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Calculating Area of a Trapezium
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Today we are going to calculate the area of a trapezium. The formula we use is: Area = (1/2) ร (base1 + base2) ร height. Can anyone help me identify the bases and height for an example trapezium?
The bases are 8cm and 12cm, and the height is 5cm.
Excellent! Now let's plug that into the formula. What do we get?
The area would be: (1/2) ร (8 + 12) ร 5 = 50 cmยฒ.
Great job! Remember, you can think of this as half of the sum of the bases multiplied by the height. Let's recap: Area = 1/2 ร (base1 + base2) ร height. Can anyone remember what we call this type of shape?
A trapezium!
Correct! That's our key point for today.
Volume of a Cylinder
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Next, let's look at cylinders. The formula for volume is Volume = ฯ ร rยฒ ร height. Who can tell me the values weโd use if we have a cylinder with a radius of 7cm and a height of 10cm?
We would use r = 7cm and height = 10cm.
Exactly! Now can someone calculate that volume?
Volume = ฯ ร (7)ยฒ ร 10 = 490ฯ cmยณ, which is approximately 1539.38 cmยณ.
Well done! To remember this formula, think of the volume of a cylinder as the base area multiplied by the height. Let's recap together: Volume = ฯ ร rยฒ ร height.
Application Problem - Calculating Tiles
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Finally, letโs solve a real-world problem. Suppose we have a floor measuring 4m by 5m, and we want to use 25cm ร 25cm tiles. How do we approach this?
First, we need to find the area of the floor in cmยฒ.
Correct! What is that area in cmยฒ?
Area = 4m ร 5m = 20mยฒ, which is 20,000 cmยฒ.
Now, how do we find out how many tiles we need?
We divide the total area by the area of one tile, which is 625 cmยฒ.
So how many tiles do we need?
We need 20,000 cmยฒ รท 625 cmยฒ = 32 tiles.
Fantastic! Remember, practical applications like this show how important mensuration is in real life. Let's review: Calculate area, convert units, apply formulas.
Introduction & Overview
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Quick Overview
Standard
In this section, students will find assessment questions that test their understanding of mensuration concepts, including calculations for area of a trapezium, volume of a cylinder, and practical applications such as determining how many tiles are needed for a given space.
Detailed
Assessment Questions in Mensuration
The assessment questions encapsulate critical aspects of mensuration, linking theoretical concepts to practical applications in geometry. Students will be tasked with calculating the area of a trapezium, employing the relevant formula:
- Area of Trapezium = (1/2) ร (base1 + base2) ร height.
Next, they delve into 3D shapes by calculating the volume of a cylinder using the formula:
- Volume of Cylinder = ฯ ร rยฒ ร height.
Lastly, the practicality of mensuration is highlighted through a real-world problem involving tiles, where students must determine the number of tiles needed to cover a specified area, practicing their skills in unit conversion and area calculation. This section reinforces key concepts from the chapter and prepares students for future practical applications.
Audio Book
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Question 1: Area of a Trapezium
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- Find the area of a trapezium with parallel sides 8cm and 12cm, height 5cm.
Detailed Explanation
To find the area of a trapezium (also called a trapezoid), you can use the formula:
\[
\text{Area} = \frac{(a + b) \times h}{2}
\]
where \( a \) and \( b \) are the lengths of the parallel sides and \( h \) is the height. In this case, the lengths of the parallel sides are 8 cm and 12 cm, and the height is 5 cm.
1. Add the lengths of the parallel sides:
- 8 cm + 12 cm = 20 cm
2. Multiply the sum by the height:
- 20 cm ร 5 cm = 100 cmยฒ
3. Divide by 2:
- 100 cmยฒ รท 2 = 50 cmยฒ.
So, the area of the trapezium is 50 cmยฒ.
Examples & Analogies
Imagine a trapezium-shaped garden bed where the top is wider than the bottom. If the top is 12 cm wide, the bottom 8 cm, and it's 5 cm tall, calculating its area helps you understand how much soil is needed to fill it or how many plants can be placed in it.
Question 2: Volume of a Cylinder
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- Calculate the volume of a cylinder with radius 7cm and height 10cm.
Detailed Explanation
To find the volume of a cylinder, the formula is:
\[
\text{Volume} = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height. In this example:
1. Find the area of the base (circle):
- \( r^2 = (7 \text{cm})^2 = 49 \text{cm}^2 \)
- So, Area = \( \pi \times 49 \approx 3.14 \times 49 \approx 153.86 \text{cm}^2 \)
2. Multiply by the height:
- Volume = 153.86 cmยฒ ร 10 cm = 1538.6 cmยณ.
Therefore, the volume of the cylinder is approximately 1538.6 cmยณ.
Examples & Analogies
Think about a tall, cylindrical glass. If the glass has a radius of 7 cm and a height of 10 cm, the volume tells you how much liquid it can hold. So, if you want to pour juice into this glass, about 1538.6 cmยณ of juice fits inside!
Question 3: Number of Tiles Needed
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- How many 25cm ร 25cm tiles needed for a 4m ร 5m floor?
Detailed Explanation
First, convert the dimensions of the floor from meters to centimeters since the tile dimensions are given in centimeters:
\[
4m = 400cm\; ext{ and }\; 5m = 500cm.
\]
Next, calculate the area of the floor:
1. Floor Area = Length ร Width = 400 cm ร 500 cm = 200,000 cmยฒ.
2. Calculate the area of one tile:
- Tile Area = 25 cm ร 25 cm = 625 cmยฒ.
3. Now, to find the number of tiles, divide the total area of the floor by the area of one tile:
- \( \text{Number of tiles} = \frac{200,000 \text{ cm}^2}{625 \text{ cm}^2} = 320 \text{ tiles}. \)
Thus, you would need 320 tiles to cover the entire floor.
Examples & Analogies
Imagine you're tiling your bathroom floor. Knowing that each tile covers a small area, you can find out how many such small tiles are needed to cover the entire floor area. Here, for a floor of size 4m ร 5m, you need enough tiles to fill it without any gaps!
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Area Calculation: Understanding area formulas for 2D shapes such as trapeziums.
-
Volume Calculation: Learning how to compute the volume of 3D shapes, particularly cylinders.
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Unit Conversion: Applying mathematics to convert between different measurement units in real-world scenarios.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
To find the area of a trapezium, use the formula: Area = (1/2) ร (base1 + base2) ร height. For bases of 8cm and 12cm with a height of 5cm, the area is 50 cmยฒ.
-
The volume of a cylinder with a radius of 7cm and a height of 10cm is calculated as Volume = ฯ ร 7ยฒ ร 10 = approximately 1539.38 cmยณ.
-
For a floor measuring 4m by 5m, converting to cmยฒ gives an area of 20,000 cmยฒ, needing 32 tiles of 625 cmยฒ each.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
-
To find the area, use base and height, a trapezium's math is just right!
๐ Fascinating Stories
-
Imagine a builder measuring a floor. He needs trapezium and cylinder calculations to ensure tiles fit well. Imagine him carrying columns into the room to base the work on the correct measurements.
๐ง Other Memory Gems
-
For Volume use V = ฯrh: 'Very Packed Rain Hats' to remember variables!
๐ฏ Super Acronyms
A for Area, V for Volume, P for Perimeter - remember the difference!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Mensuration
Definition:
The branch of mathematics dealing with the measurement of geometric figures.
-
Term: Area
Definition:
The measure of the space within a two-dimensional shape.
-
Term: Volume
Definition:
The amount of space occupied by a three-dimensional object.
-
Term: Trapezium
Definition:
A four-sided figure with at least one pair of parallel sides.
-
Term: Cylinder
Definition:
A three-dimensional shape with circular bases connected by a curved surface.