# How do you solve a surface area problem? How do you find the surface area of a triangular prism and a cylinder?

## Expert Answers The simple formula is Lateral  Surface Area + 2( area of the base).  You must remember that Lateral surface Area = perimeter times the height. For example if you have a triangular prism, your base is a triangle.   Add the length of its sides and multiply times its height.  Now you find the area of the triangular base with the formula, 1/2base x the height and multiply this times two.    Lastly you add the Lateral Surface Area to this part of the formula and you have your answer in squared units. For a cylinder, you follow the same formula, but now  the base is a cirlce.  You must use the formula for the circumference of a circle, 3.14 x diameter.  Next you multiply this with the height.  Plug in the numbers for the area of the base which for a cirlce is 3.14 x radius squared.  Multiply this by 2 and add this to the Lateral Surface Area.  The result is the Surface Area of the prism and the cylinder.

Approved by eNotes Editorial Team Hi Tuty,

To find the surface area of any prism, including cylinders because a cylinder is just a prism with circles for bases (take a regular sheet of paper which is a rectangle and roll it up and what do you get?), I teach my 7th grade students a generic formula: S=2B + Ph

S=surface area

B=area of 1 base of the prism

P=perimeter of 1 base of the prism

h=the height of the prism

First, let's make sure we understand what each of these parts are and how to identify them. A prism is a polyhedron (3 dimensional shape) that is made from connecting two congruent polygons (a 2 dimensional shape like a triangle or rectangle) bases together using 1 or more rectangles. A simple example of a common prism seen all the time is a box. A box has a top and a bottom that are the same size and shape, and all the sides of the box that connect that top and bottom are rectangles.  So in this example of the box, the base would be either the top or bottom of the box. It really doesn't matter if you use the top or bottom, because they are identical or congruent. In a cylinder, the bases would be the circular ends, because those are the top and bottom that are congruent. In a triangular prism, the triangle ends that are the same would be your bases.

The height of a prism is the length of the line that connects the congruent bases to each other. In the example of the box, the height would be the length of the edge that connects the top and bottom of the box.

So now that the vocabulary is defined, let's use the formula S=2B+Ph to find the surface area of a box that has a base that is 4in by 3in, and a height of 6in. First we find the area of the base or B. For a rectangle, that is length times width, so in this case it is 4x3 or 12in squared.

Next we find the perimeter of the base or P. Perimeter means all around the outside of a shape, so it just means add up the sides of the rectangle that is the base. In our case, it would be 4+4+3+3 or 14 inches.

Now we need the height of the prism. Make sure not to confuse the height of the prism with other heights that occur in geometry, like the height of a triangle for example. Remember, the height of a prism is the length of the line that connects 1 base to the other base. Our height is 6in as stated in the beginning of the problem.

Now that we have all the necessary numbers, let's plug them in and simplify:

S=2(12)+14(6)

S=24+84

S=108in squared (area is always units squared)

With a triangular prism, for the B, or base area, you would use the formula for the area of a triangle, or a=1/2bh (1/2 of the base of the triangle times the height of the triangle). Remember though, the h in this formula is not the height of the prism; rather it is the height of the triangle base, which is a perpendicular line (90 degrees) from the base of your triangle to the top of your triangle. To help my students avoid confusion with this, I have them draw the triangle to the side as a simple polygon, label the parts of it, and find its area. That will then serve as B in the S=2B+Ph. The P would just be the sum of the length of the 3 sides of the triangle base. The height of the prism in S=2B+Ph is the length of the line that connects the top triangle to the bottom triangle.

For a cylinder, since the bases are circles, you would use (pi*r)squared to find B, and 2*pi*r to find P. The height would just be the length of the line that connects the two circular bases.

I hope that this helps Tuty, good luck.

Approved by eNotes Editorial Team The surface area is the sum of all the areas of all the shapes that cover the surface of the object.

h is the height of the cylinder, r is the radius of the top

Surface Area = Areas of top and bottom +Area of the side

Surface Area = 2(Area of top) + (perimeter of top)* height

Surface Area = 2(pi r 2) + (2 pi r)* h

The surface area is the areas of all the parts needed to cover cylinder. That's the top, the bottom, and the part that wraps around the middle.

You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r2). Since there is both a top and a bottom, that gets multiplied by two.

The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height, the other side is the perimeter of the circle. The area of the rectangle is (2 pi r)* h.  Add those two parts together and you have the formula for the surface area of a cylinder.

Surface Area = 2(pi r 2) + (2 pi r)* h

Triangular based prism

Base shape: Triangle: base 'b', height 'h', and sides S1, S2 and S3

Area of base: ½b×h

Perimeter of base: S1+ S2 + S3

Surface area = bh + (S1+ S2 + S3)H

(pi = = 3.14)

Approved by eNotes Editorial Team As a general rule, to find the surface area of an object you have to find the area of each shape that make up the surface of the object.

There are four of steps to finding the surface area of a triangular prism.

1.  Finding the area (A) of the base by multiplying the altitude (a) of the triangle by the base (b) of the triangle and dividing that number in half.  ab/2 = A.

2.  Find the perimeter (P) of the base by adding up the lengths of the three sides. P=s1+s2+s3.

3.  Find the area of the sides(AS) by multiplying P by the height of the prism (h).  Ph= AS.

4.  Add the base area (A), multiplied by two to account for the two bases,to the area of the sides (AS) to get the surface area.  2A+AS=Surface area of a triangular prism.

A cylinder is similar with a few modifications for the circular shape.

1.  Find the area(A) of the circle base by using the formula pi multiplied by radius squared (pi*r2=A).

2.  Find the area of the sides (AS) by using perimeter (p) of the cylinder's base multiplied by the height (h) of the cylinder.

3.  Find the Surface Area (SA) by adding the area of the bases (2A), A multiplied by 2 to account for the two bases, to the area of the sides (AS).  SA= 2A+AS

Approved by eNotes Editorial Team

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