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**Surface Area of Prisms and Cylinders**

Geometry Surface Area of Prisms and Cylinders

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**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

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**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

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**Example Base Lateral Edges Lateral Face Lateral Face Base**

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**Prisms can have any polygon for its bases.**

Base is a pentagon. Base is a triangle. Pentagonal Prism Triangular Prism April 22, 2017

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**These are not prisms: …and no parallel bases.**

Lateral Faces are not parallelograms. …and no parallel bases. April 22, 2017

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**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

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**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

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Oblique Prism A prism in which lateral faces are not perpendicular to the bases. 110 April 22, 2017

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**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

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**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

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**Classifying Prisms Use the shape of the base in the name.**

Right Triangular Prism Right Rectangular Prism Right Pentagonal Prism April 22, 2017

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**Have you ever seen a regular heptagonal prism?**

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**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

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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

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**Example 20 What’s the area? A = 20 25 = 500 ? ? 4 6 4 6 25 6 6 4 4**

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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

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**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

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**The surface area is the sum of these regions.**

P h SA = 2B + Ph A = Ph B B April 22, 2017

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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

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**Alternate Method Find the area of each face separately.**

Add them together. Don’t omit any face – be careful. April 22, 2017

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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

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**Example SA = 104 ft2 2B + Ph Find the surface area. 2 ft. Base 2 ft.**

or… April 22, 2017

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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) = SA = 104 ft2 April 22, 2017

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**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

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**Example Find the Surface Area**

B = 40 P = 44 h = 16 SA = 2B + Ph SA = 2(40) + 44(16) SA = S = 784 Base 16 h 2 20 April 22, 2017

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Your Turn Find the surface area. 7 cm 6 cm 18 cm April 22, 2017

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**Solution Perimeter = 2(6 + 18) = 48 cm Base Area = 6 18 = 108 cm2**

SA = 2B + Ph = 2(108) + 48(7) = = 552 cm2 Lateral Area April 22, 2017

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**Find the Surface Area 4 6 B Area of Equilateral Triangular Base**

Hint: April 22, 2017

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Try this problem. 12 10 Find the surface area of the right, hexagonal prism. Each base is a regular polygon. April 22, 2017

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**Solution That’s the area of one base. 12 12 12 ? ? ? 6 10**

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**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

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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

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Cylinders April 22, 2017

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**Cylinder A prism with congruent circular bases.**

May be right or oblique, just like prisms. r r = radius h = height h April 22, 2017

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**Surface Area of a Cylinder**

Take a cylinder and cut it apart… You get two circles and a rectangular area. April 22, 2017

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**Surface Area of a Cylinder**

h 2r The width of the rectangle is… the circumference of the circle. April 22, 2017

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**Surface Area of a Cylinder**

2rh h 2r The area of the rectangle is… 2rh (aka Lateral Area) April 22, 2017

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**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

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**Surface Area of a Cylinder**

2rh h r2 2r r2 The surface area of the cylinder is: SA = 2r2 + 2rh April 22, 2017

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**Surface Area of a Cylinder**

Or, for easier computing… h April 22, 2017

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**Example Find the surface area.**

12 SA = 2r(r + h) SA = 2(12)( ) SA = 24(22) SA = 528 SA 10 April 22, 2017

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**Your Turn Find the surface area.**

d = 2 in. SA = 2(1)(1 + 14) SA = 2(15) SA = 30 SA in2 r = 1 in. 14 in. April 22, 2017

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**Problem. Find the height.**

4 h SA = 301.6 April 22, 2017

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**Your turn. Find the height.**

SA = 6 h April 22, 2017

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**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

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Practice Problems April 22, 2017

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