Areas of Polygons for 7th Grade

🏆Practice area of a rectangle

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.

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Test yourself on area of a rectangle!

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What is the area of the given triangle?

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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

Calculating Rectangle Area

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

S=whS=w\cdot h

A2 - Rectangle area formula

Calculating the area of any triangle

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

S=Baseh2S={{Base \cdot h}\over 2}

A3 - Triangle Area Formula

Do you know what the answer is?

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

A - area of a new right triangle

Calculating the Area of a Parallelogram

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

S=Baseh2S={{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

1 - Area of a Parallelogram

Check your understanding

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=(Base1+Base2)h2S={{(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?

Examples with solutions for Areas of Polygons for 7th Grade

Exercise #1

What is the area of the given triangle?

555999666

Video Solution

Step-by-Step Solution

This question is a bit confusing. We need start by identifying which parts of the data are relevant to us.

Remember the formula for the area of a triangle:

A1- How to find the area of a triangleThe height is a straight line that comes out of an angle and forms a right angle with the opposite side.

In the drawing we have a height of 6.

It goes down to the opposite side whose length is 5.

And therefore, these are the data points that we will use.

We replace in the formula:

6×52=302=15 \frac{6\times5}{2}=\frac{30}{2}=15

Answer

15

Exercise #2

What is the area of the triangle in the drawing?

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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

5×72=352=17.5 \frac{5\times7}{2}=\frac{35}{2}=17.5

Answer

17.5

Exercise #3

The triangle ABC is given below.
AC = 10 cm

AD = 3 cm

BC = 11.6 cm
What is the area of the triangle?

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Video Solution

Step-by-Step Solution

The triangle we are looking at is the large triangle - ABC

The triangle is formed by three sides AB, BC, and CA.

Now let's remember what we need for the calculation of a triangular area:

(side x the height that descends from the side)/2

Therefore, the first thing we must find is a suitable height and side.

We are given the side AC, but there is no descending height, so it is not useful to us.

The side AB is not given,

And so we are left with the side BC, which is given.

From the side BC descends the height AD (the two form a 90-degree angle).

It can be argued that BC is also a height, but if we delve deeper it seems that CD can be a height in the triangle ADC,

and BD is a height in the triangle ADB (both are the sides of a right triangle, therefore they are the height and the side).

As we do not know if the triangle is isosceles or not, it is also not possible to know if CD=DB, or what their ratio is, and this theory fails.

Let's remember again the formula for triangular area and replace the data we have in the formula:

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

Now we replace the existing data in this formula:

CB×AD2 \frac{CB\times AD}{2}

11.6×32 \frac{11.6\times3}{2}

34.82=17.4 \frac{34.8}{2}=17.4

Answer

17.4

Exercise #4

The width of a rectangle is equal to 15 cm and its length is 3 cm.

Calculate the area of the rectangle.

Video Solution

Step-by-Step Solution

To calculate the area of the rectangle, we multiply the length by the width:

15×3=45 15\times3=45

Answer

45

Exercise #5

Calculate the area of the trapezoid.

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Video Solution

Step-by-Step Solution

We use the formula (base+base) multiplied by the height and divided by 2.

Note that we are only provided with one base and it is not possible to determine the size of the other base.

Therefore, the area cannot be calculated.

Answer

Cannot be calculated.

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