If you are reading this page it is because you have been given some homework where you have to calculate the area of a quadrilateral, isn't it? If you don't know what a quadrangle is, don't worry, this guide will be of enormous help to you. A quadrilateral is any geometric figure that has four sides - squares, rectangles and rhombuses are just a few examples. To calculate the area, you just need to understand what kind of quadrilateral it is and use a simple formula. That's all!
Steps
Method 1 of 4: Squares, Rectangles and other Parallelograms
Step 1. Learn to recognize a parallelogram
A parallelogram is any quadrilateral that has 2 pairs of parallel sides, where the opposite sides have the same length. Parallelograms include:
- Squares: four sides, all of the same length. Four angles, all 90 degrees (right angles).
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Rectangles:
four sides; opposite sides are the same length. Four angles, all 90 degrees.
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Rhombuses:
four sides; the opposite sides are the same length. Four corners; neither of them must be 90 degrees, but the opposite angles must be the same.
Step 2. Multiply base by height to calculate the area of a rectangle
You will need two measurements to calculate the area of a rectangle: the width, or base (the longest side of the rectangle), and the length, or height (the shortest side of the rectangle). Multiply these two values to get the area. In other words:
- Area = base × height, or A = b × h In short.
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Example:
if the base of a rectangle is 10 centimeters and the height of 5, the area of the rectangle will simply be 10 × 5 (b × h) = 50 square centimeters.
- Do not forget that when calculating the area of a figure, the result will be expressed in square units (square centimeters, square meters, etc.).
Step 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 the area. But since all sides of a square are the same, you can use a shortcut and multiply one side by itself. This is equivalent to multiplying the base by the height of the square, since they have the same value. Use the following equation:
- Area = side × side or A = l2
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Example:
if one side of a square is 4 centimeters long (l = 4), the area of the square will simply be l2, or 4 x 4 = 16 square centimeters.
Step 4. Multiply the diagonals and divide by two to find the area of a diamond
Be careful in this case - to find the area of a rhombus, you can't just multiply two adjacent sides. Instead, find the diagonals (the lines connecting each pair of opposite corners), multiply them, and divide by two. In other words:
- Area = (Diag. 1 × Diag. 2) / 2 or A = (d1 × d2)/2
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Example:
if a rhombus has diagonals 6 and 8 meters long respectively, its area is calculated as (6 × 8) / 2 = 48/2 = 24 square meters.
Step 5. Alternatively, you can use the base × height formula to find the area of a rhombus
Technically, you can also use the rectangle formula to find the area of a rhombus. In this case, however, the base and height do not indicate two adjacent sides. First, choose a side that will be the base. Then, draw a line from the base to the opposite side. The line should meet both sides at a 90 degree angle. The length of this line represents the height.
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Example:
a rhombus has sides of 10 meters and 5 meters. The straight line distance between the sides of 10 meters is 3 meters. If you want to find the area of the rhombus, you should multiply 10 × 3 = 30 square meters.
Step 6. Note that the formulas for rhombuses and rectangles also apply to squares
The side × side formula mentioned above is undoubtedly the most convenient for finding the area of a square. But since squares are also rectangles and diamonds, you can use the formulas for those figures to calculate 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
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Example:
a four-sided figure has two adjacent 4-meter sides. You can calculate the area of this square by multiplying the base by the height: 4 × 4 = 16 square meters.
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Example:
the diagonals of a square both measure 10 centimeters. You can find the area of that square with the diagonal formula: (10 × 10) / 2 = 100/2 = 50 square centimeters.
Method 2 of 4: Finding the Area of a Trapezoid
Step 1. Learn to identify a trapezoid
A trapezoid is a quadrilateral with at least two parallel sides. The angles can have any value. Each side of the trapezoid can have a different length.
There are two different ways to find the area of a trapezoid, depending on the information available to you. Below, you will find both formulas
Step 2. Find the height of a trapezoid
The height of a trapezoid is the perpendicular line connecting the two parallel sides. It will usually not be the same size as the other sides, which often have a diagonal slope. You will need this data for both formulas. Here's how to find the height of a trapezoid:
- Find the shorter base between the two parallel lines. Put the pencil in the corner between that base and one of the non-parallel sides. Draw a straight line that is perpendicular to the two parallel bases. Measure the line to find the height.
- You can use trigonometric formulas to find the height if it, the base and the other side form a right triangle. You can find articles on wikiHow covering the subject.
Step 3. Find the area of the trapezoid using the height and lengths of the bases
If you know the height of the trapezoid and the length of both bases, use the following equation:
- Area = (Base 1 + Base 2) / 2 × height or A = (a + b) / 2 × h
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Example:
if you have a trapezoid with a base of 7 meters, the other of 11 and the height that connects them by 2, you can find the area like this: (7 + 11) / 2 × 2 = (18) / 2 × 2 = 9 × 2 = 18 square meters.
- If the height is 10 and the bases measure 7 and 9, you can find the area with: (7 + 9) / 2 × 10 = (16/2) × 10 = 8 × 10 = 80
Step 4. Use the half sum to find the area of a trapezoid
This is an imaginary line that runs parallel to the bases of the trapezoid and is exactly the same distance from both. Since the half sum is always equal to (Base 1 + Base 2) / 2, if you know that data you can use a shortcut in the trapezoid formula:
- Area = half sum × height or A = m × h
- In practice, this is the same formula as above, except for the substitution of "m" a (a + b) / 2.
- '' Example:' the half sum of the trapezoid of the previous example measures 9 meters. This means that we can find the area of the trapezoid simply by multiplying 9 × 2 = 18 square meters, exactly the same result as the previous formula.
Method 3 of 4: Finding the Area of a Kite
Step 1. Learn to identify a kite
A kite is quadrilateral in which two pairs of sides of the same length are adjacent to each other and not opposite. As the name suggests, these figures are reminiscent of kites.
There are two different ways to find the area of a kite, depending on the information available to you. You will find both formulas below
Step 2. Use the rhombus diagonal formula to find the area of a kite
Since a rhombus is a special type of kite where the sides are all the same length, you can use the rhombus formula for kites as well. As a reminder, diagonals are the straight lines between two opposite corners of the kite. As with diamonds, the formula for the area of a kite is:
- Area = (Diag. 1 × Diag 2.) / 2 or A = (d1 × d2)/2
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Example:
if one kite has one diagonal measuring 19 meters and the other 5 meters, its area is simply equal to (19 × 5) / 2 = 95/2 = 47.5 square meters.
- If you don't know the value of the diagonals and can't measure them, you can use trigonometry to calculate them. Try reading this wikiHow article about it.
Step 3. Use the length of the sides and the angle between them to find the area
If you know the two different values of the lengths of the sides and the angle between the two sides, you can calculate the area of the kite thanks to the principles of trigonometry. This method requires you to know the sine function (or at least have a calculator with that function available). You can find more information by searching for articles on wikiHow, or use the following formula:
- Area = (Side 1 × Side 2) × sin (corner) or A = (l1 × l2) × sin (θ) (where θ is the angle between sides 1 and 2).
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Example:
you have a kite with two sides of 6 centimeters and two sides of 4 centimeters. The angle between them is about 120 degrees. In this case, you can calculate the area like this: (6 × 4) × sin (120) = 24 × 0.866 = 20, 78 square centimeters
- Note that you have to use the lengths of the two different sides and the angle between them in this formula - if you use congruent sides you will not get the correct result.
Method 4 of 4: Solve for Any Quadrilateral
Step 1. Find the length of all four sides
Does your quadrilateral not fit into any of the categories described above (e.g. does it have four sides of different sizes that are not parallel)? Believe it or not, there are formulas that allow you to calculate the area of any quadrilateral, regardless of its shape. In this section you will find out how to use the most common. Note that this formula requires notions of trigonometry.
- First, calculate the length of the four sides of the quadrilateral. For the purposes of this article, we will define sides a, b, c, and d. The sides "a" and "c" are opposite each other, and the sides "b" and "d" are also opposite.
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Example:
If you have an oddly shaped quadrilateral that does not fit into any of the categories described above, first measure its sides. Let's assume that the measurements are worth 12, 9, 5 and 14 centimeters. In the following steps, you will use this data to find the area of the shape.
Step 2. Find the angles between "a" and "d" and between "b" and "c"
When dealing with irregular quadrilaterals, you cannot find the area with only the sides. Continue by finding the two opposite corners. For the purposes of this section, we will call "A" the angle between sides "a" and "d" and "C" the angle between sides "b" and "c". You can also find the area with the values of the other two opposite corners.
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Example:
Let's assume that, in your quadrilateral, A measures 80 degrees and C measures 110 degrees. In the next step we will use these values to find the total area.
Step 3. Use the triangle area formula to find the area of the quadrilateral
Imagine drawing a straight line from the angle between sides "a" and "b" to that between sides "c" and "d". This line would divide the quadrilateral into two triangles. Since the area of a triangle equals ab sin C, where C is the angle between sides a and b, you can use this formula twice (once for each hypothetical triangle) to calculate the total area of the quadrilateral. In other words, for all quadrilaterals:
- Area = 0, 5 Side 1 × Side 4 × sin (corner Sides 1 & 4) + 0.5 × Side 2 × Side 3 × sin (corner Sides 2 & 3) or
- Area = 0.5 a × d × sin A + 0.5 × b × c × sin C
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Example:
you already have the sides and angles you need, so we solve:
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- = 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
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- Note that if you are trying to find the area of a parallelogram, where opposite angles are equal, the equation boils down to Area = 0.5 * (ad + bc) * sin A.
Advice
- This triangle calculator can be useful for calculations in the "All quadrilaterals" section.
- For more information, you can find specific articles on geometric figure types on wikiHow.
