How do we calculate the area of complex shapes?

Area of a Rectangle - Examples, Exercises and Solutions
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How do we calculate the area of complex shapes?

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.

Two composite shapes labeled with side lengths. • Left shape: A 'house-like' figure formed by a rectangle (4 units wide and 4 units high) with a triangle on top (two equal sides of 6 units, base 4 units). • Right shape: An L-shaped polygon composed of rec

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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.

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How do we calculate the area of complex shapes?

When students hear the words "compound shapes", they usually feel uncomfortable. Just before you also ask yourself: "Oh, why this again?", you should know that there really is no reason. Describing shapes as compound doesn't really make them so. As it turns out calculating areas and perimeters of compound shapes can actually be 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. For example: 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 becomes apparent, and thus calculate the area of the compound shape. When calculating the area of compound shapes, you will often need to perform simple arithmetic operations like division and addition (especially for sides in the shape) - all based on the unique properties of each shape.

For example: Assuming the composite shape includes several different rectangles, based on the given side lengths, it will be possible to calculate the different areas. The area of a rectangle is calculated using the formula length X width. When the side lengths are visible, subtraction and addition can be performed (according to the sizes of the rectangles and their positions within the shape) of sides, and thus calculate the area of the shape, as seen in the example below.

Diagram of a composite shape divided into two labeled areas: 'A' and 'B.' Dimensions provided for the sides: 2, 3, 6, 8, 9, and 10 units.

To calculate the shape's area - we will divide it in a way that creates two rectangles. We will find the area by adding and/or subtracting rectangles.

In this division we created:

A rectangle with size 9⋅2=189 \cdot2 = 18 rectangle AA

A rectangle with size ​​​​8⋅6=48​​​​8 \cdot 6 = 48 rectangle BB

The area of the entire composite shape is:

9⋅2+8⋅6=669\cdot2+8\cdot6=66

or

48+18=6648 + 18= 66

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Test your knowledge
Question 1

Indicate the correct answer

The next quadrilateral is:

AAABBBCCCDDD

Question 2

Indicate the correct answer

The next quadrilateral is:

AAABBBCCCDDD

Question 3

Indicate the correct answer

The next quadrilateral is:

AAABBBCCCDDD

Examples with solutions for Area of a Rectangle

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=12×base×height 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

Indicate the correct answer

The next quadrilateral is:

AAABBBCCCDDD

Video Solution

Step-by-Step Solution

Initially, let us examine the basic properties of a deltoid (or kite):

A quadrilateral is classified as a deltoid if:

  • It has two distinct pairs of adjacent sides that are equal in length.

In the question's image, we observe the following:

  • There are lines connecting A to B, B to C, C to D, and D to A, suggesting a typical quadrilateral.
  • The shape, given its central symmetry (as it is formed by joining these particular points which extend equal lines), is reminiscent of a symmetric or bilaterally mirrored formation.
  • Given the symmetry, it suggests all internal angles are less than 180 degrees, confirming the figure as a convex shape.

From this analysis, the quadrilateral satisfies the characteristic of having pairs of equal adjacent sides which confirms it as a deltoid. The symmetry suggests it is not concave (which occurs when at least one interior angle is greater than 180 degrees).

Therefore, the given quadrilateral, based on its properties and symmetry, is a convex deltoid.

Answer

Convex deltoid

Exercise #3

Indicate the correct answer

The next quadrilateral is:

AAABBBCCCDDD

Video Solution

Step-by-Step Solution

To solve this problem, let's analyze the given quadrilateral ABCD by examining its geometric properties:

  • Step 1: Identifying characteristics of a deltoid
    A deltoid, or kite, is a quadrilateral that has two distinct pairs of adjacent sides that are equal. To classify a shape as a deltoid, we need to verify these properties.
  • Step 2: Examining the quadrilateral ABCD
    The deltoid can be either concave or convex. If the shape is concave, it will have an indentation, meaning at least one angle is greater than 180∘180^{\circ}. A convex deltoid does not have such an indentation.
  • Step 3: Analyze the sides of ABCD
    Looking at the segments from the given points:
    - Verify if pairs of adjacent sides are equal.
    If we cannot find two equal pairs of adjacent sides, the quadrilateral is not a deltoid.
  • Step 4: Drawing conclusions
    Having analyzed the sides of the quadrilateral, if none of the pairs of adjacent sides conform to the deltoid property as outlined—two pairs of equal adjacent sides—then ABCD is identified as not a deltoid.

Therefore, the correct answer is: Not deltoid.

Answer

Not deltoid

Exercise #4

Indicate the correct answer

The next quadrilateral is:

AAABBBCCCDDD

Video Solution

Step-by-Step Solution

To solve this problem, let's analyze the quadrilateral depicted:

  • Step 1: Analyze the given quadrilateral's shape using its geometric features, noting potential symmetry and side equivalence.
  • Step 2: Identify if the quadrilateral fulfills the characteristics of a deltoid, which involve pairs of adjacent sides being equal.
  • Step 3: Determine if it is possible to accurately categorize the quadrilateral as a convex or concave deltoid based on the given image and without explicit measurements.
  • Step 4: In the absence of direct measurable evidence, consider if categorization is feasible.

Assessing visuals alone can lead to assumptions about equal lengths or angles, but without numerical data, it's challenging to make definitive geometrical claims about sides or symmetry.

Given these limitations, it is reasonable to conclude that we cannot definitively prove whether the quadrilateral is a deltoid (convex or concave) using just the visual representation provided.

Therefore, the solution to the problem is "It is not possible to prove if it is a deltoid or not."

Answer

It is not possible to prove if it is a deltoid or not

Exercise #5

Indicate the correct answer

The next quadrilateral is:

AAABBBCCCDDD

Video Solution

Step-by-Step Solution

The problem requires determining if a given quadrilateral is a deltoid, and if so, whether it is convex, concave, or indeterminate based on the provided diagram. A deltoid, or kite, is generally defined as a quadrilateral with two pairs of adjacent sides being of equal length. Thus, a visual analysis is essential here as only diagrammatic data is available.

To address this, one must closely analyze the properties of the given quadrilateral in terms of similarity and its symmetry relative to a conventional deltoid structure:

  • Typically, you'd look for simultaneous symmetry or patterns indicating two equal-length adjacent pairs of sides.
  • After examining the diagram and the naming convention (vertices labelled A, B, C, D), see if it implies any such congruency visually or through label symmetry.
  • Lack of distinct clues for equal side pairs or diagonals prevents concluding its specific nature without additional information, especially since no specific length measures or angles are provided.

Given this and under diagram-only conditions, it's not possible to definitively prove that the shape is completely a deltoid (convex or concave). Therefore, without further data, identifying the indicated quadrilateral deltoid nature is beyond determining from the given data itself.

Consequently, the correct answer is: It is not possible to prove if it is a deltoid or not.

Answer

It is not possible to prove if it is a deltoid or not

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