When students hear the words "compound shapes", they usually feel uncomfortable. Just before you also ask yourself: "Oh, why this again?", you should be aware that there is no real reason. Describing shapes as compound doesn't really make them so. As it turns out calculating areas and perimeters of compound shapes is in fact relatively straightforward.
You will be introduced to Complex shapes only after you learn various shapes in geometry. The reason these shapes are complex is due to the fact that they are slightly different from those you've come to know. In each complex shape, additional shapes that you need to identify are hidden. Dividing the complex shape into several different (and familiar) shapes will allow you to answer the question of how to calculate the area of complex shapes.
The trick: extract a familiar shape from within the complex shape
So how do we answer the question of how to calculate the area of complex shapes? First, you need to identify familiar shapes within the complex shape. An example of this: a rectangle. As you know, each shape has properties that you are familiar with, so within the complex shape itself, you can apply the properties of the familiar shape and thus calculate areas and perimeters.
After completing the missing data (according to the properties of each shape, for example: rectangle), you can complete the "puzzle", identify additional data that is revealed to you, and thus calculate the area of the complex shape. When calculating the area of complex shapes, you will often need to perform simple arithmetic operations such as division and addition (mainly for sides in the shape) - all based on the unique properties of each shape.
The width of a rectangle is equal to 15 cm and its length is 3 cm.
Calculate the area of the rectangle.
Incorrect
Correct Answer:
45
Question 5
Calculate the area of the trapezoid.
Incorrect
Correct Answer:
Cannot be calculated.
Examples with solutions for Area of a Rectangle
Exercise #1
What is the area of the given triangle?
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:
The 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:
26×5=230=15
Answer
15
Exercise #2
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
25×7=235=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?
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:
2CB×AD
211.6×3
234.8=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
Answer
45
Exercise #5
Calculate the area of the trapezoid.
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.
Question 1
Complete the sentence:
To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.
Incorrect
Correct Answer:
the two legs
Question 2
Calculate the area of the triangle below, if possible.
Incorrect
Correct Answer:
Cannot be calculated
Question 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?
Incorrect
Correct Answer:
24 cm²
Question 4
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?
Incorrect
Correct Answer:
9 cm²
Question 5
Look at rectangle ABCD below.
Side AB is 10 cm long and side BC is 2.5 cm long.
What is the area of the rectangle?
Incorrect
Correct Answer:
25 cm²
Exercise #6
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=21×base×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 #7
Calculate the area of the triangle below, if possible.
Video Solution
Step-by-Step Solution
The formula to calculate the area of a triangle is:
(side * height corresponding to the side) / 2
Note that in the triangle provided to us, we have the length of the side but not the height.
That is, we do not have enough data to perform the calculation.
Answer
Cannot be calculated
Exercise #8
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 #9
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?
Video Solution
Step-by-Step Solution
We begin by multiplying side AB by side BC
We then substitute the given data and we obtain the following:
4.5×2=9
Hence the area of rectangle ABCD equals 9
Answer
9 cm²
Exercise #10
Look at rectangle ABCD below.
Side AB is 10 cm long and side BC is 2.5 cm long.
What is the area of the rectangle?
Video Solution
Step-by-Step Solution
Let's begin by multiplying side AB by side BC
If we insert the known data into the above equation we should obtain the following:
10×2.5=25
Thus the area of rectangle ABCD equals 25.
Answer
25 cm²
Question 1
Given the following rectangle:
Find the area of the rectangle.
Incorrect
Correct Answer:
54
Question 2
Given the following rectangle:
Find the area of the rectangle.
Incorrect
Correct Answer:
32
Question 3
Given the following rectangle:
Find the area of the rectangle.
Incorrect
Correct Answer:
10
Question 4
Given the following rectangle:
Find the area of the rectangle.
Incorrect
Correct Answer:
77
Question 5
Calculate the area of the following parallelogram:
Incorrect
Correct Answer:
30 cm²
Exercise #11
Given the following rectangle:
Find the area of the rectangle.
Video Solution
Step-by-Step Solution
We will use the formula to calculate the area of a rectangle: length times width
9×6=54
Answer
54
Exercise #12
Given the following rectangle:
Find the area of the rectangle.
Video Solution
Step-by-Step Solution
Let's calculate the area of the rectangle by multiplying the length by the width:
4×8=32
Answer
32
Exercise #13
Given the following rectangle:
Find the area of the rectangle.
Video Solution
Step-by-Step Solution
Let's calculate the area of the rectangle by multiplying the length by the width:
2×5=10
Answer
10
Exercise #14
Given the following rectangle:
Find the area of the rectangle.
Video Solution
Step-by-Step Solution
Let's calculate the area of the rectangle by multiplying the length by the width:
11×7=77
Answer
77
Exercise #15
Calculate the area of the following parallelogram:
Video Solution
Step-by-Step Solution
To solve the exercise, we need to remember the formula for the area of a parallelogram:
Side * Height perpendicular to the side
In the diagram, although it's not presented in the way we're familiar with, we are given the two essential pieces of information:
Side = 6
Height = 5
Let's now substitute these values into the formula and calculate to get the answer: