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!

Complete the sentence:

To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.

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

A parallelogram has a length equal to 6 cm and a height equal to 4.5 cm.

Calculate the area of the parallelogram.

Question 2

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

Question 3

Calculate the area of the following triangle:

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

Calculate the area of the following triangle:

Question 2

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

Question 3

AB = 5 cm

The height of the rectangle is 2 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

Given the following trapezoid:

Calculate the area of the trapezoid ABCD.

Question 2

Given the following rectangle:

Find the area of the rectangle.

Question 3

AB = 6 cm

The height of the rectangle is 2 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

AB = 17 cm

The height of the rectangle is 8 cm.

Calculate the area of the parallelogram.

Question 2

Look at the rectangle ABCD below.

Side AB is 4.5 cm long and side BC is 2 cm long.

What is the area of the rectangle?

Question 3

Given the trapezoid:

What is the area?

Examples with solutions for Areas of Polygons for 7th Grade

Exercise #1

Complete the sentence:

To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.

Step-by-Step Solution

To solve this problem, begin by identifying the elements involved in calculating the area of a right triangle. In a right triangle, the two sides that form the right angle are known as the legs. These legs act as the base and height of the triangle.

The formula for the area of a triangle is given by:

A = \frac{1}{2} \times \text{base} \times \text{height}

In the case of a right triangle, the base and height are the two legs. Therefore, the process of finding the area involves multiplying the lengths of the two legs together and then dividing the product by 2.

Based on this analysis, the correct way to complete the sentence in the problem is:

To find the area of a right triangle, one must multiply the two legs by each other and divide by 2.

Answer

the two legs

Exercise #2

A parallelogram has a length equal to 6 cm and a height equal to 4.5 cm.

Calculate the area of the parallelogram.

Video Solution

Step-by-Step Solution

To solve this problem, let's apply the formula for the area of a parallelogram:

The formula for the area of a parallelogram is $\text{Area} = \text{base} \times \text{height}$ .

Here, the base of the parallelogram is 6 cm, and the height is 4.5 cm.

Substituting these values into the formula gives:

$\text{Area} = 6 \times 4.5$

Performing the multiplication:

$\text{Area} = 27$ square centimeters.

Therefore, the area of the parallelogram is $27 \, \text{cm}^2$ .

Referring to the given multiple-choice answers, the correct choice is:

Choice 3: $27$ .

Answer

Exercise #3

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

Video Solution

Step-by-Step Solution

Remember that the formula for the area of a rectangle is width times height

We are given that the width of the rectangle is 6

and that the length of the rectangle is 4

Therefore we calculate:

6*4=24

Answer

24 cm²

Exercise #4

Calculate the area of the following triangle:

Video Solution

Step-by-Step Solution

The formula for calculating the area of a triangle is:

(the side * the height from the side down to the base) /2

That is:

$\frac{BC\times AE}{2}$

We insert the existing data as shown below:

$\frac{4\times5}{2}=\frac{20}{2}=10$

Answer

Exercise #5

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

Video Solution

Step-by-Step Solution

To solve the problem of finding the area of triangle $\triangle ABC$ , we follow these steps:

Step 1: Identify the given measurements.
Step 2: Use the appropriate formula for the area of a triangle.
Step 3: Calculate the area using these measurements.

Let's go through each step in detail:
Step 1: From the figure, the base $AB = 10$ and height $AC = 2$ .
Step 2: The formula for the area of a triangle is: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ .
Step 3: Substituting the known values into the formula, we get:

$\text{Area} = \frac{1}{2} \times 10 \times 2 = \frac{1}{2} \times 20 = 10$

Therefore, the area of triangle $\triangle ABC$ is 10.

Answer

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Areas of Polygons for 7th Grade