So, you have homework that asks you to find the area of a quadrilateral… but you don't even know what a quadrilateral actually is. Don't worry - help has arrived! A quadrilateral is any shape that has four sides - squares, rectangles, and diamonds are just a few. To find the area of a quadrilateral, all you need to do is identify the type of quadrilateral you're working with and then follow a simple formula. That is all!
Method 1 of 4: Squares, rectangles and other parallelograms
Step 1. Know how to identify a parallelogram
A parallelogram is any four-sided shape that has two pairs of parallel sides, with the opposite sides of the same length. Parallelographs include:
four sides, all with the same measurement. Four corners, all with 90 degree angles (right angles).
four sides, with opposites of equal length. Four corners, all with 90 degree angles.
four sides, with opposites of equal length. Four corners - none of them have a 90 degree angle, but all opposites must have equal angles.
Step 2. Multiply the base by the height to get the area of a rectangle
To find the area of a rectangle, you need two measurements: the width, or base (the longest side of the rectangle), and the length, or height (shortest side of the rectangle). So just multiply them to get the area. In other words:
- Area = base × height or A = b × h (from English Height).
if the base of a rectangle has a base of 10 centimeters and a height of 5 centimeters, the area of the rectangle is equal to 10 × 5 (b × h) = 50 square centimeters.
- Don't forget: when looking for the area of a shape, you need to use square units (square centimeters, square meters, square kilometers, etc.) in your answer.
Step 3. Multiply a side by itself to find the area of a square
Basically, squares are special rectangles, so you can use the same formula to figure out their area. However, since the sides of a square are all the same size, you can use the shortcut to multiply a side by itself. Performing this calculation is equal to multiplying the base of the square by its height, since both measurements will always be the same. Use the following equation:
- Area = side × side, A = s2 (from English sgo) or A = h2.
if one side of the square is 4 meters long (s = 4), its area is simply equal to s2, or 4 × 4 = 16 square meters.
Step 4. Multiply the diagonals and divide the result by two to find the area of a diamond
Be careful with this equation - when you're trying to figure out the area of a diamond, you can't just multiply two adjacent sides. Instead, find the diagonals (the lines connecting each set of opposite corners), multiply them, and divide the result by two. In other words:
- Area = (diagonal 1 × diagonal 2)/2 or A = (d1 × d2)/2.
if a diamond has diagonals with lengths equal to 6 and 8 meters, respectively, its area will be equal to (6 × 8)/2 = 48/2 = 24 square meters.
Step 5. Alternatively, use the base × height formula to find the area of a diamond
Technically, it is also possible to use the base × height formula to find out what the area of a diamond is. Here, however, “base” and “height” do not mean that it is possible to simply multiply two adjacent sides. First of all, choose a side and base it on. Then draw a line from the base to the opposite side. It should meet both sides at a 90 degree angle. The length of that side will be your height measurement.
a diamond has sides equal to 10 and 5 kilometers. The distance in a straight line that passes between the sides of 10 kilometers totals 3 kilometers. If you want to find the area of the diamond, just multiply 10 × 3 = 30 square kilometers.
Step 6. Be aware that formulas for diamonds and rectangles also work on squares
The side × side formula given above for squares is, in fact, the most convenient way to find the area of these shapes. However, since squares are also technically rectangles and diamonds, it is possible to use the formulas corresponding to these shapes for the squares and get a correct answer. In other words, for squares:
- Area = base × height or A = b × h.
- Area = (diagonal 1 × diagonal 2)/2 or A = (d1 × d2)/2.
a four-sided shape has two sides that are 4 meters long. You can find the area of this square by multiplying its base by its height: 4 × 4 = 16 square meters.
the diagonals of a square are both equal to 10 centimeters. You can find the area of this square with the diagonal formula: (10 × 10)/2 = 100/2 = 50 square centimeters.
Method 2 of 4: Finding the Area of a Trapezium
Step 1. Know how to identify a trapeze
The trapeze is a quadrilateral with at least two sides parallel to each other. Its corners can have any type of angle. Each of the four sides of a trapeze can be a different size.
There are two different ways to find the area of a trapeze, depending on what information is available. Below, you can check both
Step 2. Find the height of the trapeze
The height of a trapezoid is represented by the perpendicular line connecting both parallel sides. It will not be the same length on either side, as they are usually designed diagonally. You will need this value for both area equations. Learn here how to find the height of a trapeze:
- Find the shorter of the two baselines (parallel sides). Position your pencil in the corner between the base and one of the non-parallel sides. Draw a straight line passing from one line to the other at a right angle. Measure this line to find the height.
- Occasionally, you can also use trigonometry to determine height, when the height line, base, and other side make up a right triangle. Read our trigonometry article for more information.
Step 3. Find the trapeze area using the height and length of the bases
If you know the measurement of the height of the trapeze, as well as its bases, use the following equation:
- Area = (base 1 + base 2)/2 × height or A = (b1 + b2)/2 × h.
if you have a trapeze with a base of 7 meters, another base of 11 meters and a height of 2 meters, you can find your area as follows: (7 + 11)/2 × 2 = (18)/2 × 2 = 9 × 2 = 18 square meters.
- If the height is equal to 10 and the bases have measurements equal to 7 and 9, you can find the trapezoid area just by doing the following: (7 + 9)/2 × 10 = (16/2) × 10 = 8 × 10 = 80.
Step 4. Multiply the middle segment by two to find the area of a trapeze
The middle segment consists of an imaginary line that runs parallel between the lower and upper lines of the trapezius, at the same distance from both. Since the mean segment is always equal to (base 1 + base 2)/2, if you know its value, you can use a shortcut to the trapezoid formula.
- Area = average segment × height or A = m × h.
- Essentially, this is the same procedure as using the original formula, except that you are using “m” instead of (b1 + b2)/2.
the middle segment of the trapeze in the example above is 9 meters long. This means that we can find the area of a trapezoid just by multiplying 9 × 2 = 18 square meters, as we did earlier.
Method 3 of 4: Finding the Area of a Kite
Step 1. Know how to identify a kite
A kite is a kind of four-sided diamond, with two pairs of equal sides adjacent to each other, not opposite each other. As the name suggests, kites look like real-life kites.
There are two different ways to find the area of a kite, depending on what information is available. Below, you will learn how to use both
Step 2. Use the diamond diagonal formula to find the area of a kite
Since a diamond is just a special type of kite, in which the sides are all the same measurement, you can use the diamond area formula to find the area of a kite. As a reminder, diagonals are the lines between two opposite corners of the kite. As in the rhombus, the formula for the kite is as follows:
- Area = (diagonal 1 × diagonal 2)/2 or A = (d1 × d2)/2.
if a kite has diagonals with size equal to 19 meters and 5 meters, its area will be equal to (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 also use trigonometry to calculate them. Read the trigonometry section of our article for more information.
Step 3. Use the lengths of the sides and the angle between them to figure out the area
If you know the two different values for the lengths of the sides and the angle present at the corner between those sides, you can figure out the kite area with principles drawn from trigonometry. This method requires prior knowledge of sine functions (or at least a calculator with this function). Read our article or use the following formula:
- Area = (side 1 × side 2) × sin(angle) or A = s1 × s2) × sin(θ) - where θ is the angle between sides 1 and 2.
you have a kite with two sides equal to 6 meters and two sides equal to 4 meters. The angle between them is approximately equal to 120 degrees. In this case, you can find out your area as follows: (6 × 4) × sin(120) = 24 × 0, 866 = 20, 78 square meters.
- Note that you have to use two different sides and the angle between them - just using the set of sides with equal measurements will not work.
Method 4 of 4: Solving Any Quad
Step 1. Find the length of the four sides
Your quadrilateral may not be in any of the categories described above (if, for example, it has all sides with different measurements and no pairs of parallel sides). Believe it or not, there are formulas that can be used to find the area of any quadrilateral, regardless of its shape. In this section, you will learn how to use the most common of them. Note that this formula requires some knowledge of trigonometry - read our guide for more information.
- Initially, you must find the length of each side of your quadrilateral. For the purposes of this article, we will give them the names a, b, c and d. Sides a and c are opposite each other, as are sides b and d.
If you have an irregularly shaped quadrilateral that doesn't fall into any of the above categories, first measure its four sides. Let's say they have measurements equal to 12, 9, 5 and 14 centimeters. In the steps below, you will make use of this information to discover the area in this way.
Step 2. Find the angles between a and d and between b and c
When you're working with an irregular quadrangle, you can't figure out the area by just measuring the sides. Proceed by discovering two of the opposite angles. To solve this section, we will use angle A between sides a and d and angle C between sides b and c. However, you can also perform this procedure with the other two opposite angles.
let's say that, in its quadrilateral, A is equal to 80 degrees and that C is equal to 110 degrees. In the next step you will use these values to find the total area.
Step 3. Use the area formula for triangles to find the area of the quadrilateral
Imagine that there is a straight line that runs from the corner between a and b and to the corner between c and d. This line would divide the quadrilateral into two triangles. Since the area of a triangle is equal to ab × sin(C), where C is the angle between sides a and b, you can use this formula twice (once for each of the imaginary triangles) to get the total area of the triangle. quadrilateral. In other words, for any quadrilateral:
- Area = 0.5 side 1 × side 4 × sin(angle between sides 1 and 4) + 0.5 × side 2 × side 3 × sin(angle between sides 2 and 3) or
- Area = 0.5 a × d × sin(A) + 0.5 × b × c × sin(C).
you already have the necessary sides and angles. Let's solve the problem:
- = 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 centimeters.
- Note that if you want to find the area of a parallelogram where opposite angles are equal, the equation is reduced to Area = 0.5 × (ad + bc) × sin(A).
- This trigonometric calculator can be useful when performing the calculations in the “Solving Any Quad” step above.
- For more information, read our specific articles: How to Find the Area of a Square, How to Calculate the Area of a Rectangle, How to Calculate the Area of a Diamond, How to Calculate the Area of a Trapezoid and How to Find the Area of a Kite.