J.BRU

__**Chapter 7.1**__
• RECTANGLE: A solid figure in which all six faces are rectangles.

• Surface Area and Volumes Formulas: The surface area, S, and volume, V, of a right rectangular prism with length, &, width and height, h, are... S = 2&w + 2wh and V = &wh [|Rectangle]

• CUBE: a regular solid figure whose faces are all squares. It has six equal-area faces and 12 equal-length edges The surface area, S, and volume, V, of a cube with side s are... S = 6s² and V = s³ [|CUBE]

__**Chapter 7.2**__
• //__Altitude:__// An altitude of a prism is a segment that has endpoints in the planes containing the bases and that is perpendicular to both planes.

• //__Height:__// The height of s prism is the length of the altitude.

• Surface Area of a Right Prism: The surface area, S, of a right prism with lateral area L, base area, B, perimeter p, and the height h is S = L + 2B or S = hp + 2B

• Cavalieri's Principle: If two solids have equal heights and the cross sections formed by every plane parallel to the bases of both soilds have equal areas, then the two soilds have equal volumes. [|The Man Himself.]

• Volume of a Prism: The volume, V, of a prism with height h and base area B is... V = Bh

**Chapter 7.3**
• //__Pyramids:__// A pyramid is any three-dimensional structure where the upper surfaces are triangular and come together at one point. The base of pyramids are generally any polygon shape. meaning that a pyramid usually has three or four sides. Otherwise known as lateral faces.

• Lateral faces: Are triangles that share a single vertex, called the **__VERTEX OF THE PYRAMID.__** Each lateral face has one edge in common with that base, called a **__BASE EDGE__**. The intersection of two lateral faces is a **__LATERAL EDGE__**.

• Altitude of a pyramid: Is the perpendicular segment from the vertex to the plane of the base. and the height of a pyramid is the length of its altitude.

• REGULAR PYRAMID: In a regular pyramid the base is a regular polygon and the lateral faces are equal triangles. The altitude of a pyramid is the perpendicular distance from the vertex to the base, which intersects the base center. The length of an altitude of a lateral face of a regular pyramid is called that **__SLANT HEIGHT__** of the pyramid.

• Surface Area of a Regular Pyramid: - The surface area, S, of a regular pyramid with lateral area L, base area B, perimeter of the base p, and the slant height & is... S = L + B or S = 1/2 &p + B

• Volume of a pyramid The volume of a pyramid is equal to one third the product of the altitude and the area of the base. The volume V, of a pyramid with height h, ad the base area B is... V=1/3Bh [|Pyramids]

__**Chapter 7.4**__
• //__Cylinders:__// A **__CYLINDER__** is a soild that consist of a circular region and its translated image on a parallel planes, with a **__LATERAL SURFACE__** connecting the cricles. The faces formed by the circular region and its translated image are called the**__BASES__** of the cylinder. An **__ALTITUDE__** of a cylinder is a segment that has endpoints in the planes containing the bases and is perpendicular to both planes The **__HEIGHT__** of a cylinder is the length of a altitude. The **__AXIS__** of a cylinder is a the segment joining the centers of the two bases. If the axis of a cylinder is perpendicular to the bases, then the cylinder is a **__RIGHT CYLINDER.__** If not, it is an **__OBLIQUE CYLINDER.__** [|Cylinder]

• Surface Area of a Right Cylinder: The surface area, S, of a right cylinder with lateral area L, base area B, radius r, and height h is... S = L+2B or S = 2rh + 2 (3.14 )r²

• Volume or a Cylinder: - The volume, V, or a cylinder with the radius r, height h, and the base area B is... V=Bh or V(3.14)r³

**Chapter 7.5**
• __//Cone://__ a cone is a three-dimensional figure that consists or a circular base and a curved lateral surface that connects the base to a single point not in the plane of the base, called a vertex. The altitude of a cone is the perpendicular segment from the vertex to that plane of the base. The height of the cone is that length of the altitude. If the altitude or a cone intersects the case of the cone at its center, the cone is a right cone. If, not. its an oblique cone.

• Surface Area of a Right Cone: The surface area, S, of a right cone with lateral area, L, base of area, B, and the radius, r, and slant & is... S = L+B or S = (3.14) r&+ (3.14)r²**.**

• Volume of a Cone: The voulme, V, of a cone with radius r, height h, and base area B is... V= ¹/³ Bh or V= ¹/³ (3.14) r²h. [|Right and Oblique Cones]

__**Chapter 7.6**__
[|Sphere] • Volume of a Sphere: The volume, V, of a sphere with radius r is... V = 4/3 (3.14)r³.

• Surface Area or a Sphere: The surface area, S,of a sphere with a radius r is... S - 4 (3.14) r².

1. volume of a triangular prism: a regular triangular prism with a height of 3 and a base edge of 2 and a base height of 2 the base area is 3x2. times that by the height of 4 and you get the volume V=8x3 V=24

2. Surface area of a triangular prism: a regular triangular prism with a height of 4 and a base edge of 3 and a base height of 2 find the product of the base areas and add that to the lateral area S=(4x4x2)+(3x2x2) S=44

3. Volume of a pyramid: A regular square pyramid with a base length of 2 and a height of 3 Volume is 1/3 times the base area which is 2x2 times the height which is 3. The 1/3 and the 3 cancel out so the volume equals 4 V=1/3x2x2x3 V=2x2 V=4

4. Surface area of a pyramid: of a square pyramid, slant height of 2 and a base edge of 4 1/2 times the slant height 2 times the base perimeter of 16 + the base area, 16 SA=1/2x2x16+16 SA=16+16 SA=32

5. volume of a cylinder: base area of 10 and a height of 3 the base area of 10 times the height of 3 equals the volume V=10x3 V=30

6. surface area of a cylinder: A cylinder with a radius of 10 and a height of 5 surfface area is pi times the radius times the height times 2 + 2 times pi times 10 squared S=2 [3.14] 10x5 +2 [3.14] 100 S=42928

7. volume of a cone: A cone with a height of 3 and a slant height of 5 find the radius which is 4 square it and multiply that by 1/2 pi and 5 V=4^2x3.14x5x1/2 V=16x3.14x2.5 V=40x3.14 V=125.6

8. surface area of a cone: a cone with a height of 3 and a slant height of 5 radius is 4, multiply it by pi and by the slant height, 5. Add that to pi times 4^2 = 16 S=3.14 x 4 x 5+3.14 x 16 S=3.14 x 20 + 3.14 x 16 S=113.04

9. volume of a sphere: Sphere with a radius of 4 multiply 4/3 by 4^2 and then by pi V=4/3x4^2xpi V=4/3x16xpi V=66xpi V=210.34

10. surface area of a sphere: Sphere with a radius of 3 multiply 16 by 3 and then by pi S=16x3x3.14 S=150x3.14 S=473.26