Surface Area of Prisms and Cylinders Geometry Surface Area of Prisms and Cylinders
Goals Know what a prism is and be able to find the surface area. Know what a cylinder is and be able to find the surface area. Solve problems using prisms and cylinders. April 22, 2017
Prism A polyhedron with two congruent faces, called the bases. The bases are parallel. The other faces are parallelograms and are called lateral faces. The segments joining corresponding vertices of the bases are lateral edges. April 22, 2017
Example Base Lateral Edges Lateral Face Lateral Face Base April 22, 2017
Prisms can have any polygon for its bases. Base is a pentagon. Base is a triangle. Pentagonal Prism Triangular Prism April 22, 2017
These are not prisms: …and no parallel bases. Lateral Faces are not parallelograms. …and no parallel bases. April 22, 2017
Altitude of a Prism The perpendicular distance between the bases. We usually use the letter h for the height – the length of the altitude. h April 22, 2017
Right Prism The lateral edges are perpendicular to the bases. For clarity, in many cases we do not indicate right prisms – use common sense. April 22, 2017
Oblique Prism A prism in which lateral faces are not perpendicular to the bases. 110 April 22, 2017
Slant Height The length of a lateral edge in an oblique prism. s Generally, you can use the Pythagorean Theorem to find one or the other. Slant Height s Height h April 22, 2017
Do you know… What a prism is? What the bases are? What a lateral face is? What the lateral edges are? What a right prism is? What an oblique prism is? What the slant height is? April 22, 2017
Classifying Prisms Use the shape of the base in the name. Right Triangular Prism Right Rectangular Prism Right Pentagonal Prism April 22, 2017
Have you ever seen a regular heptagonal prism? April 22, 2017
Surface Area The sum of the areas of all the faces of a prism. Area = Area of 2 bases + all lateral faces. Contrary to the text, use the symbol SA for surface area. April 22, 2017
Example 6 The pink sides are really rectangles. They look like parallelograms because of the projection. 4 25 There are 2 bases and 4 lateral faces. All are rectangles. April 22, 2017
Example 20 What’s the area? A = 20 25 = 500 ? ? 4 6 4 6 25 6 6 4 4 April 22, 2017
Example 4 6 4 6 20 Surface Area is the sum of the lateral area (500) and the two bases (48). 25 A = 20 25 = 500 SA = 548 6 6 4 24 4 24 April 22, 2017
What we did. A = Ph P B B This measurement is the perimeter of a base. We found the rectangular area. We found the area of both bases. B B April 22, 2017
The surface area is the sum of these regions. P h SA = 2B + Ph A = Ph B B April 22, 2017
Surface Area The surface area of a right prism can be found using SA = 2B + Ph B is the area of each base P is the perimeter of a base h is the height April 22, 2017
Alternate Method Find the area of each face separately. Add them together. Don’t omit any face – be careful. April 22, 2017
Lateral Area The lateral area of a shape is the area of the lateral faces, but doesn’t include the bases. SA = 2B + Ph is total surface area. Ph is the lateral area. LA = Ph April 22, 2017
Example SA = 104 ft2 2B + Ph Find the surface area. 2 ft. Base 2 ft. or… April 22, 2017
Example Find the surface area. Alternate solution. Base 2 ft. 2 ft. 12 ft. P = 28 h = 2 B = 24 2 B + P h SA = 2(24) + 28(2) = 48 + 56 SA = 104 ft2 April 22, 2017
Example 104 ft2 24 ft2 24 ft2 24 ft2 4 ft2 4 ft2 24 ft2 2 ft. Alternate solution 2. 2 ft. 12 ft. Separate the figure into a “net”. Find the area of each face. 104 ft2 24 ft2 24 ft2 24 ft2 4 ft2 24 ft2 4 ft2 April 22, 2017
Example Find the Surface Area B = 40 P = 44 h = 16 SA = 2B + Ph SA = 2(40) + 44(16) SA = 80 + 704 S = 784 Base 16 h 2 20 April 22, 2017
Your Turn Find the surface area. 7 cm 6 cm 18 cm April 22, 2017
Solution Perimeter = 2(6 + 18) = 48 cm Base Area = 6 18 = 108 cm2 SA = 2B + Ph = 2(108) + 48(7) = 216 + 336 = 552 cm2 Lateral Area April 22, 2017
Find the Surface Area 4 6 B Area of Equilateral Triangular Base Hint: April 22, 2017
Try this problem. 12 10 Find the surface area of the right, hexagonal prism. Each base is a regular polygon. April 22, 2017
Solution That’s the area of one base. 12 12 12 ? ? ? 6 10 April 22, 2017
Solution SA = 2B + Ph SA = 2(374.1) + 72(10) SA = 748.2 + 720 12 374.1 10 The perimeter of the hexagon is 6 12 = 72, and the height is 10. April 22, 2017
Summary A prism is a polyhedron with 2 congruent bases and parallelogram lateral faces. Prisms may be right or oblique. Basic Formula: SA = 2B + Ph The Lateral Area LA = Ph April 22, 2017
Cylinders April 22, 2017
Cylinder A prism with congruent circular bases. May be right or oblique, just like prisms. r r = radius h = height h April 22, 2017
Surface Area of a Cylinder Take a cylinder and cut it apart… You get two circles and a rectangular area. April 22, 2017
Surface Area of a Cylinder h 2r The width of the rectangle is… the circumference of the circle. April 22, 2017
Surface Area of a Cylinder 2rh h 2r The area of the rectangle is… 2rh (aka Lateral Area) April 22, 2017
Surface Area of a Cylinder 2rh h r2 2r r2 The area of one circle is… r2 The area of two circles is 2r2. April 22, 2017
Surface Area of a Cylinder 2rh h r2 2r r2 The surface area of the cylinder is: SA = 2r2 + 2rh April 22, 2017
Surface Area of a Cylinder Or, for easier computing… h April 22, 2017
Example Find the surface area. 12 SA = 2r(r + h) SA = 2(12)(12 + 10) SA = 24(22) SA = 528 SA 1658.76 10 April 22, 2017
Your Turn Find the surface area. d = 2 in. SA = 2(1)(1 + 14) SA = 2(15) SA = 30 SA 94.25 in2 r = 1 in. 14 in. April 22, 2017
Problem. Find the height. 4 h SA = 301.6 April 22, 2017
Your turn. Find the height. SA = 282.74 6 h April 22, 2017
Take a clean sheet of paper… Label it Chapter 12 Formulas Add these formulas: Prism Cylinder SA = 2B + Ph SA=2r(r + h) LA = Ph LA = 2rh Everyday as you have new formulas, add them to it with a simple drawing. April 22, 2017
Practice Problems April 22, 2017