3neunic

=**This page will teach you about the surface area and volume of solid 3-D shapes.**=

Link to volume and surface area [|equations] for many different shapes.

__Section 1: Surface Area and Volume__
Link to a fun [|interactive site] to practice surface area and volume.

•Develop the concepts of maximizing volume and minimizing surface area.
 * Objective:**

•//Surface Area// -The number of all the sides of the shapes you can see on the object. •//Volume// -The number of nonoverlapping shapes in the object.
 * Vocab:**

S - surface area V - volume l - length w - width h - height s - side
 * Key:**

•Surface area for a right rectangular prism: //S = 2lw + 2wh + 2lh// •Volume for a right rectangular prism: //V = lwh// •Surface area for a cube: //S = 6s²// •Volume for a cube: //V = s³//
 * Formulas:**

•A packaging company is deciding between two different box sizes, to see which one has the greater surface area. -Box Size A has the dimensions of 8 inches high x 4 inches wide x 5 inches long. -Box Size B has the dimensions of 10 inches high x 2 inches wide x 8 inches long. Which one has the greater surface area? •Both boxes have a volume of 160 cubic inches. •To find the surface area of Box Size A: //2(8)(4) + 2 (4)(5) + 2(5)(8) = 184// square inches. •To find the surface area of Box Size B: //2(10)(2) + 2(2)(8) + 2(8)(10) = 232// square inches. •Therefore, Box Size B has the greater surface area.
 * Example:**
 * Solution:**

__Section 2: Surface Area and Volume of Prisms__
Link to a site about the area and volume of [|solids].

•Define and use a formula for finding the surface area of a right prism. •Define and use a formula for finding the volume of a right prism. •Use Cavalieri's Principle to develop a formula for the volume of a right or oblique prism.
 * Objectives:**

•//Altitude// -A segment that has endpoints in the planes containing the bases and that is perpendicular to both planes. •//Height// -The length of an altitude. •//Cavalieri's Principle// -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.
 * Vocab:**

S - Surface Area V - Volume L - Lateral Area B - Base Area p - Perimeter h - Height
 * Key:**

•Surface area of a right prism: //S = L + 2B// or //S = hp + 2B// •Volume of a prism: //V = Bh//
 * Formulas:**

•An aquarium in the shape of a right rectangular prism has dimensions of 110 x 50 x 7 feet. If 1 gallon equals about 0.134 cubic feet... How many gallons of water will the aquarium hold? •The volume of the aquarium is found using the volume formula. //V = Bh = lwh = (110)(50)(7) = 38,500// feet. •To get the exact volume in gallons, divide above answer by 0.134. //V = 38,500 ÷ 0.134 = 287,313// gallons.
 * Example:**
 * Solution:**

__Section 3: Surface Area and Volume of Pyramids__
Link to a cool [|help] site.

•Define and use a formula for the surface area of a regular pyramid. •Define and use a formula for the volume of a pyramid.
 * Objectives:**

•//Pyramid// -A polyhedron consisting of a base and three or more lateral faces. •//Base// -A polygon. •//Lateral Faces// -Triangles that share a single vertex. •//Vertex of the Pyramid// -The single vertex from above definition. •//Base Edge// -The one edge that all lateral faces have in common. •//Lateral Edge// -The intersection of two lateral faces. •//Altitude// -The perpendicular segment from the vertex to the plane of the base. •//Height// -The length of the altitude. •//Regular Pyramid// -A pyramid whose base is a regular polygon and whose lateral faces are congruent isoceles triangles. •//Slant Height// -The length of an altitude of a lateral face of a regular pyramid.
 * Vocab:**

S - Surface Area L - Lateral Area B - Base Area p - Perimeter of the Base l - Slant height
 * Key:**

•Surface area of a regular pyramid: //S = L + B// or //S = 1/2lp + B// •Volume of a pyramid: //V = 1/3Bh//
 * Formulas:**

•The roof of a building is a regular octagonal pyramid with a base edge of 4 feet and a slant height of 6 feet. Find the area of the roof. •The area of the roof is the lateral area of the pyramid. //L = 1/2lp = 1/2(6)(8 × 4) = 96// square feet.
 * Example:**
 * Solution:**

__Section 4: Surface Area and Volume of Cylinders__
Link to a [|word problem] site about cylinders.

•Define and use a formula for the surface area of a right cylinder. •Define and use a formula for the volume of a cylinder.
 * Objectives:**

•//Cylinder// -A solid that consists of a circular region and its translated image on a parallel plane, with a lateral surface connecting the circles. •//Lateral Surface// -What connects the two circles of a cylinder. •//Bases// -The faces formed by the circular region and its translated image. •//Altitude// -A segment that has endpoints in the planes containing the bases and is perpendicular to both planes. •//Height// -The length of the altitude. •//Axis// -The segment joining the centers of both bases. •//Right Cylinder// -If the axis of a cylinder if perpendicular to the bases. •//Oblique Cylinder// -If the axis of a cylinder if not perpendicular to the bases.
 * Vocab:**

S - Surface Area L - Lateral Area B - Base Area r - Radius h - Height
 * Key:**

•Surface area of a right cylinder: //S = L + 2b// or //S = 2//π//rh + 2//π//r² •//Volume of a cylinder: //V = Bh// or //V =// π//r²h//
 * Formulas:**

•A penny is a right cylinder with a diameter of 19.05 millimeters and a thickness of 1.55 millimeters. What is the surface area of the penny? •The radius of the penny is HALF the diameter. //19.05 ÷ 2 = 9.525// millimeters •Use the surface area formula to find the surface area of the penny. //S = 2//π//rh + 2//π//r² S = 2//π//(9.525)(1.55) + 2//π//(9.525)² = 663.46// square millimeters
 * Example:**
 * Solution:**

__Section 5: Surface Area and Volume of Cones__
Link to a [|cool cone] site.

•Define and use the formula for the surface area of a cone. •Define and use the formula for the volume of a cone.
 * Objectives:**

•//Cone// -A three dimensional figure that consists of a circular base and a curved lateral surface. •//Base// -The circular face on a cone. •//Lateral Surface// -Connects the base to a single point not in the plane of the base. •//Vertex// -The point not in the plane of the base. •//Altitude// -The perpendicular segment from the vertex to the plane of the base. •//Height// -The length of the altitude. •//Right Cone// -If the altitude of a cone intersects the base of the cone at its center. •//Oblique Cone// -If the altitude of a cone DOES NOT intersect the base of the cone at its center.
 * Vocab:**

S - Surface Area L - Lateral Area B - Base of Area V - volume r - Radius l - Slant Height h - Height
 * Key:**

•Surface area of a right cone: //S = L + B// or //S =// π//rl +// π//r²// //V = 1/3Bh// or //V = 1/3//π//r²h//
 * Formulas:**
 * •**Volume of a cone:

•Find the surface area of a right cone with the measurements of l = 15 and r = 7. •The lateral area is //C = 2//π//l = 30//π. •The portion of the circular region occupied by the sector is //c/C = 14//π///30//π //= 7/15.// •Calculate the lateral area. π//l² = 225//π //L = 7/15 × 225//π //= 105//π •Calculate the base area and add the lateral area. //B =// π//r² = 49//π //B + L = 49//π //+ 105//π //= 154//π //= 483.8//
 * Example:**
 * Solution:**
 * •**The circumference of the base is //c = 2//π//r = 14//π.

__Section 6: Surface Area and Volume of Spheres__
Link to a cool sphere [|help site].

•Define and use the formula for the surface area of a sphere. •Define and use the formula for the volume of a sphere.
 * Objectives:**

//•Sphere// -The set of all points in space that are the same distance from a given point. //•Annulus// -A ring shaped figure.
 * Vocab:**

V - Volume S - Surface Area r - Radius
 * Key:**

•Volume of a sphere //V = 4/3πr³// •Surface area of a sphere //S = 4πr²//
 * Formulas:**

•A certain helium balloon has a radius of 27 feet when it's fully inflated. How many cubic feet of helium air can it hold? //V = 4/3πr³ = 4/3π(27)³ = 4/3(19,683)π = 26,244π = 82,488// cubic feet
 * Example:**
 * Solution:**

__Section 7: Three-Dimensional Symmetry__
Link to a cool [|3-D] site.

•Define various transformations in three-dimensional space. •Solve problems by using transformations in three-dimensional space.
 * Objectives:**

•//Three-Dimensional Figure// -The measurement of something in three directions such as length, width, and height.
 * Vocab:**

•You are given AB with endpoints A (0, 5, 0) and B (0, 5, 5,). Describe and give the dimensions of the figure. •a. AB is rotated around the z-axis. •b. AB is rotated around the y-axis. •a. AB is rotated about the z-axis, forming the lateral surface of a cylinder with a radius of 5 and a height of 5. •b. AB is rotated about the y-axis, forming a circular region with a radius of 5.
 * Example:**
 * Solution:**