Areas of Polygons for 7th Grade - Examples, Exercises and Solutions

Question Types:
Area of a Parallelogram: Applying the formulaArea of a Parallelogram: Calculate The Missing Side based on the formulaArea of a Parallelogram: Calculating in two waysArea of a Parallelogram: Finding Area based off Perimeter and Vice VersaArea of a Parallelogram: Using additional geometric shapesArea of a Parallelogram: Using congruence and similarityArea of a Parallelogram: Using external heightArea of a Parallelogram: Using Pythagoras' theoremArea of a Parallelogram: Using ratios for calculationArea of a Parallelogram: Using variablesArea of a Parallelogram: Verifying whether or not the formula is applicableArea of a Rectangle: Applying the formulaArea of a Rectangle: A shape consisting of several shapes (requiring the same formula)Area of a Rectangle: Calculate The Missing Side based on the formulaArea of a Rectangle: Calculation using the diagonalArea of a Rectangle: Extended distributive lawArea of a Rectangle: Finding Area based off Perimeter and Vice VersaArea of a Rectangle: Subtraction or addition to a larger shapeArea of a Rectangle: Using additional geometric shapesArea of a Rectangle: Using Pythagoras' theoremArea of a Rectangle: Using ratios for calculationArea of a Rectangle: Using short multiplication formulasArea of a Rectangle: Using variablesArea of a Rectangle: Worded problemsArea of a Trapezoid: Applying the formulaArea of a Trapezoid: Calculate The Missing Side based on the formulaArea of a Trapezoid: Finding Area based off Perimeter and Vice VersaArea of a Trapezoid: Subtraction or addition to a larger shapeArea of a Trapezoid: Suggesting options for terms when the formula result is knownArea of a Trapezoid: Using additional geometric shapesArea of a Trapezoid: Using Pythagoras' theoremArea of a Trapezoid: Using ratios for calculationArea of a Trapezoid: Using variablesArea of a Triangle: Applying the formulaArea of a Triangle: Ascertaining whether or not there are errors in the dataArea of a Triangle: Calculate The Missing Side based on the formulaArea of a Triangle: Calculating in two waysArea of a Triangle: Extended distributive lawArea of a Triangle: Finding Area based off Perimeter and Vice VersaArea of a Triangle: How many times does the shape fit inside of another shape?Area of a Triangle: Identifying and defining elementsArea of a Triangle: Subtraction or addition to a larger shapeArea of a Triangle: Using additional geometric shapesArea of a Triangle: Using congruence and similarityArea of a Triangle: Using Pythagoras' theoremArea of a Triangle: Using ratios for calculationArea of a Triangle: Using variablesArea of a Triangle: Worded problems

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.

Practice Areas of Polygons for 7th Grade

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?

5557778.68.68.6

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

Calculate the area of the parallelogram based on the data in the figure:

101010444

Video Solution

Step-by-Step Solution

In this particular problem, despite being given certain measurements, the diagram lacks sufficient clarity to identify which corresponds definitively as the base and which as the perpendicular height of the parallelogram. This insufficiency means that without further context or labeling to avoid assumptions that may lead to error, it is not feasible to calculate the area confidently using the standard formula.

Thus, the answer to the problem is that it is not possible to calculate the area with the provided data.

Answer

It is not possible to calculate.

Exercise #4

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

Calculate the area of the parallelogram.

6664.54.54.5

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 Area=base×height \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:

Area=6×4.5 \text{Area} = 6 \times 4.5

Performing the multiplication:

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

Therefore, the area of the parallelogram is 27cm2 27 \, \text{cm}^2 .

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

Choice 3: 27 27 .

Answer

27

Exercise #5

Calculate the area of the parallelogram using the data in the figure:

555777

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the given information
  • Step 2: Apply the appropriate formula
  • Step 3: Perform the necessary calculations

Now, let's work through each step:
Step 1: The problem provides us with a base (bb) of 7 units and a height (hh) of 5 units, perpendicular to this base.
Step 2: We'll apply the formula for the area of a parallelogram, which is Area=b×h \text{Area} = b \times h .
Step 3: Substituting the given values, Area=7×5=35 \text{Area} = 7 \times 5 = 35 .

Therefore, the area of the parallelogram is 35 35 square units.

Answer

35

Exercise #6

Calculate the area of the parallelogram using the data in the figure:

888555

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the base and height from the information provided.
  • Step 2: Apply the formula for the area of a parallelogram.
  • Step 3: Calculate the area by multiplying the base and height.

Now, let's work through each step:
Step 1: The base of the parallelogram is given as 88 units, and the height is given as 55 units.
Step 2: We use the formula for the area of a parallelogram: Area=base×height \text{Area} = \text{base} \times \text{height} .
Step 3: Plugging in the given values, we calculate the area as follows:
Area=8×5=40 \text{Area} = 8 \times 5 = 40 .

Therefore, the area of the parallelogram is 40 40 square units, which corresponds to choice 2.

Answer

40

Exercise #7

Calculate the area of the parallelogram using the data in the figure:

999444

Video Solution

Step-by-Step Solution

To solve this problem, we must calculate the area of the given parallelogram using the formula:

Area=base×height \text{Area} = \text{base} \times \text{height}

Assuming the figure (as described) provides a base of 9 9 units and a height of 4 4 units, we substitute these values into the formula:

Area=9×4=36 square units \text{Area} = 9 \times 4 = 36 \text{ square units}

The necessary calculations have been carried out using the correct dimensions, ensuring dimensional consistency and precise arithmetical methods. Therefore, the calculated area of the parallelogram is 36 36 .

Given the multiple-choice options, the correct choice is the one specifying the area as 36 36 , confirming the answer provided in choice 3.

Answer

36

Exercise #8

Calculate the area of the parallelogram using the data in the figure:

555888333

Video Solution

Step-by-Step Solution

From the given constraints, it is impossible to confidently compute the area of the parallelogram because of insufficient and unclear relationships between provided figures and the calculations they must produce. Clarity on which numbers correspond to the height and base—as or any definitional angles—is absent.

The correct answer, aligning with acknowledged drawing limitations, is: It is not possible to calculate.

Answer

It is not possible to calculate.

Exercise #9

The triangle ABC is given below.
AC = 10 cm

AD = 3 cm

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

11.611.611.6101010333AAABBBCCCDDD

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 #10

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 #11

The width of a rectangle is equal to 18 18 cm and its length is 2  2~ cm.

Calculate the area of the rectangle.

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the given information
  • Step 2: Apply the appropriate formula
  • Step 3: Perform the necessary calculations

Now, let's work through each step:

Step 1: The problem gives us the width, W=18W = 18 cm, and the length, L=2L = 2 cm.

Step 2: We'll use the formula for the area of a rectangle: Area=Length×Width \text{Area} = \text{Length} \times \text{Width}

Step 3: Plugging in the values, we get: Area=2×18=36 square centimeters \text{Area} = 2 \times 18 = 36 \text{ square centimeters}

Therefore, the area of the rectangle is 3636 square centimeters.

In the provided answer choices, the correct choice is:

Choice 3: 3636

Answer

36

Exercise #12

Calculate the area of the trapezoid.

555141414666

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.

Exercise #13

Calculate the area of the trapezoid.

666777121212555

Video Solution

Step-by-Step Solution

To find the area of the trapezoid, we would ideally use the formula:

A=12×(b1+b2)×h A = \frac{1}{2} \times (b_1 + b_2) \times h

where b1b_1 and b2b_2 are the lengths of the two parallel sides and hh is the height. However, the given information is incomplete for these purposes.

The numbers provided (66, 77, 1212, and 55) do not clearly designate which are the bases and what is the height. Without this information, the dimensions cannot be definitively identified, making it impossible to calculate the area accurately.

Thus, the problem, based on the given diagram and information, cannot be solved for the area of the trapezoid.

Therefore, the correct answer is: It cannot be calculated.

Answer

It cannot be calculated.

Exercise #14

Calculate the area of the trapezoid.

555888333

Video Solution

Step-by-Step Solution

To solve this problem, we'll calculate the area of the trapezoid using the standard formula:

  • Step 1: Identify the given dimensions:
  • Shorter base b1=5 b_1 = 5 .
  • Longer base b2=8 b_2 = 8 .
  • Height h=3 h = 3 .

Step 2: We apply the trapezoid area formula, which is:

A=12×(b1+b2)×h A = \frac{1}{2} \times (b_1 + b_2) \times h .

Step 3: Substitute the given values into the formula:

A=12×(5+8)×3 A = \frac{1}{2} \times (5 + 8) \times 3 .

Step 4: Perform the calculations:

A=12×13×3 A = \frac{1}{2} \times 13 \times 3 .

A=12×39 A = \frac{1}{2} \times 39 .

A=19.5 A = 19.5 or 1912 19 \frac{1}{2} .

The area of the trapezoid is 1912 19 \frac{1}{2} .

Answer

19 1/2

Exercise #15

What is the area of the trapezoid ABCD?

999121212555AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Identify the given measurements: the lengths of the parallel sides (bases) and the height.
  • Use the trapezoid area formula to calculate the area.
  • Perform the necessary arithmetic to find the numerical answer.

Now, let's work through each step:
Step 1: The given measurements are Base1=9 \text{Base}_1 = 9 , Base2=12 \text{Base}_2 = 12 , and the height = 5.
Step 2: The formula for the area of a trapezoid is Area=12×(Base1+Base2)×Height \text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} .
Step 3: Substituting the numbers into the formula, we have:
Area=12×(9+12)×5 \text{Area} = \frac{1}{2} \times (9 + 12) \times 5

Calculating inside the parentheses first:
9+12=21 9 + 12 = 21

Then multiply by the height:
21×5=105 21 \times 5 = 105

Finally, multiply by one-half:
12×105=52.5 \frac{1}{2} \times 105 = 52.5

Therefore, the area of trapezoid ABCD ABCD is 52.5 52.5 .

Answer

52.5