Areas of Polygons for 7th Grade

Areas of Polygons

Polygon Definition

A polygon defines a geometric shape that is made up of sides. In other words, under the umbrella of polygons fall the following square, rectangle, parallelogram, trapezoid, and many more.

For example, a triangle has 3 sides, every quadrilateral has 4 sides, and so on.

We have already learned to calculate the areas of standard polygons. There are also non-standard polygons, for which there is no specific formula. However, their area of complex shapes can be calculated using two methods:

We can divide the area of the required polygon into several areas of polygons that we are familiar with, calculate the areas separately, and then add them together to obtain the final area.
We can try to "complete" the area of the required polygon into another polygon whose area we know how to calculate, and the proceed to subtract the area we added. This way, we can obtain the area of the original polygon.

Example

Let's demonstrate this using a simple exercise:

Diagram of a composite shape divided into two rectangles, with dimensions labeled. The left rectangle has dimensions 7 by 4 with an area (A) of 28, and the right rectangle has dimensions 3 by 6 with an area (A) of 18. The diagram illustrates how to calculate areas of composite polygons by dividing them into simpler shapes. Featured in a tutorial on calculating areas of polygons.

Here is a drawing of a polygon.

We need to calculate its area. From the start, we can see that this is not a standard polygon, so we will use the first method to calculate its area. We will divide the polygon as shown in the drawing, and we should obtain two rectangles.

According to the data shown in the drawing, in the rectangle on the right side we obtain the side lengths of 3 and 6, therefore the area of the rectangle will be 18 (multiplication of the two values). In the rectangle on the left side we obtain the side lengths of 4 and 7, therefore the area of the rectangle will be 28 (multiplication of the two values). Thus, the total area of the polygon will be the sum of the two areas we calculated separately, meaning, 18+28=46.

Detailed explanation

Start practice

Test yourself on area of a rectangle!

AB = 10 cm

The height of the rectangle is 5 cm.

Calculate the area of the parallelogram.

Practice more now

In 7th grade we focus on learning about several polygons (click on the links for in-depth reading):

How to calculate areas of polygons

The formula for calculating the area of a polygon varies according to the polygon in question. (Click on the titles to read the full articles including examples and practice)

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Test your knowledge

Question 1

AB = 12 cm

The height of the rectangle is 4 cm.

Calculate the area of the parallelogram.

Question 2

AB = 15 cm

The height of the rectangle is 6 cm.

Calculate the area of the parallelogram.

Question 3

AB = 17 cm

The height of the rectangle is 8 cm.

Calculate the area of the parallelogram.

Calculating Rectangle Area

The formula for calculating the area of a rectangle is: width X length.

$S=w\cdot h$

Calculating the area of any triangle

The formula for calculating the area of any triangle: base X height divided by 2

$S={{Base \cdot h}\over 2}$

Do you know what the answer is?

Question 1

AB = 25 cm

The height of the rectangle is 13 cm.

Calculate the area of the parallelogram.

Question 2

AB = 32 cm

The height of the rectangle is 15 cm.

Calculate the area of the parallelogram.

Question 3

AB = 3 cm

Height of the rectangle = 1.5 cm

Calculate the area of the parallelogram.

Calculating the area of a right triangle

In the case of a right triangle's area, it's the same formula, but the height is actually one of the sides

Calculating the Area of a Parallelogram

The area of a parallelogram is calculated by multiplying one of its sides by the height.

$S={{Base \cdot h}\over 2}$

For example in the drawing, you can calculate the area of the parallelogram by multiplying DC by h1 and then dividing by 2, or by multiplying BC by h2 and then dividing by 2

Check your understanding

Question 1

AB = 5 cm

The height of the rectangle is 2 cm.

Calculate the area of the parallelogram.

Question 2

AB = 6 cm

The height of the rectangle is 2 cm.

Calculate the area of the parallelogram.

Question 3

AB = 7 cm

Height of the rectangle = 3.5 cm

Calculate the area of the parallelogram.

Calculating the Area of a Trapezoid

The formula for calculating the area of a trapezoid is the sum of the two bases X the height divided by 2

$S={{(Base{_1} + Base{_2}) \cdot h}\over 2}$

A1 - How do you calculate the area of a new trapezoid

Do you think you will be able to solve it?

Question 1

ABCD is a parallelogram.

AH is the height.

DC = 6
AH = 3

What is the area of the parallelogram?

Question 2

ABCD is a parallelogram.

AH is its height.

Given in cm:

AB = 7

AH = 2

Calculate the area of the parallelogram.

Question 3

ABCD is a rectangle.

Given in cm:

AB = 7

BC = 5

Calculate the area of the rectangle.

Examples with solutions for Areas of Polygons for 7th Grade

Exercise #1

Calculate the area of the right triangle below:

Video Solution

Step-by-Step Solution

Due to the fact that AB is perpendicular to BC and forms a 90-degree angle,

it can be argued that AB is the height of the triangle.

Hence we can calculate the area as follows:

$\frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24$

Answer

24 cm²

Exercise #2

Calculate the area of the triangle ABC using the data in the figure.

Video Solution

Step-by-Step Solution

First, let's remember the formula for the area of a triangle:

(the side * the height that descends to the side) /2

In the question, we have three pieces of data, but one of them is redundant!

We only have one height, the line that forms a 90-degree angle - AD,

The side to which the height descends is CB,

Therefore, we can use them in our calculation:

$\frac{CB\times AD}{2}$

$\frac{8\times9}{2}=\frac{72}{2}=36$

Answer

36 cm²

Exercise #3

The trapezoid ABCD is shown below.

AB = 2.5 cm

DC = 4 cm

Height (h) = 6 cm

Calculate the area of the trapezoid.

Video Solution

Step-by-Step Solution

First, let's remind ourselves of the formula for the area of a trapezoid:

$A=\frac{\left(Base\text{ }+\text{ Base}\right)\text{ h}}{2}$

We substitute the given values into the formula:

(2.5+4)*6 =
6.5*6=
39/2 =
19.5

Answer

$19\frac{1}{2}$

Exercise #4

The trapezoid ABCD is shown below.

Base AB = 6 cm

Base DC = 10 cm

Height (h) = 5 cm

Calculate the area of the trapezoid.

Video Solution

Step-by-Step Solution

First, we need to remind ourselves of how to work out the area of a trapezoid:

Formula for calculating trapezoid area

Now let's substitute the given data into the formula:

(10+6)*5 =
2

Let's start with the upper part of the equation:

16*5 = 80

80/2 = 40

Answer

40 cm²

Exercise #5

What is the area of the triangle in the drawing?

Video Solution

Step-by-Step Solution

First, we will identify the data points we need to be able to find the area of the triangle.

the formula for the area of the triangle: height*opposite side / 2

Since it is a right triangle, we know that the straight sides are actually also the heights between each other, that is, the side that measures 5 and the side that measures 7.

We multiply the legs and divide by 2

$\frac{5\times7}{2}=\frac{35}{2}=17.5$

Answer

17.5

Start practice

Areas of Polygons for 7th Grade