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Five Methods:Helpful Guidance3-4-5 Pythagorean Triple MethodDirect Proportional AdjustmentCartesian Graph ApproachUse Proportions

You will learn how, in a few steps, to find the area of a rectangle by knowing that its diagonal is the hypotenuse of the rectangle's width and height. Two methods are limited in scope to a single Pythagorean Triple and thus are not universally applicable.

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EditMethod 1 of 5: Helpful Guidance

  1. 1
    Make use of helper articles when proceeding through this tutorial:
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EditMethod 2 of 5: 3-4-5 Pythagorean Triple Method

  1. 1
    Let the diagonal = c^2 and let the height and width of the rectangle = a^2 and b^2, respectively.
  2. 2
    Let the diagonal = 225, and let its square = 60,025; you know that a Pythagorean Triple is 3-4-5. with the hypotenuse equal  to 5, so now you want to know how to get the other sides, given that information.
  3. 3
    Reason the answer is going to be =(3*x^2)^2+(4*x^2)^2, or 3^2x^4 +4^2x^4. Since 3 and 4 are close to each other, try a number between 3^2 and 4^2, like 12.5 and various x's to the 4th power. Try 6. 6^4 * 12.5 = 16,228.512 which is not bad, considering we averaged, but it's low.
  4. 4
    Try 7 because 60,025/5^2=7^4=2401:   7^4 *9 + 7^4 *16 = 60,025 -- the square of the diagonal! You did it! 3^2 = 9 and 4^2 = 16 and x^4 = 7^4.
  5. 5
    Learn that the shorter side is sqrt(3* 7^2)^2 = 147, and the longer side is sqrt(4 * 7^2)^2 = 196. 147^2 + 196^2 = 60,025 = c^2. Its square root is 225, the diagonal.
  6. 6
    Deduce that the area of the rectangle then is 196 * 147 = 28,812
    • You are warned, however, that this method works because many Pythagorean Triples are related to the 3-4-5 triple, but not all. However, there are various other triples, since all that is required of the triangle is that it have a 90 degree angle in it.
    • Still, with diligence, one may discover the given relation for a given rectangle by close observation of how long and thin it is.
    • Direct relative measurement of the sides of the rectangle also helps in finding its area, but error can creep in with direct measurement so formulation is needed to provide a more exact answer, typically.

EditMethod 3 of 5: Direct Proportional Adjustment

  1. 1
    Try Direct Proportional Adjustment of the 3-4-5 Triple. If you are given c = 7.5, then divide 7.5/5 to arrive at 1.5; then multiply 4*1.5 to arrive at b = 6 and multiply 3* 1.5 to arrive at a = 4.5; 4.5^2 + 6^2 = 56.25 and c=7.5^2 = 56.25 also, so side a= 4.5 and side b = 6 and the area = 4.5* 6 = 27 square units.

EditMethod 4 of 5: Cartesian Graph Approach

  1. 1
    • Try also the Cartesian Graph approach, which is a very reliable method:
    • Draw the diagonal hypotenuse starting at point {0, 0} using a ruler, and allowing for approximation that may be required. If the length of the diagonal is 7.2 units, then its slope depends upon a and b of a^2 + b^2 = (7.2)^2 = 51.84, making a and b about 5 and 5.18 if the rectangle is close to a square; therefore, slope m = (y2-y1) / (x2-x1) = (5.18-0)/(5-0) = 5.18/5 = 1.036; or just a shade greater than 1. If the rectangular form is given and it is not close to a square, proportion it with a ruler, and make adjustments accordingly.
    • Draw the surrounding rectangle with very straight lines;
    • Let 5^2 + 5.18^2 = 51.8324 whereas 7.2^2 = 51.84 -- you are probably close enough - check the problem requirements for tolerance/accuracy/exactitude required.
    • You're done! The area of the rectangle is 5 * 5.18 = 25.9

EditMethod 5 of 5: Use Proportions

  1. 1
    You are given the length of the diagonal. Use your compass and rule to make a line equal to the diagonal in a separate area.
  2. 2
    Use your compass and rule to place straight lines equal to sides a and b alongside diagonal c.
  3. 3
    Use proportions to measure the lengths of sides a and b. That is, find the relation a:b:c where c is of given length longer than a or b and a is less than b. Indirectly measure a and b and fill in the proportion to obtain lengths a and b.
  4. 4
    Check your measurement by using the Pythagorean Theorem.
  5. 5
    Calculate the area of the rectangle as side a times side b.
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Categories: Mathematics | Calculating Volume and Area

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