Find the Surface Area of Prisms

In geometry, a prism is a three-dimensional shape with the same polygon at its ends. The sides are usually rectangles or parallelograms, though cylinders and other shapes can also be considered prisms. To find the surface area of different prisms, follow these instructions.

Steps

  1. Label as many sides as you can. The nice thing about prisms is that the matching end pairs make it easy to determine a lot of the individual measurements without having to solve for anything. Any measurement that is given for one of the end shapes (ex. height of the triangle, radius of the circle) also applies to the matching shape at the other end. Moreover, if you know the length of one side of the prism, you know the length of all the sides. Distribute these numbers as necessary around your prism to see what you have to work with.

Find the Area of One of the Matching Ends

  1. If the matching ends are triangles, find the area of one triangle using this guide.
  2. If the matching ends are squares or rectangles, multiply base by height. The base and height are simply the measurements of two perpendicular sides of the square or rectangle; for a square, this two measurements are the same. Simply multiply both numbers together to find the area of the end.
  3. If the matching ends are circles, multiply pi by the radius squared. The radius is the length from the center of the circle to the outside edge. Square this number (i.e. multiply it by itself) and then multiply the result by pi (3.14159…). This gives you the area of the end.
    • If you’ve been given the diameter (i.e. the length across the entire circle), divide this number in half to find the radius.
    • If you’ve been given the circumference (i.e. the length of the outside edge of the circle), divide this number by pi and divide that result by 2 to find the radius.
  4. If the matching ends are parallelograms, multiply base by height. Parallelograms are slanted squares (like open-ended boxes that have been pushed to one side); they have two pairs of parallel sides but none of the corners are right angles. The base of a parallelogram is simply the length of one of the two long, misaligned sides; the height, however, is the distance between these two sides, not the length of one of the angled sides. If this height isn’t already given to you, the problem will ask you to solve for it by turning on of the angled sides into a right triangle and giving you the length of two of the sides of the triangle. To solve for height this way:
    • Use the Pythagorean Theorem, which is A^2 + B^2 = C^2. The hypotenuse of the triangle, or C, is simply the side of the triangle opposite the right angle. We’ll call the other side that’s been given B. To solve for the height, which we’ll call A, rearrange the formula to A^2 = C^2 – B^2. Multiply C by itself, then multiply B by itself. Subtract the second result from the first to get A^2; to then solve for A, find the square root of this sum. This is the height of the parallelogram, which you can now multiply by the base to find the total area.
  5. If the matching ends are another polygon, break the shape down into triangles to solve. A pentagon, for example, can be broken down into 5 equal triangles; a hexagon can be broken down into 6; and so on. When you’ve finished drawing the triangles, solve for the area of one triangle using this guide. When you’re done, multiply that area by the total number of equal triangles you’ve drawn.
    • If the polygon can’t be broken down into perfect triangles, break it down into triangles and squares. Find the area of each shape individually using the shape guides above and then add them together to find the total area of the polygon.
  6. Note the area of this end on your paper and leave it alone for now. You’ll come back to it later.

Find the Perimeter of One of the Matching Ends

  1. Solve for any missing sides. After solving for area, you may already know the length of every side of the shape at the end of your prism. If not, solve using one of the following methods:
    • If the matching ends are triangles, solve for all sides by using the Pythagorean Theorem. The Pythagorean Theorem is A^2 + B^2 = C^2: A and B are the base and height of a right triangle and C is the hypotenuse, which is simply the side opposite the right angle.
      • If you have been given A and B, use the formula C^2 = A^2 + B^2. Multiply A by itself, multiply B by itself, and add the two numbers together; this gives you C^2. To then solve for C, simply find the square root of this sum.
      • If you have been given C and B: use the formula A^2 = C^2 – B^2. Multiply C by itself, multiply B by itself, and subtract the second result from the first; this gives you A^2. To then solve for A, simply find the square root of this sum.
      • If you have been given C and A: use the formula B^2 = C^2 – A^2. Multiply C by itself, multiply A by itself, and subtract the second result from the first; this gives you B^2. To then solve for B, simply find the square root of this sum.
    • If the matching ends are circles, find the circumference. The formula for circumference is C = D x pi: C is circumference and D is diameter. If you have the radius, simply multiply this by 2 to find the diameter.
    • If the matching ends are another polygon, break the shape down into triangles and/or squares as before and find the outside edges by solving for these shapes individually. Use the shape guides above if necessary.
  2. Mark the perimeter measurements on your paper. You can then use them to determine the area of the sides of the parallelogram.

Find the Area of Each Side

  1. Note the length of the prism. This is the distance between the two matching ends of the prism. Since the ends of the prism are parallel, this distance will be uniform throughout – even if the matching ends are both angled. This means that if you know the length of one side, you know the length of all the sides.
  2. Find the area of each side. Each side will either be a square/rectangle or a parallelogram. Parallelograms are slanted squares (like open-ended boxes that have been pushed to one side); they have two pairs of parallel sides but none of the corners are right angles.
    • To find the area of a square/rectangle, multiply base by height. The base and height are simply the measurements of two perpendicular sides of the square or rectangle; for a square, this two measurements are the same. Simply multiply both numbers together to find the area of the end.
    • To find the area of a parallelogram, multiply base by height. Note that the base of a parallelogram is simply the length of one of the two long, misaligned sides; the height, however, is the distance between these two sides, not the length of one of the angled sides. If you only know the length of the angled side but not the parallelogram’s true height, draw a line through either side of the parallelogram, turning it into a perfect square/rectangle with a triangle on either end. To find the height using this triangle:
      • Use the Pythagorean Theorem, which is A^2 + B^2 = C^2. The hypotenuse of the triangle, or C, is simply the side of the triangle opposite the right angle. We’ll call the other side that’s been given B. To solve for the height, which we’ll call A, rearrange the formula to A^2 = C^2 – B^2. Multiply C by itself, then multiply B by itself. Subtract the second result from the first to get A^2; to then solve for A, find the square root of this sum. This is the height of the parallelogram, which you can now multiply by the base to find the total area.
    • If the prism is cylindrical, find the area of the sides by multiplying circumference, which you solved for in the previous section, by the total height. (Image the cylinder has a piece of paper wrapped around, which, when unwrapped, forms a perfect square or rectangle. The circumference can then be thought of as the length of that piece of piece of paper, which can be solved like any square by multiplying length times height.)

Find the Total Area

  1. Multiply the area of the end of the prism by 2. Find the number you marked down when you solved for the area of one of the matching ends and double it to account for the other end.
  2. Add up the areas of the sides of the prism. If your prism has triangular ends, you will be adding up three sides; if they’re pentagonal, you will be adding five sides; etc. If the prism is cylindrical, you don’t have to add anything as there’s only one “side”.
  3. Add the total area of the ends to the total area of the sides. This gives you the total surface area of the prism.

Tips

  • Treating each face of the shape as a separate entity will keep the math simple and approachable; instead of seeing it as an overwhelming multi-faced object, just think of it as a few squares and triangles.
  • Labeling your subtotals is extremely important so you can total them all!
  • To deal with a trapezoidal end shape: the formula is A = h*(b1 + b2)/2, where b1 and b2 are the two base edges of that trapezoid and, incidentally, the dividing by 2 actually finds the average of the two bases to account for there being two different bases in a trapezoid (which helps to understand and remember why that formula is not A = b*h).

Warnings

  • Do 'NOT' try to make this process into one formula.

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Sources and Citations

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