Total Surface Area - the trick to getting it right



How to Find the Surface Area of a Box

Three Methods:

Finding the surface area of a box is easy as long as you know the length of the sides. Once you know how long the sides are, you simply have to plug them into a basic equation to get your answer. You can even find the surface area of cylindrical boxes with a few simple measurements.

Steps

Rectangular Boxes

  1. Add together the area of each side to get the surface area of the box.Surface area is the total area of each side. As long as you know how to find the area of a regular rectangle, which is simply the length times the height, you can find each side and add them together. However, there is also a simplified formula that will do this for you if you know the measurements of the box:SurfaceArea=2lw+2lh+2wh{\displaystyle SurfaceArea=2lw+2lh+2wh}
    • lis the length of the box, or the longest side.
    • his the height of the box.
    • wis the width of the box.
  2. Measure the length of the box.This is your longest side. There should be 4 identical lines equally long across the whole box. Lay the box down on its longest side to make it easier to measure.
    • Example:The length of the box is 5 feet.
  3. Measure the height of the box, or the distance from the floor to the top.Make sure you don't measure the same side as the length.
    • Example:The height of the box is 4 feet.
  4. Measure the width of the box.This is the side directly perpendicular (it forms a right angle, or L) with the length. Make sure you are not remeasuring the height.
    • Example:The width of the box is 2 feet.
  5. Ensure that you didn't measure the same side twice.The easiest way to prevent this is to check that all three lines meet at some point. Find a corner and measure the three sides coming off of it -- this ensures you have the right measurements.
    • The sides may be the same measurement. You just want to make sure you're measuring three unique sides, even if two or three of them are all identical.
  6. Input your measurements into the formula to solve the equation.Once you've got the numbers, the rest is easy. Simply input the measurements and add it all up.
    • SurfaceArea=2lw+2lh+2wh{\displaystyle SurfaceArea=2lw+2lh+2wh}
    • SurfaceArea=2(5)(2)+2(5)(4)+2(2)(4){\displaystyle SurfaceArea=2(5)(2)+2(5)(4)+2(2)(4)}
    • SurfaceArea=20+40+16{\displaystyle SurfaceArea=20+40+16}
    • SurfaceArea=76{\displaystyle SurfaceArea=76}
  7. Express your answer in "units squared" when done.This lets people know how you measured everything, and is a crucial part of your answer. Luckily, all you need to do is use the units provided in the question. For this example, since the units were in feet, you would put "feet squared" at the end of your answer:
    • You have a box 5 feet long, 4 feet high, and 2 feet wide, what is the total surface area?
    • Answer:76feet2{\displaystyle 76feet^{2}}
  8. Break complex boxes into smaller pieces to find surface area.For example, say you had an "L" shaped box. Instead of finding everything by hand, simply think of two separate boxes, the vertical half on top and the horizontal half on the bottom. Find the surface area of both, then add them together to get the total surface area.For example, if you have a U shaped box:
    • Say the bottom has a surface area of 12 units squared.
    • Say both sides have a surface area of 15 units squared.
    • The total area would be42 units squared, since12+15+15=42{\displaystyle 12+15+15=42}.

Cylindrical Boxes (Tubes)

  1. Add the area of the bases to the height times circumference to find the surface area of a cylinder.This only works with right cylinders, meaning they are not slanted. The proper formula isSurfaceArea=2B+hC{\displaystyle SurfaceArea=2B+hC} For example, if the area of the base is 3, the height is 5, an the circumference is 6, what is the surface area? Answer =36 units squared.
    • Bis the area of the base.
    • his the height of the cylinder.
    • Cis the circumference of the base.
  2. Find the area of the base.The base is the circular end of the cylinder. It can be found with the formulaArea=pi∗r2{\displaystyle Area=pi*r^{2}}Ris the radius of the circle.Piis the constant, rounded to 3.14 for simple problems. You can also leave it as "pi" if you do not have a calculator.
    • Example:The radius of the base in a cylinder is 2. What is the area of the base?
    • pi∗(2)2{\displaystyle pi*(2)^{2}}
    • B =4pi{\displaystyle 4pi}
  3. Calculate the circumference if the base.The circumference is the distance around the edge of the circle. It is found with the formulaCircumference=2∗r∗pi{\displaystyle Circumference=2*r*pi}Continuing the previous example:
    • 2∗pi∗(2){\displaystyle 2*pi*(2)}
    • C =4pi{\displaystyle 4pi}
  4. Find the height of the cylinder by measuring the space between the two bases.The height of a cylinder is just another way to figure out how long it is. It is the straight line between the center of the two bases.
    • Example:In the same cylinder, with radius 2 inches, the height is 5 inches.
    • h=5{\displaystyle h=5}
  5. Input your smaller parts into the formula to find your surface area.Once you've got your base area, circumference, and height, all you need to do is plug the numbers into your formula to get the right answer.
    • SurfaceArea=2B+hC{\displaystyle SurfaceArea=2B+hC}
    • SurfaceArea=2(4pi)+(5)(4pi){\displaystyle SurfaceArea=2(4pi)+(5)(4pi)}
    • SurfaceArea=8pi+20pi{\displaystyle SurfaceArea=8pi+20pi}
    • SurfaceArea=28pi{\displaystyle SurfaceArea=28pi}
  6. Express your answer in units squared.All problems of surface area need units to give them scale. Is the area in inches, or feet, for example. The units always match the units given to you in the problem. If no units are given, you should just write the phrase "units squared," or "units2{\displaystyle units^{2}}"
    • In the example, the units were given as inches. So the final answer would be28pi{\displaystyle 28pi}inches2{\displaystyle inches^{2}}

Practice Problems

  1. Try out some practice problems with rectangles.To see the answers, highlight right after the arrow:
    • L = 10, W = 3, H = 2, →112 units squared
    • L = 6.2, W = 2, H = 5.4 →113.36 units squared
    • On a rectangular box, the top is 5x3x2, the bottom 6x2x2. →118pi units squared
  2. Try out some practice problems with cylinders.To see the answers, highlight after the arrow:
    • Base = 3, Height = 10, Circumference = 1.5 →21 units squared
    • Base = 25pi, Height = 3, Circumference = 10pi →80pi units squared
    • Radius = 3, height = 3 →36pi units squared

Community Q&A

Search
  • Question
    I'm painting a box 2' wide by 3' long by 2' high. If I'm not painting the bottom, how do I calculate the surface area I need to cover?
    Top Answerer
    The top is 2 x 3; two sides are each 3 x 2; and the other two sides are each 2 x 2.
    Thanks!
  • Question
    What is the surface area of a box if the length is 4m the width is 2m and the height is 2m?
    Top Answerer
    Multiply the length by the width and double it. Multiply the length by the height and double it. Multiply the width by the height and double it. Add those three amounts together.
    Thanks!
  • Question
    What is the surface area of a box open on both ends?
    Top Answerer
    The surface area equals the height multiplied by the sum of twice the length and twice the width.
    Thanks!
  • Question
    How do I find the area of a long, rectangular, hollow box with no thickness?
    Top Answerer
    First, we'll assume you're looking for the total surface area of a rectangular (or rectilinear) box, and we'll define that box as having a length (L), a width (W), and a depth (D). Find twice the product of L x W; to that add twice the product L x D; to that add twice the product W x D. The sum is the total surface area.
    Thanks!
  • Question
    How do I increase 10 percent of 50?
    Top Answerer
    If you're asking how to increase 50 by 10%, calculate 10% of 50, and add that number to 50.
    Thanks!
  • Question
    V is 180 cubic centimeters and L is 15 centimeters what is h?
    Top Answerer
    You can't find "h" without also knowing "w".
    Thanks!
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Quick Summary

To find the surface area of a box, start by calculating the area of each side using the formula a = lh, where l is the length and h is the height. Once you know the area of every side, add them all together to get the surface area of the box. If the box is cylindrical, you'll need to use the formula: surface area = 2b + hc, where b is the area of the base of the box, h is the height, and c is the circumference of the base.

Did this summary help you?
  • If you are measuring a real box, you might want to try measuring different edges that should be the same length, and then use the average.

Things You'll Need

  • A box and a way to measure it.
  • Measurements of an existing or hypothetical box.

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Sources and Citations

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Date: 05.12.2018, 01:13 / Views: 64174