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**Surface Area Introduction and SA Formula for Rectangular Prisms**

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**REMEMBER THE PARTS OF SOLIDS?**

Prisms and cylinders have 2 congruent parallel bases. A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has at least one nonrectangular lateral face.

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An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude.

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**Surface area is the total area of all faces and curved**

surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces.

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l w h Notice from the net there are two of each rectangle. The area of 2 of them will be lw, 2 will be wh, and 2 will be lh.

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**Surface Area of a Rectangular Prism**

The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as S = 2ℓw + 2wh + 2ℓh. NOTE: Add this formula onto your notes sheet for PRISMS – under the Volume formula. Head it “Surface Area of a Rectangular Prism”

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**Surface Area of Right Prisms**

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**Examine the net of the prism**

Examine the net of the prism. The base of the rectangle is equal to the perimeter of the base of the prism.

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B a b c P = a + b + c NOTE: Add this formula (S = Ph + 2b) onto your notes sheet for PRISMS – under the Surface Area formula for rectangular prisms. Head it “Surface Area of a Right Prism”

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**The surface area formula is only true for right prisms**

The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces. Caution!

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Example 1 Find the surface area of the rectangular prism with the following dimensions: l = 14, w = 2, h = 15 P = =32 B=14*2=28 SA = 15*32+2*28=536 units2

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Example 2 Find the surface area of the rectangular prism with the following dimensions: l = 3’, w = 6’, h = 2.5’ P = 3*2+6*2=18 B=3*6=18 SA = 2.5*18+2*18=81ft2

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**Equilateral base with 6” sides**

Example 3 Find the surface area of the regular triangular prism with the following dimensions: Equilateral base with 6” sides Prism height 14” Use triangles to find the height of the triangular base. B = 0.5*6*3√3=9√3 P = 6+6+6=18 SA = 14*18+2*9√3= in2

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**Example 4 Work backwards to solve for the unknown information.**

In a rectangular prism, SA = 560ft2, and the base is 7 ft by 8 ft. What is the height? 560=h* 560=30h+112 448=30h h=14.93 ft

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**Example 5 Work backwards to solve for the unknown information.**

In a regular hexagonal prism, the base sides are 18 cm, and the SA = cm2. What is the height of the prism? Use triangles to find the apothem of the hexagon (9√3) =h*108+2*.5*9√3*108 =108h H= = 25 cm

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**SA = 197.87 in2. What is the height of the prism?**

Example 6 Work backwards to solve for the unknown information. In a isosceles triangular prism, the base legs are 4 in, and base is 6 in; and the SA = in2. What is the height of the prism? Split the triangular base in half and use the Pythagorean Theorem (or trig) to solve for the height (√7) 197.87=h*14+2*3√7 12.999=h=13in

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A face of a prism or pyramid that is not a base An edge of a prism or pyramid that is not an edge of a base A prism whose lateral faces are all rectangles.

A face of a prism or pyramid that is not a base An edge of a prism or pyramid that is not an edge of a base A prism whose lateral faces are all rectangles.

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