11 2 Surface Area of Prisms and Cylinders

today's lesson is on surface areas of prisms and cylinders take minute to read over The Learning goal and scale find where you are on the scale before we begin the lesson to find the surface area of three-dimensional figure we find the sum of the areas of all the surfaces of the figure prism is polyhedron with two congruent parallel faces called bases the other faces are called lateral faces we name prism using the shape of its bases because the base in this prism is pentagon it is pentagonal prism because the base in this prism is triangle it is triangular prism an altitude of prism is perpendicular segment that joins the planes of the bases the height of prism is the length of an altitude the prism may be either right or oblique here we have right prisms and here we have oblique prisms in example one we will use net to find surface area of prism what is the surface area of the rectangular prism use net let's start by drawing the net the net will have two bases the top and the bottom both with Bass length 4 and height 3 our lateral faces we have two lateral faces the front and the back with base length of four and height of 5. and then we have lateral faces the right and the left with base link 3 and height of 5. let's begin by finding the area of the bases our base has length of four and height of three there are two of those now let's find the area of the larger rectangular lateral faces both of these have length of four and height of five now let's look at the smaller rectangular lateral faces both of these have length of three and height of five now that we have the area of all six faces let's add them up we have 12 plus 12. 20 plus 20 and 15 plus 15. 12 plus 12 is 24 20 plus 20 is 40. 15 plus 15 is 30. which gives us surface area of 94 centimeters squared pause the video and do you try number one what is the surface area of the triangular prism use net let's start by drawing the net we have one rectangle with base 6 and height 12. second rectangle with base 5 and height 12 and third rectangle with base 5 and height 12. we also have two triangular bases with base 6 and the height that we'll need to find let's start by finding the area of the two triangular bases since we don't know the height of the triangle we're going to have to use the Pythagorean theorem notice we have an isosceles triangle so if we draw that perpendicular height it is going to bisect this six unit side and make it 3 here and the hypotenuse will be five so let's use three squared plus squared equals 5 squared to find the length of the height so our height will be four centimeters which will give us an area of 12 centimeters squared for each of the Triangular bases now we have two rectangular lateral faces with base length 5 and height 12. and we have one rectangular lateral face with base length six and height 12. so we will add our two triangles with 12 centimeters squared each our two rectangles with 60 centimeters squared each and our rectangle that is 72 centimeters squared 12 plus 12 is 24 60 plus 60 is 120. plus 72 will give us 216 centimeters squared we can find formulas for the lateral and surface areas of prism by using net remember the formulas are there to help you if you cannot remember formula you really don't need to use it the lateral area of right prism is the product of the perimeter of the base and the height of the prism lateral area equals perimeter times height the surface area of right prism is the sum of the lateral area and the area of the two bases so surface area equals lateral area plus two times the area of the base in example two we will use formulas to find surface area of prism what is the surface area of the prism we know that surface area is the lateral area plus the sum of the area of the bases we also know that to find the lateral area we take the perimeter of the base times the height to find the perimeter of the base we are going to have to add all three sides however we do not know the length of the hypotenuse so we need to use the Pythagorean theorem squared plus squared equals squared to find this third side length now that we know the length of the hypotenuse is five centimeters we can add three plus four plus five to get the perimeter of the base now let's substitute 12 in for the perimeter and 6 in for the height we now need to find the area of the base since the base is triangle we're going to take base times height divided by 2. 3 times 4 divided by 2. so we will substitute 6 in for the area of our bases now let's simplify our formula 12 times 6 is 72 plus 2 times 6 is 12. 72 plus 12 is 84 so the surface area of our triangular prism is 84 centimeters squared pause the video and do you try number two for part what is the lateral area of the prism we know that lateral area is the perimeter of the base times the height since the base of this prism is hexagon to find the perimeter we will multiply 6 times the length of the side 6. so the perimeter is 6 times 6 or 36. the height of our prism is 12. to find the lateral area we will multiply 36 times 12 for an area of 432 square meters for Part what is the area of base in simplest radical form since our bases are hexagons with side length 6 meters we know that we're going to have six congruent isosceles triangles with six meter base we will need to find the height of the triangle by drawing in that perpendicular segment that will give us right triangle with base that is half the length of the original triangle base 3 and since our vertex angle of this triangle is 60 degrees remember six triangles form circle so 360 divided by 6 is 60 that means this angle will be half of 60 or 30. we now have 30 60 90 triangle with the short leg being three so the long leg or the height will be three radical three so the area of our hexagonal base is going to be six triangles with base length of six height of 3 radical three divided by two let's start with six times three is eighteen divided by two is nine times six will give me 54 radical three for part what is the surface area of the prism rounded to the nearest whole number the surface area is the lateral area plus two times the area of our base the lateral area is 432 meters squared and the area of one base is 54 radical three meters squared this gives us 432 plus 108 radical three using calculator we will get approximately 619 meters squared cylinder is solid that has two congruent parallel bases that are circles an altitude of cylinder is perpendicular segment that joins the planes of the bases the height of the cylinder is the length of the altitude in right cylinder the segment joining the centers of the bases is the altitude in an oblique cylinder the segment joining the centers is not perpendicular to the bases to find the area of the curved surface of cylinder visualize unrolling it like label on soup can the area of the resulting rectangle is the lateral area of the cylinder the surface area of cylinder is the sum of the lateral area and the area of the two circular bases the formula for lateral area of cylinder is 2 pi times the height because the base length of this rectangle is actually the circumference of the base we can also use the formula pi times the diameter times the height to find the surface area of right cylinder we will take the sum of the lateral area or 2 pi times the height plus the sum of each circular base so 2 because there are two bases pi squared in example three we will find the surface area of cylinder the radius of the base of cylinder is four inches and its height is 6 inches what is the surface area of the cylinder in terms of Pi like to start by drawing the cylinder and labeling the radius and the height to find the surface area we need to start by finding the lateral area the lateral area is the circumference of the circle times the height 2 times 4 is 8 times 6 is 48 times pi is 48 pi so the lateral area of the cylinder is 48 Pi inches squared to find the surface area we need to add the lateral area to the area of the circular bases since the lateral area is 48 Pi we will substitute that in for lateral area since the radius is 4 we will substitute that into the formula for the area of the base so we have 4 squared which is 16 times 2 which is 32 times pi which is 32 pi now let's add the lateral area to the area of the two bases so 48 pi plus 32 Pi is 80 pi so the surface area of our cylinder is 80 Pi inches squared pause the video and do you try number three cylinder has height of 9 centimeters and radius of 10 centimeters what is the surface area of the cylinder in terms of Pi let's start by finding the lateral area which is the circumference times the height 2 times 10 is 20 times 9 is 180 times pi is 180 pi to find the surface area of the cylinder we're going to add the lateral area to the area of the two circular bases 10 squared is 100 times 2 is 200 times pi is 200 pi 180 pi plus 200 Pi is 380 pi so the surface area of the cylinder is 380 Pi centimeters squared in example 4 we will find lateral area of cylinder you are using the cylindrical stencil roller below to paint patterns on your floor what area does the roller cover in one full turn lateral area is the circumference times the height in this case we are given the diameter so we'll use the formula pi for the circumference of the ruler 2.5 times 6 is 15 times pi and here we'll actually use the pi key on the calculator is 47.1238898 so the area of the roller covers in one turn is about 47 inches squared pause the video and do you try number four smaller stencil roller has height of 1.5 inches and the same diameter as the other roller to the nearest tenth what area does the smaller roller cover in one turn again we're going to use lateral area lateral area will be the circumference times the height 2.5 times 1.5 is 3.75 times pi and let's go ahead and use the pi key on the calculator is 11.78097245 so rounded to the nearest tenth since 8 is larger than 5 we will have approximately 11.8 centimeters squared now's your chance to see how well you understand the lesson pause the video and do the lesson check don't forget to check your answers on the next slide if you have any questions about the lesson check be sure to ask me in class now that you know all about surface area of prisms and cylinders go ahead and do the challenge now take another minute to reread the learning goal and scale have you climbed any higher on the scale than where you were at the beginning of the lesson if you did the challenge here's the answer