Find the Area of a Quadrilateral

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So you've been assigned homework that requires you to find the area of a quadrilateral ... but you don't even know what a quadrilateral is. Don't worry—help is here! A quadrilateral is any shape with four sides — squares, rectangles, and diamonds are just a few examples. To find a quadrilateral's area, all you have to do is to identify the type of quadrilateral you're working with and follow a simple formula. That's it!

Steps

Area of Square, Rectangle, and Rhombus Cheat Sheets

Doc:Area of a Square Diagram,Area of a Rectangle Diagram,Area of a Rhombus Diagram

Area of Trapezoid and Kite Cheat Sheets

Doc:Area of a Trapezoid Diagram,Area of a Kite Diagram

Squares, Rectangles and Other Parallelograms

  1. Know how to identify a parallelogram. A parallelogram is any four-sided shape with two pairs of parallel sides where the sides across from each other are the same length. Parallelograms include:
    • Squares: Four sides, all the same length. Four corners, all 90 degrees (right angles).
    • Rectangles: Four sides; opposite sides have same lengths. Four corners, all 90 degrees.
    • Rhombuses: Four sides; opposite sides have same lengths. Four corners; none have to be 90 degrees but opposite corners must have the same angles.
  2. Multiply base times height to get the area of a rectangle. To find the area of a rectangle, you need two measurements: the width, or base (the longer side of the rectangle), and the length, or height (the shorter side of the rectangle). Then, just multiply them together to get the area. In other words:
    • Area = base × height, or A = b × h for short.
    • Example: If the base of a rectangle has a length of 10 inches and the height has a length of 5 inches, then the area of the rectangle is simply 10 × 5 (b × h) = 50 square inches.
    • Don't forget that when you're finding a shape's area, you will use square units (square inches, square feet, square meters, etc.) for your answer.
  3. Multiply one side by itself to find the area of a square. Squares are basically special rectangles, so you can use the same formula to find their area. However, since a square's sides all have the same length, you can use the shortcut of just multiplying one side's length by itself. This is the same as multiplying the square's base by its height because the base and height are simply always the same. Use the following equation:[1]
    • Area = side × side or A = s2
    • Example: If one side of a square has a length of 4 feet, (t = 4), then the area of this square is simply t2, or 4 x 4 = 16 square feet.
  4. Multiply the diagonals and divide by two to find the area of a rhombus. Be careful with this one — when you're finding the area of a rhombus, you can't simply multiply two adjacent sides. Instead, find the diagonals (the lines connecting each set of opposite corners), multiply them, and divide by two. In other words: [2]
    • Area = (Diag. 1 × Diag. 2)/2 or A = (d1 × d2)/2
    • Example: If a rhombus has diagonals with a length of 6 meters and 8 meters, then its area is simply (6 × 8)/2 = 48/2 = 24 square meters.
  5. Alternatively, use base × height to find the area of a rhombus. Technically, you can also use the base times height formula to find the area of a rhombus. Here, "base" and "height" don't mean you can just multiply two adjacent sides, however. First, pick one side to be the base. Then, draw a line from the base to the opposite side. The line should meet both sides at 90 degrees. The length of this side is what you should use for height.
    • Example: A rhombus has sides of 10 miles and 5 miles. The straight-line distance between the {{safesubst:#invoke:convert|convert}} sides is {{safesubst:#invoke:convert|convert}}. If you want to find the area of the rhombus, you would multiply 10 × 3 = 30 square miles.
  6. Be aware that the rhombus and rectangle formulas work for squares. The side × side formula given above for squares is by far the most convenient way to find the area for these shapes. However, because squares are technically both rectangles and rhombuses as well as squares, you can use those shapes' area formulas for squares and get the correct answer. In other words, for squares:
    • Area = base × height or A = b × h
    • Area = (Diag. 1 × Diag. 2)/2 or A = (d1 × d2)/2
    • Example: A four-sided shape has two adjacent sides with lengths of 4 meters. You can find the area of this square by multiplying its base times its height: 4 × 4 = 16 square meters.
    • Example: A square's diagonals are both equal to 10 centimeters. You can find this square's area with the diagonal formula: (10 × 10)/2 = 100/2 = 50 square centimeters.

Finding the Area of a Trapezoid

  1. Know how to identify a trapezoid. A trapezoid is a quadrilateral with at least two sides that run parallel to each other. Its corners can have any angles. Each of the four sides on a trapezoid can be a different length.
    • There are two different ways you can find the area of a trapezoid, depending on which pieces of information you have. Below, you'll see how to use both.
  2. Find the height of the trapezoid. The height of a trapezoid is the perpendicular line connecting the two parallel sides. This will not usually be the same length as one of the sides, because the sides are usually pointed diagonally. You will need this for both area equations. Here's how to find the height of a trapezoid:[3]
    • Find the shorter of the two base lines (the parallel sides). Place your pencil at the corner between that baseline and one of the non-parallel sides. Draw a straight line that meets the two base lines at right angles. Measure this line to find the height.
    • You can also sometimes use trigonometry to determine the height if the height line, the base, and the other side make a right triangle. See our trig article for more information.
  3. Find the area of the trapezoid using the height and the length of the bases. If you know the height of the trapezoid as well as the length of both bases, use the following equation:
    • Area = (Base 1 + Base 2)/2 × height or A = (a+b)/2 × h
    • Example: If you have a trapezoid with one base of 7 yards, another base of 11 yards, and the height line connecting them is 2 yards long, you can find its area like this: (7 + 11)/2 × 2 = (18)/2 × 2 = 9 × 2 = 18 square yards.
    • If the height is 10 and the bases have the lengths of 7 and 9, then you can find the area simply by doing the following: (7 + 9)/2 * 10 = (16/2) * 10 = 8 * 10 = 80
  4. Multiply the midsegment by two to find the area of a trapezoid. The midsegment is an imaginary line that runs parallel to the bottom and top lines of the trapezoid and is exactly the same distance from each. Since the midsegment is always equal to (Base 1 + Base 2)/2, if you know it, you can use a shortcut for the trapezoid formula:
    • Area = midsegment × height or A = m × h
    • Essentially, this is the same as using the original formula except that you're using "m" instead of (a + b)/2.
    • 'Example:' The midsegment of the trapezoid in the example above is 9 yards long. This means we can find the area of the trapezoid simply by multiplying 9 × 2 = 18 square yards, just like before.

Finding the Area of a Kite

  1. Know how to identify a kite. A kite is a four-sided shape with two pairs of equal-length sides that are adjacent to each other, not opposite each other. Like their name suggests, kites resemble real-life kites.
    • There are two different ways to find the area of a kite depending on which pieces of information you have. Below you will find how to use both.
  2. Use the rhombus diagonal formula to find the area of a kite. Since a rhombus is just a special kind of kite where the sides are the same length, you can use the diagonal rhombus area formula to find a kite's area as well. As a reminder, diagonals are the straight lines between two opposite corners on the kite. Like a rhombus, the kite area formula is:
    • Area = (Diag. 1 × Diag 2.)/2 or A = (d1 × d2)/2
    • Example: If a kite has diagonals with lengths of 19 meters and 5 meters, then its area is simply (19 × 5)/2 = 95/2 = 47.5 square meters.
    • If you don't know the lengths of the diagonals and can't measure them, you can use trigonometry to calculate them. See our kite article for more information.
  3. Use the lengths of the sides and the angle between them to find the area. If you know the two different values for the lengths of the sides and the angle at the corner between those sides, you can solve for the area of the kite with the principles of trigonometry.[4] This method requires you to know how to do sine functions (or at least to have a calculator with a sine function). See our trig article for more information or use the formula below:
    • Area = (Side 1 × Side 2) × sin (angle) or A = (s1 × s2) × sin(θ) (where θ is the angle between sides 1 and 2).
    • Example: You have a kite with two sides of length 6 feet and two sides of length 4 feet. The angle between them is about 120 degrees. In this case, you can solve for the area like this: (6 × 4) × sin(120) = 24 × 0.866 = 20.78 square feet
    • Note that you need to use the two different sides and the angle between them here — using the set of sides with the same length won't work.

Solving for Any Quadrilateral

  1. Find the lengths of all four sides. Does your quadrilateral not fall into any of the tidy categories above (for instance, does it have sides with all different lengths and zero parallel sets of sides?) Believe it or not, there are formulas you can use to figure out the area of any quadrilateral, regardless of its shape. In this section, you will find how to use the most common one. Note that this formula requires knowledge of trigonometry (once again, here is our basic trig guide.
    • First, you must find lengths of each of the quadrilateral's four sides. For the purposes of this article, we will label them a, b, c and d. Sides a and c are opposite from each other and sides b and d are opposite each other.
    • Example: If you have an oddly-shaped quadrilateral that doesn't fit in any of the categories above, first, measure its four sides. Let's say that they have lengths of 12, 9, 5, and 14 inches. In the steps below, you'll use this information to find the shape's area.
  2. Find the angles between a and d and b and c. When you're working with an irregular quadrilateral, you can't find the area from the sides alone. Continue by finding two of the opposite angles. For the purposes of this section, we'll use angle A between sides a and d, and angle C between sides b and c. However, you can also do this with the two other opposite angles.
    • Example: Let's say that in your quadrilateral, A is equal to 80 degrees and C is equal to 110 degrees. In the next step, you'll use these values to find the total area.
  3. Use the triangle area formula to find the area of the quadrilateral. Imagine that there is a straight line from the corner between a and b to the corner between c and d. This line would split the quadrilateral into two triangles. Since the area of a triangle is absinC, where C is the angle between sides a and b, you can use this formula twice (once for each of your imaginary triangles) to get the total area of the quadrilateral. In other words, for any quadrilateral:
    • Area = 0.5 Side 1 × Side 4 × sin(Side 1&4 angle) + 0.5 × Side 2 × Side 3 × sin (Side 2&3 angle) or
    • Area = 0.5 a × d × sin A + 0.5 × b × c × sin C
    • Example: You already have the sides and angles you need, so let's solve:
      = 0.5 (12 × 14) × sin (80) + 0.5 × (9 × 5) × sin (110)
      = 84 × sin (80) + 22.5 × sin (110)
      = 84 × 0.984 + 22.5 × 0.939
      = 82.66 + 21.13 = 103.79 square inches
    • Note that if you're trying to find the area of a parallelogram, in which the opposite angles are equal, the equation reduces to Area = 0.5*(ad + bc) * sin A.

Tips

Related Articles

Sources and Citations

  1. ↑ http://www.dummies.com/how-to/content/how-to-calculate-the-area-of-a-quadrilateral.html
  2. ↑ http://www.mathopenref.com/rhombusarea.html
  3. ↑ http://www.mathgoodies.com/lessons/vol1/area_trapezoid.html
  4. ↑ http://www.mathopenref.com/kitearea.html
  5. ↑ http://www.handymath.com/cgi-bin/irregangle12.cgi?submit=Entry
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