Volume & Surface Area MATH 102 Contemporary Math S. Rook

Overview Section 10.4 in the textbook: – Volume – Surface area

Volume & Surface Area

Volume & Surface Area in General Recall that perimeter and area are measurements of two-dimensional figures – e.g. rectangles, triangles, etc. These same measurements have counterparts for three-dimensional figures Surface Area: a measurement of space on the outside of a three-dimensional figure Volume: a measurement of the space inside a three- dimensional figure – In general V = A x h where A is the area of the base and h is the height (third dimension)

Volume & Surface Area in General (Continued) What follows on the next few slides are the formulas for volume and surface area for common three-dimensional figures – Except for the volume of a rectangular solid and a cube, you are not required to memorize these formulas – However, you should know how to apply them to solve problems

Rectangular Solids & Cubes Recall the formula for area of a rectangle Given a rectangular solid with length l, width w, and height h: – SA = 2lw + 2lh + 2wh – V = lwh Recall the formula for area of a square Given a cube with length l (length, width, and height are the same for a cube): – SA = 6l 2 – V = l 3

Cylinder A cylinder is a three-dimensional extension of a circle with an added height Recall the formula for area of a circle Given a cylinder with radius r and height h: – S.A. = 2πrh + 2πr 2 Think about “opening up” the cylinder and lying it down flat as a rectangle and then adding in the area for the two circular bottoms – V = πr 2 h

Cone A right circular cone has a height h that extends from the tip and is perpendicular to the circular base of the cone Given a right circular cone with a height h and radius of its circular base r: –

Sphere A sphere is a three-dimensional extension of a circle with radius r – Think of a ball that can be cut into circles – The radius is measured from the center of the sphere – The Earth is essentially a sphere Given a sphere with radius r:

Volume & Surface Area (Example) Ex 1: What is the minimum area of wrapping paper required to completely cover a box with dimensions 12.4” x 11.9“ x 7.4“?

Volume & Surface Area (Example) Ex 2: A packing crate in the shape of a rectangle has dimensions of 12 ft x 8 ft x 60 in. How many cubic packages with sides of length 3 ft can fit into the crate?

Volume & Surface Area (Example) Ex 3: An ice-cream cone in the shape of a right- circular cone has a radius of 4 cm and a height of 8 cm. a) How much ice cream can the cone hold if we completely fill it? b) After filling the cone, a company decides to wrap it for packaging. How much wrapping is required?

Volume & Surface Area (Example) Ex 4: A punch bowl is in the shape of a hemisphere (half a sphere) with a radius of 9 inches. The cup part of the ladle in the bowl is also in the shape of a hemisphere with a diameter of 4 inches. If the punch bowl is filled completely, how many full ladles of punch are in the bowl?

Summary After studying these slides, you should know how to do the following: – Be familiar with the different formulas for surface area & volume of common three-dimensional figures Additional Practice: – See problems in Section 10.4 Next Lesson: – Introduction to Counting Methods (Section 13.1)