How to Find the Area of a Quadrilateral

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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!

Table of Contents

Squares, rectangles and parallelograms

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Know how to distinguish parallelograms. A parallelogram is a four-sided figure with two pairs of parallel sides, opposite sides of equal length. Parallelograms include:

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.

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Multiply the base by the height to get the area of the rectangle. To find the area of a rectangle, you need the measure of the lengths of: length (longer side) and width (shorter side). Then multiply the two values to get the area. In other words:

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 …).

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Multiply the length of a side by itself to find the area of the square. Basically a square is a special rectangle, so you can use the same formula to calculate the area. However, since the four sides of the square are the same length, you only need to multiply the length of one side by itself. This is similar to multiplying the base by the height because the square has the same base and height. Use the following equation: ^{[1] X Research Source}

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 .

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Multiply the lengths of the diagonals and then divide by 2 to find the area of the rhombus. Be careful with this figure – when you’re trying to find the area of a rhombus, it’s not possible to multiply the lengths of the two adjacent sides. Instead you have to find the length of the diagonal (the line connecting pairs of opposite angles), multiply them, and then divide by two. In other words: ^{[2] X Research Source}

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.

Calculate the area of a trapezoid

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Know how to distinguish trapezoids. A trapezoid is a quadrilateral that has at least one pair of parallel sides. A trapezoid has no angle regulation. Each side of a trapezoid can have different lengths.

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.

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Find the trapezoidal altitude. The altitude of a trapezoid is a line connecting and perpendicular to two parallel sides. The elevation line is usually not the same length as the two side edges because these edges often run in the oblique direction. You need the elevation line length for both area formulas. Here is how to calculate the length of the altitude of a trapezoid: ^{[3] X Research Source}

Find the shorter of the two parallel base edges. Place the pen at the corner between that bottom edge and a non-parallel 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.

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Find the area of a trapezoid given the lengths of the altitude and the two bases. If you know the length of the altitude as well as the two bases of the trapezoid, use the following equation:

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

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Multiply the median by 2 to find the area of the trapezoid. The median is an imaginary line that runs parallel to the two bases of the trapezoid and is equidistant from them. Since the moving average is always equal to (Low 1 + Bottom 2)/2 , if you know its length you can use the following formula:

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

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Know how to distinguish black images. A black figure is a four-sided figure with two pairs of sides of equal length, and two equal sides next to each other, not facing each other . In general, the black image looks like a kite in real life.

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.

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Use the rhombus diagonal formula to find the area of the black figure. Since a rhombus is a special form of a rhombus when all four sides are the same length, you can use the formula for the area of a rhombus diagonally to find the area of the rhombus. Remember that the diagonal is the line connecting the two opposite corners of the black figure. Like a rhombus, the formula for calculating the area of a black figure is:

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.

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Use the lengths of the sides and the angle between them to find the area. If you know the lengths of the pairs of sides and the angle between them, solve for the area of the black figure using the principle of trigonometry. ^{[4] X Research Source} This method requires you to know how to use the sine function (or at least a calculator with a sine function). See the article on trigonometry for more information or use the following formula:

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

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Find the lengths of all four sides. Does your quadrilateral do not belong to any of the above groups (for example, all four sides have different lengths and no pair of sides are parallel)? There are actually many formulas for calculating the area of any quadrilateral, regardless of its shape. In this section you will learn how to use the most common formulas. Note that this formula requires you to know how to use trigonometry.

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.

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Find the angles between a with d and b with c . When the problem is for an asymmetrical quadrilateral, you cannot find the area from the lengths of the sides. You have to find two of the opposite corners. For this part, we will use angle A between sides a and d , and angle C between sides b and c . However, you can also use the other two opposite corners.

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.

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Use the formula for the area of a triangle to find the area of the quadrilateral. Imagine there is a line connecting the angle between sides a and b with the angle between sides c and d . This line divides the quadrilateral into two triangles. Since the area of the triangle is ab sin C , where C is the angle between sides a and b , you can use this formula twice (for one triangle each time) to get the area of the entire quadrilateral . In other words, for any quadrilateral:

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.

Steps

Squares, rectangles and parallelograms

Calculate the area of a trapezoid

Calculate the area of the black figure ta

How to solve for any quadrilateral

Advice

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