Problem of the Day Which figure has the longer side and by how much, a square with an area of 81 ft 2 or a square with perimeter of 84 ft? A square with a perimeter of 84 ft; by 12 ft 9 21

Surface Area Prisms and Pyramids

surface area: the sum of the areas of a solid figure net: a two-dimensional drawing of a three-dimensional figure lateral area: area of the faces connected to the base(s) Our Objective? Learn to find the surface areas of prisms and pyramids.

Finding the Surface Area of a Prism Find the surface area (S)or SA of the prism. Method 1: Use a net. A net may help you see each face of the prism. Think about each face as a separate piece. Since the faces are all rectangles, use the formula A = lw to find the area of each face.

One Step at a Time!!! A: A = 5  2 = 10 B: A = 12  5 = 60 C: A = 12  2 = 24 D: A = 12  5 = 60 E: A = 12  2 = 24 F: A = 5  2 = 10 SA = ( ) = 188 in 2. Add the areas of each face. Letters have been used to identify each face. in 2.

A: A = 5  2 = 10 B: A = 12  5 = 60 C: A = 12  2 = 24 D: A = 12  5 = 60 E: A = 12  2 = 24 F: A = 5  2 = 10 The second method will take the values from the net and fit them into a formula that works for all prisms. in 2. This represents the height of the prism…the distance between the two bases. Of course, the base area is listed twice. What other value is repeated in all other steps? So, what do the 5, 2, 5, and 2 represent? Now let’s put this into a formula!

Pre-Algebra Formula for Surface Area of Prisms SA = 2B + LA SA = 2B + ph This means find the area of the surfaces connecting the bases and add it to the area of both bases. LA stands for lateral area……you find the perimeter of the base and multiply that by the height…… WAIT.. This is easier than it sounds. We’ll rewrite LA as ph. This shows you how to find the LA. I chose the bottom part as the base. This means that 2 in. will be the height of the prism. (notes ) SA = LA + 2B SA = ph + 2B SA = 2(12 + 5)[2] + 2(5)12 Perimeter(height) SA = 34[2] Bases(2) SA = SA = = 188 in 2.

f & b: 9  7 = 63 t&b: 9  5 = 45 sides: 7  5 =  2 =  2 =  2 = 70 SA = = 286Add the areas of each face. The surface area is 286 cm 2. Finding the Surface Area of a Prism from a 3-D drawing..

SA = LA + 2B SA = ph + 2B SA = 2(9 + 5)7 +5(9)2 SA = 28(7) + 90 SA = SA = 286 square cm. Double the area of the bases. Find the Perimeter of the base Multiply it by the height. I chose 9 x 5 as the base, so 7 is my height.

3 in. 11 in. 6 in. SA = LA + 2B SA = ph + 2B SA = 2(6 + 11) 3 + 2(6)11 I chose the 6 x 11 as my base. These are the top and bottom. 3 = h Height of the shape SA = 34(3)+ 2(66) SA = 234 square inches Why is this easier???? It’s generic. It works with all prisms. Find the base area and double it, add this to the (perimeter)height product.

Using a Net A: A = 6  3 = 18 B: A = 11  6 = 66 C: A = 11  3 = 33 D: A = 11  6 = 66 E: A = 11  3 = 33 F: A = 6  3 = 18 S = = 234 Add the areas of each face. The surface area is 234 in in. 6 in. 3 in. A B C DE F

11 in. 6 in. Front & back: 3 x 11 x 2 = 66 Sides: 3 x 6 x 2 = 36 Top & Bottom: 11 x 6 x 2 = 132 Surface area: 234 square inches Using a 3-D Picture Does anyone see yet another way????? 6(11) +3(11) +6(3)2[ ]= 2[ ]= 2[117] = 234 sq. inches This works for all rectangular prism surface area.

What is the height of a rectangular prism with a square base that has a side length of 8 cm and a surface area of 512 sq. cm? -128 ___ ____ 32 32

6 in 5 in 7 in 4 in 7 in Bases: ½ (6)(5) 2 = 30 sq. in. Faces: 4(7) + 7(4) + 4(6) = 80 sq. in. SA = 110 sq. in. 4 in Nets and triangular pyramids

The formula stays the same, even if the prism changes! SA = 2B + LA SA = (2B + ph) Locate the bases and prism height. 6 in. 4 in. 5 in. 7 in. SA = SA = 110 in 2 Important Note!! The height we are referring to in the formula is the height of the prism, NOT the height of the triangular base!! This height is dealt with in the B portion of the formula.

12m 4m 3m 4 ft 3 ft 4 ft 5 ft SA = 2B + LA SA = 2B + ph SA = LA + 2B SA = 2(12)3 + 4(30) SA = ( ) m 2 SA = 192 m 2 SA = ph + 2B SA = (12)4 + 2(12) 2 SA = 60 ft 2 6ft. 7ft. 5ft. X

Okay…..now what? You have practiced a variety of ways to find the surface area of a prism. The formula will work for ANY prism…… Let’s practice prism surface area before going to pyramids.

The surface area of a pyramid equals the sum of the area of the base and the areas of the triangular faces. To find the surface area of a pyramid, you could think of its net. base face

With a pyramid, we have one base. The lateral faces are triangles that slant as they merge into a single vertex…… This gives us a new term……. …slant height. It refers to the height of the triangular face as it “slants” toward the top vertex of the pyramid. The letter l stands for slant height. SA = B + LA

Finding the Surface Area of a Pyramid Find the surface area SA of the pyramid. SA = 49 ft + 4(28) ft SA = 49 ft ft SA = B + n(bl) 2 __ SA = (7ft)  7(8) 2 ___ ft 2 SA = 161 ft Choose your plan. Write the formula.. SA = B + LA SA = B + pl SA = 7(7) + 4(7)8( ) SA = ( ) ft 2 SA = 161 ft 2. Perimeter of base(half slant height)

This is not in your notes. How can we find the surface area?

10m 6m 9m SA = LA + B SA = pl(1) + 6(10) 2 SA = 32(9) SA = [(144) + 60] m 2 SA = 204 m 2 The base of this pyramid is not a square. This is supposed to be a right triangle with a base height of 5 cm.* The hypotenuse is rounded. 9cm 5cm