Surface Area of Triangular Prisms - Nets

Transcription

1 U2B L6: Students will determine the surface area of a triangular prism Surface Area of Triangular Prisms - Nets Surface area - the area of all the faces of the prism Triangular prism - made up of three rectangles and two identical triangles Surface area of triangular prisms = Area of 3 rectangles + Area of 2 triangles Find the area of each rectangle, then find the total area of the rectangles. o In the case above all three rectangles have the same area, but this is not always the case for a triangular prism. o A = lw = 12cm x 3cm = 36cm x 3 rectangular faces = 108 sq cm total Find the area of the triangular faces. A = 1/2bh o (These will always be the same.) o A = 1/2bh = ½(3cm x 2.6 cm) = 3.9 sq cm x 2 triangular faces = 7.8 sq cm total Add the total area of the 3 rectangles to the total area of the 2 triangles. o 108 sq cm sq cm = sq cm Triangular Prisms with Right Triangular Bases What shapes are identical in the net? Notice that the three rectangles do not always have the same area. In this case, one of them has a different area.

2 Find the area of each rectangle. A = lw o 29 cm * 21 cm = 609 cm 2 x 2 rectangular faces = 1218 cm 2 o 29 cm x 29.7 cm =861.3 cm 2 Then add all rectangular areas together. o 1218 cm cm 2 = cm 2 Find the area of the triangular faces. A = 1/2bh o A = ½(21 x 21) = x 2 triangular faces = 441 cm 2 Add the total area of the 3 rectangles to the total area of the 2 triangles. o cm cm 2 = cm 2 Drawing Conclusions about a Prism Given a 3-Dimensional figure instead of a net requires us to draw conclusions about a figure. Things to note: The prism has equilateral triangular bases. Since all three sides of the triangles are the same, the three rectangles will be equivalent. Surface Area of Triangular Prism = Area of 3 rectangles + Area of 2 triangles Find the area of each rectangle. A = lw o 16 in * 8 in = 128 in 2 x 3 rectangular faces = 384 in 2 Find the area of the triangular faces. A = 1/2bh o A = ½(8 x 6.9) = 27.6 x 2 triangular faces = 55.2 in 2 Add the total area of the 3 rectangles to the total area of the 2 triangles. o 384 in in 2 = in 2 Unit 2B L6 RETEACH

3 U2B L7: Students will solve real world problems involving surface area of triangular prisms Surface Area of Triangular Prisms Real World Problems Read the problem carefully and envision what is being asked Write the process that will be needed to determine the surface area. Calculate the area of each face and then add them together o If you have any identical bases, you can use multiplication as a shortcut. Example: A family is going camping and is anticipating rain. In order to be prepared and ensure that their belongings do not get wet, they are going to take a heavy-duty plastic covering to cover the entire tent. What is the minimum amount of plastic they should take for the tent? Notice that the sides of the triangle are equivalent and each measures 100 inches. If all of the sides of a triangle are equivalent, then the rectangles in the prism are equivalent. Surface Area of Triangular Prism = Area of 3 rectangles + Area of 2 triangles Find the area of each rectangle. A = lw o 120 in * 100 in = 12,000 in 2 x 3 rectangular faces = 36,000 in 2 Find the area of the triangular faces. A = 1/2bh o A = ½(100 x 78.1) = 3,905 x 2 triangular faces = 7,810 in 2 Add the total area of the 3 rectangles to the total area of the 2 triangles. o 36,000 in 2 + 7,810 in 2 = 43,810 in 2

4 Surface Area Minus One Face Math Study Guide Imagine we want to cover the doghouse below with a pitched roof, how many faces are we actually covering? How much material will be needed to make the roof? Surface Area = Area of 2 rectangles + Area of 2 triangles *Note: We only need calculate the area of 2 rectangles since the bottom of the prism is the top of the house. Find the area of each rectangle. A = lw o 50 in * 50 in = 2,500 in 2 x 2 rectangular faces = 5,000 in 2 Find the area of the triangular faces. A = 1/2bh o A = ½(80 x 30) = 1,200 x 2 triangular faces = 2,400 in 2 Add the total area of the 3 rectangles to the total area of the 2 triangles. o 5,000 in 2 + 2,400 in 2 = 7,400 in 2 Surface Area of More Than One Triangular Prism Congruent prims identical in form; coinciding exactly when superimposed Calculate the surface area of one and simply multiply that area by the requested number of items

5 A family is going to build three identical ramps while constructing a new skateboard park. The top of the ramp measures 10 feet by 4 feet. The bottom of the ramp measures 8 feet by 4 feet, and the back of the ramp measures 6 feet by 4 feet To calculate the amount of material needed, we must calculate the area of each rectangle and then calculate the area of the two equivalent triangles and add it all together. SA = Area of top rectangle + Area of bottom rectangle + Area of back rectangle + Area of 2 triangles SA = lw + lw + lw + 2( 1 / 2bh) Substitute the measurements for the length and width of each rectangle and the base and height of the triangle. SA = ( 1 / 2 6 8) Make the calculations for each shape in the formula. SA = SA = 144 ft 2 Since a family is building three identical ramps, we must multiply the surface area for one ramp by 3: SA = SA = 432 ft 2 The family will need a total of 432 square feet of material for the three ramps. Lesson 7 RETEACH Volume of a Triangular Prism & Cube What is a Triangular Prism?

6 U2B L8: Students will determine the volume of a triangular prism. Volume of a Triangular Prism V = ( 1 / 2bh) h Volume is the amount of cubic units that will fit inside of a shape. The volume of a triangular prism can be calculated by finding the area of the base and multiplying it by the height of the prism. V = (area of the base) h Sometimes the area of a base will be represented by a capital B V = Bh The base of the prism is a triangle, which means we will substitute the formula for area of a triangle into the portion of the formula for area of the base. V = ( 1 / 2 bh) h * If you notice, the formula has two h measurements listed. The first part of the formula is the area of a triangle, so the h in this part is the height of the triangle, or 10 inches. The second h in the formula refers to the height of the prism, which in this example is 25 inches. The base of the triangle is 15 inches. V = ( 1 / ) 25 Perform the calculations, working carefully with the fraction. V = V = 1,875 in 3 The volume of the triangular prism is 1,875 in 3.

7 Triangular Prisms with Right Triangular Bases V = ( 1 / 2 bh) h The formula does not change. V = ( 1 / ) 5 V = (3.125) 5 V = ft 3 Finding a Missing Dimension Given the Volume *notice that we are given all of the parts in the volume of a triangular prism formula except the height of the prism Start with the formula for volume of a triangular prism. V = ( 1 /2bh) h Let s substitute the information that we know. We know the volume, which means we will substitute 320 cubic inches for the variable V. We also have the base and height of the triangular bases. The unknown variable is the height of the prism. 320 = ( 1 /2 8 8) h We will continue to make the calculations and simplify. 320 = (32) h Finally, we will divide both sides of the equation by 32 to isolate the h for the height = (32)h = h We determined that the length of the missing height of the prism is 10 inches.

8 Lesson 8 RETEACH U2B L9: Students will solve real world problems involving volume of triangular prisms. Volume of Triangular Prisms in Real World Problems For fun, let s calculate the number of balloons that we would need to fill a tent to surprise a friend. If we remember how to find the volume of a triangular prism, we will take the area of the base and multiply it by the height of the prism. We will use the following formula: V = (Area of the base) h The base of the prism is a triangle, which means we will substitute the formula for area of a triangle into that portion of the formula. Now, our formula looks like this: V = ( 1 / 2bh) h Let s give the formula a try. In the tent we want to fill, the base of the triangular base is 3 feet, the height of the triangle is 4 feet, and the length or height of the tent is 10 feet. Now, let s substitute the measurements into the formula and make the calculations. V = ( 1 / 2 3 4) 10 V = 6 10 V = 60 ft 3 The volume of the tent is 60 cubic feet. That means we would need 60 cubic feet of balloons to fill the tent

9 Calculating Volume for Many Congruent Triangular Prisms Find the volume of a prism and then multiply the volume by a number of items We are going to start a candle company and create candles shaped like right triangular prisms. How much wax will we need to create 100 candles? If each candle costs $0.39 per cubic inch, how much will it cost for the wax to make 100 candles? 1) Find the total volume of one candle V = ( 1 / 2bh) h From the diagram above, the base of the triangle is 4, the height of the triangle is 3, and the height of the rectangle is 6 2) Find the total volume of 100 candles V = ( 1 / 2 3 4) 6 V = 6 6 V = 36 in 3 V 100 = V 100 = 3,600 in 3 3) Find the cost of making 100 candles..cost = 3, Cost = $1,404

10 Volume of Objects Made of More than One Figure Sometimes, a three-dimensional figure is made up of more than one type of prism. Maybe part of it is a triangular prism and the other is a rectangular prism, like a house. To calculate the volume we will need to find the volume of each shape individually and then add the volumes together. A house shaped cake requires a thin rectangular cake pan, a square cake pan, and a triangular prism cake pan. We are going to determine the amount of cake batter needed to fill all of the cake pans. The dimensions of each cake pan are labeled in the diagrams below In order to calculate the amount of batter for the bottom two cake pans, we must use the formula for volume of a rectangular prism. Now, we will use the original formula for volume of a triangular prism to calculate the amount of batter needed for the rooftop of the house cake. We will continue to make the calculations within the formula. The last step is to calculate the amount of batter for all three cake pans. Lesson 9 Reteach: