone: The lateral area of a right cone with radius r and slant height l is L= πrl. The surface area of a right cone with lateral area l and base area b is S=L+B, or S=πr+πr2

The volume of a three dimensional figure is the number of non-overlapping unit cubes of a given size that will exactly fill the interior.

A prism is formed by two parallel congruent polygonal faces called bases connected by faces that are parallelograms.

Three-Dimensional figures, or solids can be made up of flat or curved surfaces. Each flat surface is called a face. An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more faces.

Volume of a Prism: The volume of a prism with base area B and height h is V=Bh. The volume of a right rectangular prism with length l, width w, and height h is V=lwh. The Volume of a cube with edge length s is V=s3

Volume of a Cylinder: The volume of a cylinder with base area B, radius R, and Height H is V=Bh or, Vπr2h.

The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown. The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube.

Volume of a Pyramid: The volume of a pyramid with base area B and height H is V=1/3Bh.