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= = GeAn210 -- Link to Chapter 9 Wiki.

7.1 - __Surface Area and Volume__ S = 2lw + 2wh + 2l V = lwh.
 * Surface Area of a right rectangular prism:**
 * The Volume of a right rectangular prism:**

Where l is the length, w is the width, and h is the height.

Click [|here] for problems using surface area for a rectangular prism.

S = 6s² V = s³ Where "S" stands for the number of sides.
 * The Surface Area of a cube is:**
 * The Volume for a cube is:**

To see a net for a cube click [|here.]

**7.2** **-** __Surface Area and Volume of Prisms__
An __altitude__ of a prism is a segment that has endpoints in the planes containing the bases and that is perpendicular to both planes.

The __height__ of a prism is the length of an altitude.

S = L + 2B or S hp + 2B
 * Surface Area of a Right Prism:**

If two solids have equal heights and the cross sections formed by every plane parallel to the bases of both solids have equal areas, then the two solids have equal volumes.
 * Cavalieri's Principle:**

V = Volume, h Height, and B Base area. To find the volume of a prism, plug in the equation: V = Bh.
 * Volume of a Prism:**

**7.3** - __Surface Area and Volume of Pyramids__
A **pyramid** is a polyhedron consisting of a **base,** which is a polygon, and three more **lateral faces**. The lateral faces are triangles that share a single vertex, which is known as the **vertex of a pyramid.** Each lateral face has a **base edge,** which is one common edge with the lateral face and base. The intersection of two lateral faces is a **lateral edge**. The **altitude** of a pyramid is the perpendicular segment from the vertex to the plane of the base. The **height** of a pyramid is the length of its altitude. A **regular pyramid** is a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. The length of an altitude of a lateral face of a regular of a regular pyramid is called the **slant height.**

S = L + B or S 1/2lp + B. V = 1/3Bh.
 * Surface Area of a Regular Pyramid:**
 * Volume of a Pyramid:**

To see an example of a pyramid click [|here.]

**7.4 -** __Surface Area and Volume of Cylinders__
A **cylinder** is a solid that consists of a circular region and its translated image on a parallel plane, with a **lateral surface** connecting the circles. The faces formed by the circular region and it's translated image are called the **bases**. 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 an altitude. The **axis** of a cylinder is 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 it's not a **right cylinder,** then it is an **oblique cylinder.

Surface Area of a Right Cylinder:** S = L + 2B or S 2πrh + 2πr²

V = Bh or V πr²h
 * Volume of a Cylinder:**

Where l is the length, b is the base, h is the height, and π is Pi.

**7.5 -** __Surface Area and Volume of Cones__
A **cone** is a 3-D figure that consists of a circular **base** and a curved **lateral surface** that connects the base to a single point not in the plane of the base, called the **vertex.** The The **height** of the cone is the length of the altitude. If the altitude of a cone intersects the base of the cone at its center, the cone is a **right cone**. If it's not a right cone, then it's an
 * altitude** of a cone is the perpendicular segment from the vertex to the plane of the base.
 * oblique cone.** Below is a picture of an oblique cone:



S = L + B or S πrl + πr²
 * Surface Area of a Right Cone:**

V = 1/3Bh or V 1/3πr²
 * Volume of a Cone:**

**7.6 -** __Surface Area and Volume of Spheres__
A **sphere** is the set of all points in a space that are the same distance, r, from a given point known as the center of the sphere.

V = 4/3πr³ S = 4πr²
 * Volume of a Sphere:**
 * Surface Area of a Sphere:**

Where r is the radius and π is Pi.

[|Surface Area of a Sphere] [|Volume of a Sphere]



**Examples:**
1. V = l x w x h Example: = 12 x 7 x 10 = 840 units³

2. SA = 2lh + 2lw + 2wh = 2x7x10 + 2x7x4 + 2x4x10 = 140 + 56 + 80 = 276units³

3. V = 1/3BH Example: = 1/3 X 39 X 17 = 12.999 X 17 = 220.9996 units

4. SA = 1/2lp + B Example: = 1/2 X 6 X 5 + 20 = 3 X 5 X 20 = 15 X 20 SA = 300 units²

5. V = πr²h Example: = π7²12 = π X 49 X 12 = 588π units

6. SA = 2πrh + 2πr² = 2π9x10 + 2π9² = 2π90 + 2π81 = 565.2 + 508.68 = 1,073.88 units²

7. V = 1/3Bh Example: 1/3πr²h = 1/3π X 1² X 1/2 = 1/6π units

8. SA = πrl + πr² Example: = π X 7 X 15 + π 7² = 105π + 49π = 154π units²

9. V = 4/3πr³ = 4/3π33³ = 14,374π / 3 = 47,916π units³

10. SA = 4πr² = 4π x 8² = 4π x 64 = 256π²