# Surface Area Introduction and SA Formula for Rectangular Prisms

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Surface Area Introduction and SA Formula for Rectangular Prisms

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.

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.

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.

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.

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”

Surface Area of Right Prisms

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.

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”

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!

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

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

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

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

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

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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