in this video we're going to focus on calculating the surface area of a triangular prism so let me draw a picture first so let's call this a b c d e and f so let's say that c f is 30 and let's say that a b is 10 ac is 21 and e f is 17. and let's say the height of the triangle is 8 units long calculate the lateral area of the prism and then the total surface area of the prism now if you wish to calculate the lateral area it's simply the perimeter of the base times the height of the prism the height is 30. the base is the triangle so the perimeter triangle is just the length of a b plus bc plus ac so a b is 10 bc is the same as ef that's 17 and ac is 21. so the perimeter is going to be 10 plus 21 plus 17 and the height that's 30. 10 plus 21 plus 17 that's 48 and 48 times 30 is 1440. so the lateral area is 1440. the lateral area is basically the area of b e f c plus a c f d plus a b e d so those three faces combined represents the lateral area so you gotta add up the area of those three faces now let's calculate the total surface area the total surface area is the area of the base plus the lateral area the base represents a triangle so we need to calculate the area of triangle abc and def which those two will be the same now to calculate the area of a triangle we could simply use one half base times height so we have the base of the triangle that's 21 and the height is eight half of a is four and four times 21 is 84. so that's the area of just one triangle since we have two triangles abc and def we need to multiply 84 by 2. so the total area of the base is 2 times 84. for the triangle in the front and the triangle back and then the lateral area is 1440 so 2 times 84 is the 168 plus 1440 so we have a total surface area of one thousand six hundred and eight and then it's going to be square units so anytime you wish to calculate the surface area of a prism it's simply the area of the base plus the latter area and the lateral area is the perimeter times the height now let's try another example you can pause the video if you want to try this one so let's use the same letters to describe this triangle so this time we have an isosceles triangle so a b and b c have the same measure and let's say ac is 10 inches cf is 15 inches and you're told that the height of the triangle is 12 inches go ahead and calculate the lateral area and the surface area so let's focus on the triangle for now so the height of the triangle is 12. and we know the base is 10. so let's call this a b c and let's add a new point d so a d has to be five and d c has to be five and this is basically the 5 12 13 triangle so a b and b c are both 13. and you can calculate the missing side using the pythagorean theorem if you don't know the special right triangles so it's going to be 5 squared plus 12 squared equals c squared so 5 squared is 25 12 squared is 144 and then these two add up to 169. so if you take the square root of both sides you can see that the hypotenuse is going to be 13. so now we have everything that we need so to calculate the lateral area it's going to be the perimeter times the height so ac is 10 a b is 13 bc is 13. so the perimeter is going to be 10 plus 13 plus 13 and then we need to multiply by the height which is 15. so the lateral area comes out to be in 540 square inches for those of you out there who may not be familiar with this formula you can also get the answer this way so the area of bcfe that's going to be 13 times 15. you could say length times width and 13 times 15 that's uh 195. now for adf c we have a length of 15 and a width of 10. so 15 times 10 that's 150 and for surface a b e d we still have a length of 15 but basically width of 13 and 15 times 13 once again is 195. so if we add up 195 plus 195 plus 150 that gives us 540. so you could do it mentally in your head step by step if you want to do it that way or you could just simply multiply the perimeter by the height of the prism so now let's calculate the surface area of the prism so it's the area of the base plus the lateral area the area of the base is going to be base times height multiplied by a half that's the area of the triangle so it's one half base times height the base of the triangle is 10 the height is 12. so half of 10 is 5 and 5 times 12 is 60. so 60 is the area of the triangle but keep in mind we have two triangles so it's going to be 2 times 60 plus the lateral area of 540 so that's 120 plus 540 which is 660 square inches so that's the total surface area of this particular figure let's work on one more example so let's say if we're given all three sides of the triangle and also the height of the prism let's say that's 15 inches so with this information calculate the surface area of the prism and also the lateral area well if we have all three sides the lateral area is going to be straightforward it's the perimeter times the height so the perimeter is going to be 6 plus 7 plus 8 is the sum of all three sides of the triangle and the height of the prism is a 15. so 6 and 7 is 13 plus 8 that's 21 and 21 times 15 is 315. and so that's going to be the lateral area it's 315 square units so now for the surface area so let's calculate the area of the base so how can we calculate the area of a triangle if we're given all three sides we need to use something called heron's formula so first we need to calculate s which is basically one half of the perimeter of the triangle so it's going to be six plus seven plus eight divided by two and six plus seven is thirteen plus eight that's a twenty one and twenty one divided by two is ten point five so that's s now once you have s we need to do is use this formula so the area is going to be the square root of s times s minus a times s minus b times s minus c so s is 10.5 and then 10.5 minus a a is 6. so 10.5 minus 6 is 4.5 and then b is 7. so 10.5 minus 7 that's three point five and c is eight ten point five minus eight is two point five so go ahead and type that in your calculator and then take the square root of the result so the area is 20.33 square units and that's just for one triangle so the total surface areas can be the area of the base plus the lateral area capital b is the area of two triangles so it's two times twenty point thirty three don't forget about the back side and then plus the lateral area which is three hundred fifteen so the total surface area is about 355.7 square units you