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Math resources Geometry Area

Area of Irregular Shapes

Area of irregular shapes

Here you will learn about the area of irregular shapes, including what irregular shapes are, and questions involving rectilinear shapes, missing side lengths and unit conversion.

Students will first learn about finding the area of an irregular shape as part of geometry in $3$ rd grade. Students will deepen their understanding of area of irregular shapes in $4$ th grade.

What is the area of irregular shapes?

The area of irregular shapes (also referred to as irregular polygons or irregular figures) is the amount of space inside a shape composed of sides and angles of different lengths and sizes.

It is measured in square units $(\mathrm{cm}^{2},\mathrm{~m}^{2},\mathrm{~mm}^{2} etc.).$

To find the area of irregular shapes, you will decompose the irregular shape into regular shapes, such as triangles or regular polygons, then find the area of each of the regular shapes and add them together.

Some of the common regular shapes used when decomposing irregular shapes are rectangles, squares, triangles, circles or semicircles, and trapezoids.

For example,

This irregular shape was split into two shapes, a rectangle (shape $A$ ) and a square (shape $B$ ). Find the area of the rectangle and the area of the square, and then add the two areas together.

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Area of irregular shapes using a square grid

Sometimes when finding the area of irregular shapes, they will be presented to you on a square grid. You can find the area of shapes using the square grid by simply counting the squares.

For example, to find the area of the given shape below, you can just count the number of squares, or space, the shape takes up.

The area of this irregular shape is $7$ square units.

What is the area of irregular shapes?

Common Core State Standards

How does this relate to $3$ rd grade math and $4$ th grade math?

Grade 3 – Measurement and Data (3.MD.7, 3.MD.7.d)
Relate area to the operations of multiplication and addition; Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

Grade 4 – Measurement and Data (4.MD.3)
Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

How to find the area of irregular shapes

In order to find the area of irregular shapes:

Break down the irregular shape into regular shapes.
Find the area of each basic shape.
Add the areas.
Write down your final answer with square units.

Area of irregular shapes examples

Example 1: using a square grid

Find the area of the given irregular shape.

Break down the irregular shape into regular shapes.

This shape can be broken down into $2$ rectangles.

2Find the area of each basic shape.

Find the area of each shape by counting the square units present.

3Add the areas.

\text{Area}=A+B=28+32=60

4Write down your final answer with square units.

$60$ square units.

Example 2: using a square grid

Find the area of the given irregular shape.

Break down the irregular shape into regular shapes.

This shape can be broken down into three smaller shapes, a square, a rectangle, and a triangle.

Find the area of each basic shape.

Find the area of each shape by counting the square units present. Remember that two half squares make $1$ square.

Add the areas.

\text{Area}=A+B+C=4+20+8=32

Write down your final answer with square units.

$32$ square units.

Example 3: word problem

Bridges High School is putting in a flower bed in the courtyard. The sketch below shows the flower bed in square feet. What is the area of the new flower bed?

Break down the irregular shape into regular shapes.

This shape can be broken down into three shapes, two triangles and one rectangle.

Find the area of each basic shape.

Find the area of each shape by counting the square units present.

Add the areas.

\text{Area}=A+B+C=4.5+24+4.5=33

Write down your final answer with square units.

33\mathrm{~ft}^2

Example 4: irregular shapes (all sides given)

Find the area of the irregular shape below.

Break down the irregular shape into regular shapes.

We can split the shape into $2$ rectangles.

The length of $13 \, cm$ is no longer correct as it has been split into two separate lengths, matching the vertical faces on the opposite side of the rectangles

Find the area of each basic shape.

Use the formula for finding the area of a rectangle: $A=l \times w$

The area of $A$ is: $A=l\times{w}=10\times{6}=60$

The area of $B$ is: $A=l\times{w}=11\times{3}=33$

Add the areas.

\text{Area}=A+B=60+33=93

Write down your final answer with square units.

93\mathrm{~cm}^2

Example 5: irregular shapes (missing sides)

Find the area of the irregular shape below.

Break down the irregular shape into regular shapes.

This shape can be broken down into $4$ smaller shapes, $2$ squares and $2$ rectangles.

The two vertical lengths of $8\mathrm{~in}$ and $10\mathrm{~in}$ must change as these lengths have been split up.

First, the height of $A$ must be $2\mathrm{~in}$ as the vertical edges sum to $10\mathrm{~in} \, (10-8=2).$

As the height of $C$ is $3\mathrm{~in},$ the height of $B$ is $5\mathrm{~in} \, (8-3=5).$

Next, relabel the vertical side of $10\mathrm{~in}$ into the three component lengths: $2\mathrm{~in},~5\mathrm{~in},~3\mathrm{~in}.$

Finally, calculate the width of $B$ using the horizontal components $(7+3=10).$

Find the area of the basic shapes.

\begin{aligned}& A=7\times{2}=14 \\\\ & B=10\times{5}=50 \\\\ & C=3\times{3}=9 \\\\ & D=3\times{3}=9 \end{aligned}

Add the areas.

\begin{aligned}\text{Area}&=A+B+C+D \\\\ &=14+50+9+9 \\\\ &=82 \end{aligned}

Write down your final answer with square units.

82\mathrm{~in}^2

Example 6: word problem

Emma built a dog run that has two dog lanes and space in the middle for a water area. She divided the dog run as seen below. The shape has a horizontal and vertical line of symmetry. What is the total area of her new dog run?

Break down the irregular shape into regular shapes.

As the shape has a horizontal and vertical line of symmetry, label the sides that do not require calculation:

The missing horizontal lengths are:

$(17-12)\div{2}=2.5\mathrm{~ft}.$

Label each of the three rectangles:

Find the area of the basic shapes.

\begin{aligned}& A=17\times{6}=102 \\\\ & B=8\times{12}=96 \\\\ & C=17\times{6}=102 \end{aligned}

Add the areas.

\text{Area}=A+B+C=102+96+102=300

Write down your final answer with square units.

300\mathrm{~ft}^2

Teaching tips for area of irregular shapes

While area of irregular shapes worksheets are necessary to allow for student practice, providing students with visual aids, such as grid paper and colored shapes, can help students visualize the decomposition of irregular shapes into regular shapes.

When creating lesson plans, make sure to incorporate the many different technology tools that exist that can help students find the area of irregular shapes.

Have students work collaboratively to create study materials, such as a step-by-step guide or examples that have been solved. These can assist students by giving them something to study outside of the classroom, and refer back to as often as needed.

Easy mistakes to make

Miscounting squares when using a square grid
Take care when counting squares using a square grid. Find a way to mark each square so you don’t end up counting the same square more than once, or forgetting a square all together. It’s also important to note any partial squares that can be added as fractions.

Using the wrong formulas when calculating area
Label each type of basic or regular shape based on its properties. It’s important to be familiar with how to find the area of each of these shapes, including the area of the triangle.

Forgetting to add up all individual shapes
Labeling each individual shape with a letter or number can help ensure that you do not forget a shape when calculating the total area.

Writing an equation with each of the shapes $($ for example, $\text{Area}=A+B+C)$ before starting any math calculations can help make sure you account for each area.

Practice area of irregular shapes questions

$28$ square units

$48$ square units

$20$ square units

$52$ square units

Split the irregular shape into smaller, regular shapes, and then count the squares within the shape to find the area.

Area of irregular shapes 26 US

$8+12+28=48$ square units

$144$ square units

$48$ square units

$80$ square units

$96$ square units

Split the irregular shape into smaller, regular shapes, and then count the squares within the shape to find the area.

Area of irregular shapes 28 US

$32+16+48=96$ square units

$40$ square meters

$77$ square meters

$45$ square meters

$85$ square meters

Divide the shape into $2$ rectangles, and find the area of each rectangle.

Area of irregular shapes 30 US

A=3\times{15}=45

B=5\times{8}=40

\text{Area}=45+40=85\mathrm{~m}^2

96\mathrm{~cm}^2

12\mathrm{~cm}^2

94.5\mathrm{~cm}^2

87\mathrm{~cm}^2

Divide the shape into a rectangle and a triangle, then use the appropriate formulas to find the area of both shapes.

Area of irregular shapes 32 US

A=7\times{12}=84

B=(3\times{7})\div{2}=10.5

84+10.5=94.5\mathrm{~cm}^2

117\mathrm{~ft}^2

54\mathrm{~ft}^2

213\mathrm{~ft}^2

288\mathrm{~ft}^2

Divide the areas into $3$ rectangles and find the area of each. As the shape has two lines of symmetry, write the missing side lengths, including any calculated length.

(13-9)\div{2}=2\mathrm{~ft}

Area of irregular shapes 34 US

A=13\times{9}=117

B=9\times{6}=54

C=13\times{9}=117

\text{Area}=117+54+117=288\mathrm{~ft}^2

330\mathrm{~ft}^2

571\mathrm{~ft}^2

196\mathrm{~ft}^2

45\mathrm{~ft}^2

Find the area of the three regular shaped areas to find the total area of the irregular shape.

Playground:

\text{Area}=l\times{w}=22\times{15}=330

Picnic Area:

\text{Area}=l\times{l}=14\times{14}=196

Garden:

\text{Area}=\cfrac{1}{2}\times{b}\times{h}=\cfrac{1}{2}\times{6}\times{15}=45

\begin{aligned}\text{Total area}&=\text{Playground}+\text{Picnic area}+\text{Garden} \\\\ &=330+196+45 \\\\ &=571\mathrm{~ft}^2 \end{aligned}

Area of irregular shapes FAQs

What is an irregular shape?

An irregular shape is any shape that does not have equal sides or angles. These shapes cannot be classified by any standard geometric names, like squares or triangles. Some examples of irregular shapes include polygons with different side lengths and shapes with curved boundaries.

How do you find the area of composite figures?

You can find the area of composite figures, or shapes, similarly to how you find the area of irregular shapes. You can break the composite shape into basic shapes, find the area of each individual shape, and then add the sums together to find the total area of the composite shape.

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