TERM AUFSTELLEN – für Umfang und Flächeninhalt, mit Variablen, Rechteck Figur

MathemaTrick

10 Sept 202005:36

Summary

TLDRThis video script offers a detailed tutorial on calculating the perimeter and area of geometric shapes. It begins by explaining the concept of perimeter, emphasizing the need to sum the lengths of all outer sides of a figure. It then demonstrates the process with an example, calculating the perimeter by adding specific side lengths. The script proceeds to discuss area calculation, suggesting breaking the figure into rectangles for easier computation. It illustrates the method by using an example, showing how to multiply the lengths of two sides to find the area of each rectangle and then summing them up. The tutorial aims to clarify the steps involved in these calculations, encouraging viewers to ask questions for further understanding.

Takeaways

📏 The script discusses how to calculate the perimeter and area of a figure by adding the lengths of all outer sides for the perimeter.
🔍 It starts with identifying the sides of the figure that are included in the perimeter calculation, such as the side labeled '8'.
📐 The script explains that some parts of the figure can be calculated by adding lengths of known segments, like the '8' and '3' to get '8'.
🧩 The process involves combining different parts of the figure to find the total perimeter, such as adding '8', '13', '15', and '3' to get '39'.
🔢 The script simplifies the calculation by using algebraic expressions, like '2x' for the perimeter, where 'x' represents a side length.
➗ The area calculation involves breaking down the figure into simpler shapes, such as rectangles, whose areas can be calculated by multiplying the lengths of two adjacent sides.
📐 For the area, the script uses the formula for the area of a rectangle, which is the product of two sides, and applies it to each part of the figure.
🔄 The script emphasizes the importance of using parentheses in algebraic expressions to ensure the correct order of operations when calculating areas.
📝 It provides a step-by-step guide on how to calculate the area by multiplying the lengths of sides and adding the results of different parts of the figure.
📉 The script simplifies the algebraic expression for the area by multiplying the numbers inside the parentheses and adding the constants outside.
🤔 The presenter encourages viewers to ask questions in the comments if they have any, indicating an interactive approach to teaching.

Q & A

What is the first step in calculating the perimeter of a figure as described in the script?
-The first step is to add up the lengths of all the outer sides of the figure.
What is the perimeter of the figure if the sides are 8, 3, 5, and the top piece is also 8?
-The perimeter is calculated by adding the lengths of the sides: 8 (bottom) + 3 (top piece) + 5 (missing side) + 8 (top), which equals 24.
How is the side labeled 'ixs' involved in the perimeter calculation?
-The side labeled 'ixs' is part of the top piece and is combined with the number 2 to contribute to the perimeter calculation.
What is the total perimeter after adding all the lengths together in the script?
-The total perimeter is the sum of 8, 13, 15, 3, and 20, which equals 59.
How does the script suggest simplifying the expression for the perimeter?
-The script suggests simplifying the expression by combining like terms and using variables to represent repeated values.
What does the script propose for calculating the area of a figure?
-The script proposes breaking down the figure into rectangles and calculating the area by multiplying the lengths of the sides of each rectangle.
How many sides are needed to calculate the area of the first part of the figure in the script?
-Two sides are needed: one with a length of 3 and the other with a combined length of 2 plus 'ixs'.
What is the expression for calculating the area of the second part of the figure?
-The expression for the second part is simpler, involving multiplying the side with a length of 5 by the other side.
How does the script simplify the area calculation for the second part of the figure?
-The script simplifies the calculation by directly multiplying the known side lengths without additional steps.
What is the final expression for the area of the figure according to the script?
-The final expression for the area is the sum of the areas of the two parts, which involves multiplying the lengths of the sides and adding the results.
What advice does the script give for dealing with complex expressions in geometry problems?
-The script advises to simplify expressions by using variables and combining like terms to make the calculation process easier.

Outlines

00:00

🧮 Calculating the Perimeter of a Shape

In this paragraph, the speaker explains how to calculate the perimeter of a geometric figure. The process involves adding up the lengths of all the outer sides of the shape. The speaker begins by identifying each side and its length. They highlight that certain segments lack direct measurements but can be determined through subtraction or addition based on the total lengths provided. For example, if a longer segment is known to be 8 units and a part of it is 3 units, the remaining segment must be 5 units. The speaker carefully walks through the calculations, ensuring that all lengths are included in the final perimeter equation. This involves a straightforward summation of the lengths, with a resulting expression that combines both numerical and variable terms, ultimately arriving at '20 + 2x' as the formula for the perimeter.

05:00

📏 Calculating the Area of a Composite Shape

This paragraph focuses on determining the area of the same geometric figure by breaking it down into smaller, more manageable parts. The figure is divided into two rectangles, and the speaker describes how to compute the area for each rectangle separately. For the first rectangle, they multiply the base (3 units) by the height, which is a combination of 2 units plus a variable 'x.' This necessitates careful multiplication, requiring the use of parentheses to ensure the correct application of the distributive property. The second rectangle's area is simpler to calculate, involving a multiplication of its known side lengths. Finally, the speaker adds both areas to formulate the total area of the shape, simplifies the expression, and provides a combined equation '6 + 3x + 5x' that equals '6 + 8x.' The speaker concludes by encouraging viewers to reach out with any questions they might have about the process.

Mindmap

Keywords

💡Perimeter

The term 'Perimeter' refers to the total length around a two-dimensional shape, which is calculated by adding together the lengths of all its sides. In the video, the concept is central to the theme of calculating the dimensions of a shape. The script provides an example of calculating the perimeter by adding the lengths of different sides of a figure, such as 8, 3, 5, and so on, to understand the total distance around it.

💡Area Content

Area Content, or simply 'Area,' is the measure of the space enclosed within a two-dimensional shape. It is a key concept in the video, where the presenter explains how to calculate the area of shapes by using the lengths of their sides. The script mentions breaking down the figure into rectangles to calculate the area, using expressions like '2 plus x times 3' to illustrate the process.

💡Addition

Addition is the mathematical operation of combining two or more numbers to find their total or sum. In the context of the video, addition is used to calculate the perimeter by summing the lengths of the sides of a figure. The script demonstrates this with phrases like 'adding together the lengths of all outer sides'.

💡Figure

A 'Figure' in geometry refers to a shape or form with distinct properties, such as sides and angles. The video script uses the term to describe the object whose dimensions are being calculated. For example, the sides of the figure are mentioned as having specific lengths that contribute to the perimeter and area calculations.

💡Sides

In geometry, 'Sides' are the straight line segments that form the boundaries of a two-dimensional shape. The script discusses the lengths of the sides as essential components in determining the perimeter and area of a figure, mentioning specific side lengths like 'eight' and 'three'.

💡Calculation

Calculation is the process of computing or determining something, often using mathematical methods. The video's theme revolves around performing calculations to find the perimeter and area of a figure. The script provides a step-by-step guide on how to perform these calculations with given side lengths.

💡Length

Length is a measure of distance or the extent of something from end to end. In the script, the length of the sides of a figure is crucial for calculating its perimeter and area. The video mentions adding the lengths of the sides to find the total perimeter.

💡Rectangles

Rectangles are a type of quadrilateral with four right angles and opposite sides equal in length. The video script suggests breaking down a figure into rectangles to simplify the calculation of its area content. This approach is exemplified by the script's mention of multiplying the lengths of sides to find the area of each rectangle.

💡Expression

In mathematics, an 'Expression' is a combination of variables, numbers, and operators that represents a value. The script uses expressions to represent the formulas for calculating the perimeter and area of a figure, such as '2x + 3' for the perimeter and '(2 + x) * 3' for the area.

💡Simplification

Simplification in mathematics refers to making a complex expression or problem easier to understand or solve. The video script mentions simplifying the expressions for perimeter and area to make the calculations more straightforward, such as by multiplying the numbers within the expressions.

💡Commentary

The term 'Commentary' in the script refers to the invitation for viewers to ask questions or provide feedback in the comments section if they have any doubts or need clarification on the calculations presented in the video.

Highlights

Introduction to calculating the perimeter of a figure by adding the lengths of all outer sides.

Explanation of how to include each side in the calculation of the perimeter.

Specific example given for a figure with sides labeled for clarity.

Calculation of the unnamed side by using the known lengths of other sides.

Summation of all sides to find the total perimeter of the figure.

Introduction to calculating the area of a figure by breaking it into rectangles.

Method of calculating area by knowing two sides of a rectangle.

Use of brackets to ensure correct multiplication for the area calculation.

Simplification of the area calculation term for easier understanding.

Multiplication of the sides of the rectangles to find the total area.

Inclusion of all parts of the figure to calculate the complete area.

Explanation of how to handle the addition within the calculation for area.

Final summation of all numbers to get the total area of the figure.

Emphasis on the importance of correctly using brackets in area calculations.

Encouragement for viewers to simplify the term if not required for further simplification.

Invitation for viewers to ask questions in the comments for further clarification.