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Math resources Geometry

Area of a quadrilateral

Area of a quadrilateral

Here you will learn about the area of a quadrilateral, including how to find the area of a rectangle, square, parallelogram, and trapezoid.

Students will first learn about the area of a quadrilateral as part of measurement and data in third grade when they learn about the area of a rectangle.

They expand on this knowledge into 6th grade when they learn how to calculate the area of more complex shapes, such as parallelograms and trapezoids.

What is the area of a quadrilateral?

The area of a quadrilateral is the amount of space inside a quadrilateral.

To find the area of a quadrilateral, you can either count the number of unit squares within a shape or use the appropriate area formula for that shape.

Area is measured in square units. For example, square feet (ft^2), square inches (in^2), square meters (m^2), etc.

Area of a rectangle or area of a square

Area of a rectangle or area of a square = \text { base } \times \text { height }

Area of a Quadrilateral image 7 US

For example,

Area of a Quadrilateral table image 1

This rectangle contains 15 squares.

\begin{aligned} \text{Area of a rectangle} &= \text{base} \times \text{height} \\ &= 5 \times 3 \\ &= 15cm^2 \end{aligned}

Area of a parallelogram

Area of a parallelogram = \text { base } \times \text { height }


Area of a Quadrilateral image 8 US

Area of a Quadrilateral table image 2


If we chop one end off of a parallelogram and rearrange it,
we get a rectangle.

This is why the formula for area of a parallelogram is the
same as the area of a rectangle.

Step-by-step guide: Area of a parallelogram

Area of a trapezoid

A trapezoid is a quadrilateral with only one pair of parallel sides.

In order to find the area of a trapezoid, you need to decompose the shape into a rectangle and two triangles (or a rectangle and one triangle if it is a right trapezoid).

Area of a Quadrilateral image 10 US

Step-by-step guide: Area of a trapezoid

What are quadrilaterals?

A quadrilateral is a closed 2 dimensional shape with 4 straight sides and 4 vertices.

Regular quadrilaterals have sides that are all the same length and interior angles that are the same size.

Irregular quadrilaterals have sides that have different lengths and interior angles that have different sizes.

There are several different types of quadrilaterals.

For example,

Square

A square is a closed 2 dimensional shape with four straight sides of equal length and four right angles (90 ^{\circ}) .

Area of a Quadrilateral image 2 US

Rectangle

A rectangle is a closed 2 dimensional shape with four straight sides and four right angles (90^{\circ}) . It has two pairs of parallel sides that are equal.

Area of a Quadrilateral image 3 US

Parallelogram

A parallelogram is a closed 2 dimensional shape with four straight sides. The opposite sides of a parallelogram have the same lengths and are parallel.

Area of a Quadrilateral image 4 US

Rhombus

A rhombus is a parallelogram with four equal straight sides.

Area of a Quadrilateral image 5 US

Trapezoid

A trapezoid, also known as a trapezium, is a closed 2 dimensional shape with four straight sides and only one pair of parallel sides.

Area of a Quadrilateral image 6 US

Step-by-step guide: Area of rhombus

What is the area of a quadrilateral?

What is the area of a quadrilateral?

Common Core State Standards

How does this relate to 6th grade math?

  • Grade 6 – Geometry (6.G.1)
    Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

How to calculate the area of a quadrilateral

In order to calculate the area of a rectangle, area of a square, or the area of a parallelogram:

  1. Substitute the values into the formula. (Make sure the units are the same for all measurements.)
  2. Complete the calculation.
  3. Add the correct units.

In order to calculate the area of a trapezoid:

  1. Decompose the shape into a rectangle and two triangles (or one triangle if it is a right trapezoid).
  2. Find the area of each shape.
  3. Add the areas together.
  4. Add the correct units.

[FREE] Area Check for Understanding Quiz (Grade 4 to 6)

[FREE] Area Check for Understanding Quiz (Grade 4 to 6)

[FREE] Area Check for Understanding Quiz (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of area. 15+ questions with answers covering a range of 4th, 5th and 6th grade area topics to identify areas of strength and support!

DOWNLOAD FREE
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[FREE] Area Check for Understanding Quiz (Grade 4 to 6)

[FREE] Area Check for Understanding Quiz (Grade 4 to 6)

[FREE] Area Check for Understanding Quiz (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of area. 15+ questions with answers covering a range of 4th, 5th and 6th grade area topics to identify areas of strength and support!

DOWNLOAD FREE

Area of a quadrilateral examples

Example 1: area of a square

Find the area of the square:

Area of a Quadrilateral image 11 US

  1. Substitute the values into the formula. (Make sure the units are the same for all measurements.)

Here the height is 8 . Since it is a square, the base is also 8 .

\begin{aligned}\text { Area } &=\text { base } \times \text { height } \\\\ \text { Area }&=8 \times 8\end{aligned}

2Complete the calculation.

\begin{aligned} & \text { Area }=8 \times 8 \\\\ & \text { Area }=64\end{aligned}

3Add the correct units.

The measurements on this square are in cm so the area will be measured in {cm}^2 .

\text { Area }=64 \mathrm{~cm}^2

Example 2: area of a rectangle, different units

Find the area of the rectangle:

Area of a Quadrilateral image 12 US

Substitute the values into the formula. (Make sure the units are the same for all measurements.)

Complete the calculation.

Add the correct units.

Example 3: area of a parallelogram

Calculate the area of the parallelogram:

Area of a Quadrilateral image 14 US

Substitute the values into the formula. (Make sure the units are the same for all measurements.)

Complete the calculation.

Add the correct units.

Example 4: area of a rhombus

Calculate the area of the rhombus:

Area of a Quadrilateral image 15 US

Substitute the values into the formula. (Make sure the units are the same for all measurements.)

Complete the calculation.

Add the correct units.

Example 5: area of a trapezoid

Find the area of the following trapezoid:

Area of a Quadrilateral image 16 US

Decompose the shape into a rectangle and two triangles (or one triangle if it is a right trapezoid).

Find the area of each shape.

Add the areas together.

Add the correct units.

Example 6: area of a right trapezoid

Find the area of the following trapezoid:

Area of a Quadrilateral image 18 US

Decompose the shape into a rectangle and two triangles (or one triangle if it is a right trapezoid).

Find the area of each shape.

Add the areas together.

Add the correct units.

Teaching tips for area of a quadrilateral

  • Provide students with study material to help them keep track of the different types of quadrilaterals and how to find the area of each one.

  • Once students are comfortable finding the area of a quadrilateral using the area formula, challenge them by asking them to find the area of a quadrilateral where all sides are decimals or fractions.

  • Be sure to add a real-world context to area of a quadrilateral problems as often as possible to provide students with a deeper understanding of the concept.

Easy mistakes to make

  • Calculating perimeter instead of area
    Students often confuse the terms area and perimeter and will sometimes add all sides of a shape in order to find the perimeter instead of finding the area.

  • Using a side length as the height of a parallelogram
    Students may mistake the side length of a parallelogram for its height.

    Area of a Quadrilateral image 20 US

  • Not converting different units so all units are the same
    It is very important to make sure the units are the same for each length (for example, all centimeters).

Practice area of a quadrilateral questions

1. Quadrilateral ABCD , shown below, is a square. Find the area of the quadrilateral.

 

 

Area of a Quadrilateral image 21 US

9 \mathrm{~m}^{2}
GCSE Quiz True

12 \mathrm{~m}^{2}
GCSE Quiz False

6 \mathrm{~m}^{2}
GCSE Quiz False

27 \mathrm{~m}^{2}
GCSE Quiz False

You can use the formula to find the area of the quadrilateral.

 

Since you know it is a square, you know all sides are the same length.

 

This means that since its height is 3{~m} , its base is also 3{~m} .

 

You can use these measurements and the area formula base \times height to calculate the area.

 

\begin{aligned}\text{Area }&=\text{ base }\times \text{ height}\\\\ &= 3 \times 3\\\\ &=9 \mathrm{~m}^{2}\end{aligned}

2. The base of the rectangle shown below is times greater than its height. Find the area of the rectangle.

 

Area of a Quadrilateral image 22 US

112 \mathrm{~yd}^{2}
GCSE Quiz False

70 \mathrm{~yd}^{2}
GCSE Quiz False

196 \mathrm{~yd}^{2}
GCSE Quiz True

3,136 \mathrm{~yd}^{2}
GCSE Quiz False

First, you need to find the height of the rectangle.

 

The problem states that the base of the rectangle is 4 times greater than its height.

 

This means you can divide the base measurement by 4 to get the height measurement.

 

28 \div 4=7

 

Once you know the height, you can use the formula for the area of a rectangle, which is base \times height .

 

\begin{aligned}\text { Area }& =\text { base } x \text { height } \\\\ & =28 \times 7 \\\\ & =196 \mathrm{~yd}^{2} \end{aligned}

3. Find the area of the parallelogram. Give your answer in square centimeters.

 

Area of a Quadrilateral image 23 US

420 \mathrm{~cm}^{2}
GCSE Quiz False

42 \mathrm{~cm}^{2}
GCSE Quiz True

36 \mathrm{~cm}^{2}
GCSE Quiz False

4200 \mathrm{~cm}^{2}
GCSE Quiz False

First, you need to make the units the same.

 

Since 1 {~cm} = 1 {~mm} , you can divide 70 {~mm} by 10 to get its value in cm.

 

70 {~mm} = 7 {~cm} .

 

Then you can use the area formula, base \times height .

 

\begin{aligned} \text{Area }&=\text{ base }\times \text{ height}\\\\ &= 7 \times 6\\\\ &=42 \mathrm{~cm}^{2} \end{aligned}

4. Find the area of the rhombus.

 

54 \mathrm{~ft}^{2}
GCSE Quiz False

48 \mathrm{~ft}^{2}
GCSE Quiz True

30 \mathrm{~ft}^{2}
GCSE Quiz False

23 \mathrm{~ft}^{2}
GCSE Quiz False

\begin{aligned} \text{Area }&=\text{ base }\times \text{ height}\\\\ &= 6 \times 8\\\\ &=48 \mathrm{~ft}^{2} \end{aligned}

5. Find the area of the trapezoid:

 

Area of a Quadrilateral image 25 US

27 \mathrm{~m}^{2}
GCSE Quiz False

24 \mathrm{~m}^{2}
GCSE Quiz False

54 \mathrm{~m}^{2}
GCSE Quiz False

42 \mathrm{~m}^{2}
GCSE Quiz True

To find the area of the trapezoid, you must first decompose it into a rectangle and two triangles.

 

Use the area of a rectangle formula and the area of a triangle formula to find the area of each shape.

 

Then you can add the areas together to find the total area of the trapezoid.

 

Area of a Quadrilateral image 26 US

 

A (triangle):

 

\begin{aligned} & \text { Area }=\cfrac{1}{2} \, b h \\\\ & \text { Area }=\cfrac{1}{2} \, (2 \times 6) \\\\ & \text { Area }=\cfrac{1}{2} \, \times 12 \\\\ & \text { Area }=6 \end{aligned}

 

B (triangle):

 

\begin{aligned} & \text { Area }=\text { base } \times \text { height } \\\\ & \text { Area }=5 \times 6 \\\\ & \text { Area }=30 \end{aligned}

 

C (triangle):

 

\begin{aligned} & \text { Area }=\cfrac{1}{2} \, b h \\\\ & \text { Area }=\cfrac{1}{2} \, (2 \times 6) \\\\ & \text { Area }=\cfrac{1}{2} \, \times 12 \\\\ & \text { Area }=6 \end{aligned}

 

\text { Total area }=6+30+6=42 \mathrm{~m}^2

6. Find the area of the right trapezoid:

 

Area of a Quadrilateral image 27 US

180 \mathrm{~ft}^{2}
GCSE Quiz False

170 \mathrm{~ft}^{2}
GCSE Quiz True

196 \mathrm{~ft}^{2}
GCSE Quiz False

54 \mathrm{~ft}^{2}
GCSE Quiz False

First, you need to decompose the right trapezoid into a rectangle and a triangle.

 

Area of a Quadrilateral image 28 US

 

Then, find the area of the rectangle and the area of the triangle using the formulas.

 

A (rectangle):

 

\begin{aligned} & \text { Area }=\text { base } \times \text { height } \\\\ & \text { Area }=16 \times 10 \\\\ & \text { Area }=160 \end{aligned}

 

B (rectangle):

 

\begin{aligned} & \text { Area }=\cfrac{1}{2} \, b h \\\\ & \text { Area }=\cfrac{1}{2} \, (2 \times 10) \\\\ & \text { Area }=\cfrac{1}{2} \, \times 20 \\\\ & \text { Area }=10 \end{aligned}

 

Add the area of the rectangle and the area of the triangle. Then add the correct units.

 

{Area} = 160 + 10

 

{Area} = 170 \mathrm{~ft}^{2}

Area of a quadrilateral FAQs

What is a quadrilateral?

A quadrilateral is a closed 2 dimensional shape with 4 straight sides and 4 vertices.

How do you find the area of a quadrilateral?

To find the area of a quadrilateral, such as a rectangle, square, or parallelogram, use the area formula which is base \times height .

The next lessons are

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