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Finding the area of a quadrilateral is an important concept in geometry, as it allows us to determine the amount of space enclosed by these foursided figures. A quadrilateral can take on various shapes and sizes, such as squares, rectangles, parallelograms, trapezoids, or even irregular polygons. Regardless of its specific properties, there are general methods that can be applied to find the area of any quadrilateral. In this guide, we will explore these methods step by step, providing you with the tools and understanding necessary to calculate the area of a quadrilateral accurately and efficiently.
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You are given a homework assignment asking you to calculate the area of a quadrilateral, but you don’t even know what a quadrilateral is. Don’t worry – this article will help you! A quadrilateral is any shape with four sides, such as a rectangle, square, and rhombus. To calculate the area of a quadrilateral, all you have to do is distinguish the type of quadrilateral and follow a simple formula. That is all!
Steps
Squares, rectangles and parallelograms
 Square: Four sides of equal length. Four 90 degree angles (right angles).
 Rectangle: Four sides, equal lengths of opposite sides. Four 90 degree angles.
 Rhombus: Four sides, the lengths of opposite sides are equal. Four angles, none of which are 90 degrees but opposite angles must be equal.
 Area = length × width , or A = b × h .
 Example: If the length of a rectangle is 10 cm long and the width is 5 cm long, the area of the rectangle is 10 × 5 (b × h) = 50 square centimeters .
 Remember to use square units for the results you find when calculating the area of any shape (square centimeter, square decimeter, square meter …).
 Area = side × side or A = s ^{2}
 Example: If a side of a square is 4 meters long (t = 4), then the area of the square is t ^{2} , or 4 x 4 = 16 square meters .
 Area = (Diagonal 1 × Diagonal 2)/2 or A = (d _{1} × d _{2} )/2
 Example: If a rhombus has 2 diagonals with lengths of 6 meters and 8 meters, then its area is (6 × 8)/2 = 48/2 = 24 square meters.
 Example: A rhombus has side lengths 10 km and 5 km. The length of the line perpendicular to the pair of sides 10 km is 3 km. If you want to find the area of this rhombus, you take 10 × 3 = 30 square kilometers .
 Area = base × height or A = b × h
 Area = (Diagonal 1 × Diagonal 2)/2 or A = (d _{1} × d _{2} )/2
 Example: A tetrahedron has two adjacent sides that are 4 meters long. You can find the area of this square by multiplying the base by the height: 4 × 4 = 16 square meters .
 Example: The diagonals of a square of equal length are 10 centimeters. You can calculate the area of this square using the formula: (10 × 10)/2 = 100/2 = 50 square centimeters .
Calculate the area of a trapezoid
 There are two ways to calculate the area of a trapezoid, depending on the information you have. Here are two ways to calculate the area of a trapezoid.
 Find the shorter of the two parallel base edges. Place the pen at the corner between that bottom edge and a nonparallel side. Draw a line perpendicular to both bottom edges. Measure this line to find the length of the altitude.
 You can also sometimes use trigonometry to calculate the length of the altitude if the altitude, base, and other side form a square. See the article on trigonometry for more information.
 Area = (Bottom 1 + Bottom 2)/2 × height or A = (a+b)/2 × h
 Example: If a trapezoid has two base sides 7 meters and 11 meters long respectively, and the altitude connecting the two bottom sides is 2 meters long, you can find the area like this: (7 + 11)/2 × 2 = (18)/2 × 2 = 9 × 2 = 18 square meters .
 If the length of the altitude is 10 and the base sides are 7 and 9, you can find the area by simply doing the following: (7 + 9)/2 * 10 = (16/2) * 10 = 8 * 10 = 80
 Area = mean × altitude or A = m × h
 This formula is essentially the same as the original, but you use “m” instead of (a + b)/2.
 Example: The median of the trapezoid in the example above is 9 meters long. That is, we can calculate the area of a trapezoid by taking 9 × 2 = 18 square meters , the same result as the first way.
Calculate the area of the black figure ta
 There are two ways to calculate the area of the black figure, depending on the information you have. Here are two ways to calculate the area of the black figure.
 Area = (Diagonal 1 × Diagonal 2)/2 or A = (d _{1} × d _{2} )/2
 Example: If a black figure has 2 diagonals with lengths of 19 meters and 5 meters, then its area is (19 × 5)/2 = 95/2 = 47.5 square meters .
 If you do not know and cannot measure the length of two diagonals, you can use trigonometry to calculate. See the article about black images for more information.
 Area = (Side 1 × Side 2) × sin(angle) or A = (s _{1} × s _{2} ) × sin(θ) (where θ is the angle between side 1 and side 2).
 Example: You have a black figure with one pair of sides that are 6 meters long and the other pair of sides are 4 meters. The angle between them is 120 degrees. In this case, you can solve for the area like this: (6 × 4) × sin(120) = 24 × 0.866 = 20.78 square meters
 Note that in this case you must use two different sides and the angle between them — using pairs of equal lengths will give the wrong result.
How to solve for any quadrilateral
 First you have to find the lengths of each side of the quadrilateral. For this article, we call the edges a , b , c and d . Side a is opposite side c and side b is opposite side d .
 Example: If you have a quadrilateral with an odd shape and do not belong to any of the above groups, you must first measure the lengths of the four sides. Let’s say they are 12, 9, 5 and 14 centimeters long. In the following section you will use this information to find the area of the quadrilateral.
 Example: Suppose in your quadrilateral A is equal to 80 degrees and C is equal to 110 degrees. In the next step you will use these values to find the area.
 Area = 0.5 Side 1 × Side 4 × sin(Angle between side 1&4) + 0.5 × Side 2 × Side 3 × sin(Angle between side 2&3) or
 Area = 0.5 a × d × sin A + 0.5 × b × c × sin C
 Example: You already have the required edges and angles, or solve like this:

 = 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’re finding the area of a parallelogram with equal opposite angles, the equation will be simplified to Area = 0.5*(ad + bc) * sin A .
Advice
 This triangle area calculator is very handy for calculations in the above “Any Quadrilateral” method. ^{[5] X Research Sources}
 For more information, see the articles about specific shapes: How to Find the Area of a Square, How to Calculate the Area of a Rectangle, How to Calculate the Area of a Rhombus, How to Find the Area of a Trapezoid, and How to Find the Area of the Black Shape.
This article is coauthored by a team of editors and trained researchers who confirm the accuracy and completeness of the article.
The wikiHow Content Management team carefully monitors the work of editors to ensure that every article is up to a high standard of quality.
This article has been viewed 264,868 times.
You are given a homework assignment asking you to calculate the area of a quadrilateral, but you don’t even know what a quadrilateral is. Don’t worry – this article will help you! A quadrilateral is any shape with four sides, such as a rectangle, square, and rhombus. To calculate the area of a quadrilateral, all you have to do is distinguish the type of quadrilateral and follow a simple formula. That is all!
In conclusion, finding the area of a quadrilateral involves different methods depending on the type of quadrilateral. For a rectangle or square, one can simply multiply the length and width. For a parallelogram or rhombus, the base is multiplied by the height. A trapezoid requires adding the lengths of the parallel sides and multiplying the sum by half the height. Lastly, for an irregular quadrilateral, the most accurate method is to divide it into triangles and calculate the area of each triangle separately before summing them up. It is essential to note that finding the area of a quadrilateral requires accurate measurements and a good understanding of the properties of different types of quadrilaterals.
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