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9.2. Area of a Polygon
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Good morning, class! Today, we will learn how to find the area of polygons by breaking them into simpler shapes. Does anyone remember how we find the area of triangles?
Yes! We use the formula area equals one-half times base times height.
Exactly! And we can use triangles to help us find the area of more complex shapes, like polygons. For example, we can take a quadrilateral and divide it into two triangles.
Could you show us how that works?
Of course! Let’s take a quadrilateral ABCD and draw diagonal AC. Now we have two triangles, ABC and ACD. Can anyone tell me how we would find the area of quadrilateral ABCD?
We just find the area of triangle ABC and add it to the area of triangle ACD!
Correct! Remember, this idea of decomposing shapes into triangles is essential for determining the area of any polygon.
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Now that we understand using triangles, let's talk about trapeziums. Who can remind us of the area formula for a trapezium?
It's one-half times the sum of the parallel sides times the height!
Great! Now, if I wanted to find the area of a polygon that includes a trapezium like shape, how would I start?
We would break it into a trapezium and possibly triangles, right?
Exactly! Let’s visualize this with a polygon and see how we can compute its area by identifying the trapezium within it.
So we look for two parallel sides in the shape?
Yes! And after calculating the area of each shape we identify, we sum them up to find the total area of the polygon. Let’s practice this with some examples.
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Create a free accountLet’s apply our knowledge with some practical examples. For instance, if you have a trapezium-shaped field with given lengths, how would you go about calculating its area?
We need to apply the trapezium area formula, right?
That’s right! And what if we had a more complex shape, say a pentagonal park?
We would divide it into triangles and trapeziums too!
Exactly! After we sum each individual area, we will have the total area of the park. Let’s calculate together using specific values from our example.
Can we do more than one way to divide it?
Of course! Different methods can offer insights into our understanding of area.
Overview
Short Summary
This section covers methods to calculate the area of various polygons through decomposition into simpler shapes like triangles and trapeziums.
Medium Summary
In this section, we explore the process of determining the area of polygons by dividing them into simpler shapes such as triangles and trapeziums. Various examples, exercises, and practical applications are included to reinforce understanding.
Detailed Summary
Area of a Polygon
In this section, we learn how to calculate the area of polygons by breaking them down into simpler shapes. For instance, we can divide a quadrilateral into triangles, allowing us to apply previously learned area formulas. The area of a polygon can often be determined through identifying its parts, calculating their individual areas, and then summing these values.
Key Concepts:
- Decomposition: The primary strategy involves cutting complex shapes into familiar ones like triangles and trapeziums. This method makes calculations feasible and often easier to visualize.
- Areas of Basic Shapes: Understanding how to calculate the area of triangles and trapeziums is crucial. We utilize formulas such as:
- Area of a triangle = 1/2 * base * height
- Area of a trapezium = 1/2 * (sum of parallel sides) * height
- Practical Applications: The section includes tasks to find the area of irregular polygons by applying the decomposition strategy, illustrated through several examples, such as finding the area of a trapezium-shaped field or a pentagonal park.
- Exercises: The section is rich in exercises that challenge students to apply their knowledge, ensuring comprehension of the area concept and calculation methods.
Thus, the area of polygons can be effectively determined using decomposition techniques and well-understood formulas, fostering a strong geometric intuition.
Example 2:
The area of a trapezium-shaped garden is 600 m², the distance between two parallel sides is 10 m and one of the parallel sides measures 25 m. Find the other parallel side.
Solution: Let one of the parallel sides of the trapezium be m, and the other parallel side be , height m.
The given area of the trapezium = 600 m²:
So,
\
\
\
Hence the other parallel side of the trapezium is 95 m.
Reference YouTube Videos
Audio Book
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Create a free accountWe split a quadrilateral into triangles and find its area. Similar methods can be used to find the area of a polygon.
Detailed Explanation
To find the area of a polygon, we can use the method of dividing the shape into simpler components, like triangles. This is because calculating the area of a triangle is straightforward using the formula: Area = 1/2 × base × height. By adding the areas of all the triangles, we can find the total area of the polygon.
Examples & Analogies
Think of a large field (which could be any polygonal shape) that you want to plant grass in. Instead of measuring the entire field at once, you could divide it into smaller triangular sections. After measuring the area of each triangle, you can add them up to find out how much space you will need for the grass seed.
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Create a free accountBy constructing one diagonal AD and two perpendiculars BF and CG on it, pentagon ABCDE is divided into four parts.
Detailed Explanation
For a pentagon, we can draw diagonals which divide the shape into several triangles and trapeziums. In this case, diagonal AD splits the pentagon into two triangles and two additional shapes formed by the perpendiculars. Each section can then have its area calculated separately and then summed up for the total area of the pentagon.
Examples & Analogies
Imagine trying to cover a flat roof that has a pentagon shape. By drawing lines from one corner to the opposite corner (diagonals) and dropping straight lines down to the base, you can calculate the area of smaller sections, making it easier to figure out how much material you'll need.
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Create a free accountArea of polygon ABCDE = area of ∆ ABC + area of right angled ∆ AFB + area of trapezium BFGC + area of right angled ∆ CGD + area of ∆ AED.
Detailed Explanation
To compute the area of polygon ABCDE, we break it down into smaller, manageable shapes. Each area can be calculated using relevant formulas. For instance, triangles can be calculated using the triangle area formula, and trapeziums can be calculated using the trapezium area formula. By summing these areas, we obtain the total area of the polygon.
Examples & Analogies
Consider a park shaped like a pentagon where different sections contain gardens, paths, and playgrounds. By determining the area of each garden (triangles) and walkway (trapezium), the park manager can understand how much ground needs to be maintained, similar to calculating areas for lawn care.
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Create a free accountTRY THESE: (i) Divide the following polygons into parts to find out its area. (ii) Polygon ABCDE is divided into parts as shown. Find its area if AD = 8 cm, AH = 6 cm, AG = 4 cm, AF = 3 cm and perpendiculars BF = 2 cm, CH = 3 cm, EG = 2.5 cm.
Detailed Explanation
In exercises focusing on area, students learn to apply the concepts discussed. For polygon ABCDE, with given dimensions, the approach involves calculating the area of each constituent shape (like triangles and trapeziums), then adding those areas together to find the total area. This practical application helps reinforce theoretical knowledge through hands-on problem solving.
Examples & Analogies
Think about an architect designing a building with an irregular shape. They will need to calculate the area of different sections of the building (like the main hall, rooms, and balconies) to estimate how much flooring material they will need. By breaking down the building’s footprint into recognizable shapes, they can calculate total area accurately.
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Create a free accountExample: The area of a trapezium shaped field is 480 m², the distance between two parallel sides is 15 m and one of the parallel side is 20 m. Find the other parallel side.
Detailed Explanation
To solve for the unknown length of the other parallel side in the trapezium, we can use the formula for the area of a trapezium: Area = 1/2 × (a + b) × h, where 'a' and 'b' are the lengths of parallel sides, and 'h' is the height. Substituting known values and rearranging the equation allows us to isolate and find the length of the unknown parallel side.
Examples & Analogies
Suppose you are placing a new fence around a trapezium-shaped garden. Knowing the area you want to cover and one side's length helps you plan and budget for materials. The calculation ensures that you have enough fencing and helps determine the space allocated for planting.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Decomposition: The primary strategy involves cutting complex shapes into familiar ones like triangles and trapeziums. This method makes calculations feasible and often easier to visualize.
Areas of Basic Shapes: Understanding how to calculate the area of triangles and trapeziums is crucial. We utilize formulas such as:
Area of a triangle = 1/2 * base * height
Area of a trapezium = 1/2 * (sum of parallel sides) * height
Practical Applications: The section includes tasks to find the area of irregular polygons by applying the decomposition strategy, illustrated through several examples, such as finding the area of a trapezium-shaped field or a pentagonal park.
Exercises: The section is rich in exercises that challenge students to apply their knowledge, ensuring comprehension of the area concept and calculation methods.
Thus, the area of polygons can be effectively determined using decomposition techniques and well-understood formulas, fostering a strong geometric intuition.
Example 2:
The area of a trapezium-shaped garden is 600 m², the distance between two parallel sides is 10 m and one of the parallel sides measures 25 m. Find the other parallel side.
Solution: Let one of the parallel sides of the trapezium be m, and the other parallel side be , height m.
The given area of the trapezium = 600 m²:
So,
\
\
\
Hence the other parallel side of the trapezium is 95 m.
Examples
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