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

ladyvols1 | High School Teacher | (Level 3) Senior Educator

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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)

megboland | High School Teacher | (Level 2) Adjunct Educator

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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

tboneson1 | Middle School Teacher | (Level 2) Adjunct Educator

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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.

tiaraton | Middle School Teacher | (Level 1) Adjunct Educator

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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.

sweetskittles24 | Student, Grade 11 | (Level 1) Salutatorian

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Easy way to remember a solution to any surface area problems : calculate the area of a face one by one . Then add all surface areas together !

revolution | College Teacher | (Level 1) Valedictorian

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To find the surface area of triangular prism:

1. Find the base and the height of the prism
2. Find the base area of the prism, which is 1/2*b*h
3. There is a top and bottom similar base area, so multiply the answer by 2, which turns out to be b*h
4. Next, find the perimeter of the triangular base area. Let's name in L1, L2 and L3, so p= L1+L2+L3
5. Multiply height with the perimeter, so H(p1+p2+p3) equals to the lateral area
6. Add the lateral area and base area up so: H(p1+p2+p3)+b*h gives you the surface area o prism

Now for surface area of cylinder:

1. Find the surface area of the circular area at the bottom which is pi*radius^2
2. There is two similar circular area one at the top and the other at the bottom so it is 2*pi*radius^2
3. Next, find the circumference of the circular area which is: Pi*diameter or 2*pi*radius
4. Multiply the perimeter with the height which is h*2*pi*r or h*pi*d, the lateral area.
5. Next, add the lateral and base area together which is: h*pi*d+2*pi*r^2= total area of surface of cylinder

Note that pi=3.142 or 22/7 or just use calculator pi.

zarghamenotes | Student, Grade 10 | (Level 1) Honors

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Imagine the whole universe to be a room.Now, there are different things to be placed inside this room.Every object is made of matter and has a boundary or surface separating the matter from the surrounding atmosphere.The area of the boundary is known as the surface area.It is calculated by simply adding the area of all the surfaces.

freedomfreedom | Student, Grade 10 | (Level 1) Honors

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Surface Area of a Cube = 6 a2

(a is the length of the side of each edge of the cube)

In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a 2 . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

(a, b, and c are the lengths of the 3 sides)

In words, the surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.

The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.

Surface Area of Any Prism

(b is the shape of the ends)

Surface Area = Lateral area + Area of two ends

(Lateral area) = (perimeter of shape b) * L

Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

Surface Area of a Sphere = 4 pi r 2

Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h

(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

In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label 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 of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So 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

Tip! Don't forget the units.

These equations will give you correct answers if you keep the units straight. For example - to find the surface area of a cube with sides of 5 inches, the equation is:

Surface Area = 6*(5 inches)2

= 6*(25 square inches)

= 150 sq. inches

Sources:

ngyunhui | Student | (Level 1) Honors

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A typical problem involving the volume or surface area of a cylinder gives us the volume or height and/or radius of the cylinder. We then need to calculate the unknown quantities based on the information given about the others. Suppose that the height of a cylinder is 30 cm and its volume is 750p cm3. Find its radius and surface area.
To get started, we need to organize as much of the given information as possible into a known formula. Since the volume (750p cm3) and height (30 cm) of the cylinder are given, we will start with the equation for volume.
V = pr2h = 30pr2
V = 750p
750p = 30pr2
r = 5 cm
Now we will find the surface area of the cylinder using our values for the radius and height.
S = 2pr2 + 2prh
S = 2p(25) + 2p(5)(30)
S = 50p + 300p
S = 350p cm2

nikasho0100 | Student, Grade 12 | (Level 1) Salutatorian

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Surface area of a cylinder is

Itz done by adding the area of the 2 circles @ the bottum & up of the cylinder and adding the curved surface area of the cylinder.

In replacement of 22/7.....3.14 can be used or the pie sign in the calculator.

I have no idea to find the surface area of a prism.....

rosememe | Student | (Level 1) eNoter

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The surface area of any prism equals the sum of the areas of its faces, which include the floor, roof and walls. Because the floor and the roof of a prism have the same shape, the surface area can always be found as follows:

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

poems for funerals

sharonsub | Middle School Teacher | (Level 1) eNoter

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Generally, the formula for surface area is:  SA = Bh, where B is area of the base and h is the height.   You need to understand what your base is.

Triangular prism:  1) the base of a triangular prism is the triangle, so the B in your formula is area of a triangle, or (base x height)/2.

2) Make sure that the numbers you use to find B are the numbers/dimensions of the TRIANGLE, base and height of the triangle, and divided by 2

3)  Once you have B, then multiply it by the height of the PRISM, thus solving the Bh.

Cylinder:  1) the base of a cylinder is a circle, so in your Bh formula, the B will be equal to the area of a circle or pi x radius^2 (or 3.14 x radius x radius)

2)  Once you have solved this, you have your B, and now you multiply it by your h, or height of the cylinder

melonsmasher | Student, Grade 9 | (Level 1) Valedictorian

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To solve any Volume and Surface area problems you have to remember some formulas:-

1) Surface Area of a Cuboid= 2(l*b+b*h+l*h)

2) Surface Area of a Cube=   6a2

a=side. In a cube the four sides will be equal.

Good luck for Maths

vikas17 | College Teacher | (Level 1) eNoter

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simple for triangular prism.

1.  Find the area (A) of the base ab/2 where a is the altitude and b is the base.

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.

For cylinder same as we did for triangular prism.

1. calculate the aread of the circle at the bottom. pi*R^2.

2. calculate the circumference of the circle 2*pi*r.

Now the surface area would be 2*pi*r +2* pi* r^2.

xrayx | Student, Grade 10 | (Level 1) eNoter

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In the majority of the cases , the surface area is the sum of all the areas of all the shapes that cover the surface of the object. So if you want to find the total surface area of a cube with sides all measuring 5 in, you would find the area of one side which = 25 and add the area of the rest of the sides. Your answer will be 210

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