GCSE Maths - How to find the Area of Compound Shapes #106

Summary

This educational video teaches viewers how to calculate the area of irregular shapes by breaking them down into simpler geometric figures like rectangles, triangles, parallelograms, and trapeziums. The script illustrates the process with examples, showing how to find areas by using basic formulas for these shapes. It guides through calculating the area of a rectangle and a triangle combined, and then a rectangle and a trapezium, highlighting the importance of understanding how to decompose complex shapes into manageable parts for area calculation.

Takeaways

• 📏 To find the area of irregular shapes, they must be divided into smaller, recognizable shapes like rectangles, parallelograms, triangles, and trapeziums.
• 🔢 The formulas for calculating areas of simple shapes are essential: length x width for rectangles, base x height / 2 for triangles, and (a + b) / 2 x height for trapeziums.
• 🔍 For composite shapes, identify and calculate the area of each simple shape component and then sum them to get the total area.
• 📐 When a shape includes a triangle, calculate its height if not given by subtracting the height of any included rectangles from the total height.
• 📈 The example in the script demonstrates how to decompose a shape into a rectangle and a triangle, calculate their areas separately, and then add them to find the total area.
• 📉 Another example shows how to handle a shape with a trapezium on top by first calculating the area of the rectangle and then the trapezium using the average of the parallel sides.
• 📝 The script emphasizes the importance of understanding how to break down complex shapes into simple ones for area calculation.
• 🎯 The video provides a practical approach to solving area problems by visually demonstrating the process with specific examples.
• 💡 The tips shared are useful for students or anyone needing to calculate areas of irregular shapes for mathematical, architectural, or artistic purposes.
• 📚 The video encourages viewers to visit a live past paper website for additional educational resources, enhancing learning opportunities.

Q & A

What is the main challenge when calculating the area of irregular shapes?

The main challenge is that irregular shapes don't have their own formulas like rectangles or triangles do, so they need to be divided into smaller, familiar shapes to calculate their areas.

What are the four basic shapes whose area formulas we need to know for this process?

The four basic shapes are rectangles, parallelograms, triangles, and trapeziums, as they have known area formulas.

How can you split a complex shape into simpler shapes to calculate its area?

You can split a complex shape by drawing lines to create rectangles, triangles, trapeziums, or parallelograms, whose areas can be calculated using known formulas.

What is the formula for the area of a rectangle?

The formula for the area of a rectangle is length times width.

How do you calculate the area of a triangle?

The area of a triangle is calculated using the formula one half times base times height.

In the example given, why is the height of the triangle 3 centimeters?

The height of the triangle is 3 centimeters because the total height of the shape is 7 centimeters, and the rectangle at the bottom takes up 4 centimeters of that height.

How do you find the total area of a compound shape after calculating the areas of its parts?

You find the total area of a compound shape by adding together the areas of each of its simpler shapes.

What is the area of the rectangle in the second example of the script?

The area of the rectangle in the second example is 180 square centimeters, calculated by multiplying the length (15 cm) by the width (12 cm).

How is the area of a trapezium calculated according to the script?

The area of a trapezium is calculated by finding the average of the two parallel sides (a + b)/2 and then multiplying by the height.

In the script, what is the total area of the second compound shape after breaking it down into a rectangle and a trapezium?

The total area of the second compound shape is 205 square centimeters, which is the sum of the rectangle's area (180 cm²) and the trapezium's area (25 cm²).

What additional resource is mentioned at the end of the script for further practice?

At the end of the script, a live past paper website is mentioned where viewers can practice similar problems.

Outlines

00:00

📏 Finding Areas of Complex Shapes

This video segment introduces a method for calculating the area of irregular shapes by breaking them down into simpler, familiar geometric shapes like rectangles, parallelograms, triangles, and trapeziums. The segment emphasizes the importance of knowing the area formulas for these basic shapes and demonstrates how to decompose a complex shape into a rectangle and a triangle. The process involves identifying the dimensions of the simpler shapes, calculating their areas using the respective formulas, and summing these areas to find the total area of the complex shape. An example is given where a shape is divided into a rectangle (5 cm by 4 cm) and a triangle (base 5 cm, height 3 cm), and their areas are calculated and added to get a total area of 27.5 square centimeters.

Keywords

💡Area

Area refers to the amount of space enclosed within a shape or a surface. In the context of the video, area is the main focus as it explains how to calculate the area of complex shapes by breaking them down into simpler shapes whose areas can be calculated using known formulas. For instance, the video demonstrates how to find the area of a shape composed of a rectangle and a triangle by calculating the area of each simple shape separately and then summing them up.

💡Shapes

Shapes are defined geometric figures with distinct properties. The video discusses various types of shapes, including simple ones like rectangles and triangles, and more complex or 'strange-looking' shapes that do not have straightforward area formulas. The script uses shapes to illustrate the process of breaking down complex figures into simpler ones for the purpose of area calculation.

💡Rectangle

A rectangle is a quadrilateral with four right angles. Its area is calculated by multiplying its length by its width. In the video, rectangles are used as examples of simple shapes whose areas can be easily determined. The script shows how to calculate the area of a rectangle at the bottom of a complex shape and then adds it to the area of another shape to get the total area.

💡Triangle

A triangle is a polygon with three edges and three vertices. The area of a triangle is found using the formula (1/2) * base * height. The video explains how to determine the area of a triangle that is part of a larger, irregular shape by first calculating the height of the triangle based on the overall shape's dimensions.

💡Formulas

Formulas are mathematical statements that show the relationship between quantities. In the video, formulas are essential tools for calculating the areas of simple shapes. The script lists and explains formulas for rectangles, parallelograms, triangles, and trapeziums, which are then applied to decompose and calculate the area of more complex shapes.

💡Parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. The area of a parallelogram is calculated by multiplying the base by the height corresponding to that base. Although the script does not directly calculate the area of a parallelogram in the provided examples, it mentions the formula as part of the set of formulas needed to handle various shapes.

💡Trapezium

A trapezium (or trapezoid in American English) is a quadrilateral with at least one pair of parallel sides. The area of a trapezium is calculated using the formula ((a+b) * h) / 2, where a and b are the lengths of the parallel sides and h is the height. The video demonstrates this by showing how to calculate the area of a trapezium that is part of a complex shape.

💡Compound Shapes

Compound shapes, also known as composite shapes, are formed by combining two or more simple shapes. The video's main theme revolves around breaking down compound shapes into their simpler components to calculate their areas. The script uses the term to describe shapes that cannot be directly measured using simple formulas and require a step-by-step breakdown into rectangles, triangles, and other simple shapes.

💡Measurements

Measurements are the process of determining the size of something, such as length, area, or volume. In the video, measurements are crucial for calculating areas. The script provides examples of how to take measurements from a shape to determine dimensions like length, width, and height, which are then used in area formulas.

💡Horizontal Line

A horizontal line in geometry is a straight line that extends left and right and is parallel to the horizontal axis. In the video, drawing a horizontal line is a strategy used to divide complex shapes into simpler parts. The script describes how to draw a horizontal line across a shape to separate a rectangle from a triangle, facilitating the calculation of their individual areas.

💡Calculation

Calculation refers to the process of computing or estimating something, often using mathematical techniques. The video is centered around calculations needed to find the areas of various shapes. The script walks through step-by-step calculations, demonstrating how to apply geometric formulas to determine the area of simple and compound shapes.

Highlights

Introduction to finding the area of complex shapes without direct formulas.

Explanation of breaking down complex shapes into simple shapes with known area formulas.

Recap of the main formulas for rectangles, parallelograms, triangles, and trapeziums.

Terminology distinction between simple and compound shapes.

Step-by-step demonstration of dividing a shape into a rectangle and a triangle.

Calculation of the rectangle's area using length and width.

Determination of the triangle's height by subtracting the rectangle's height from the total height.

Use of the triangle area formula with calculated base and height.

Summation of areas to find the total area of the compound shape.

Approach to simplifying complex shapes by cutting off protruding parts.

Calculation of a rectangle's area with a new set of dimensions.

Explanation of finding the average length for a trapezium's area calculation.

Application of the trapezium area formula using the average length and height.

Final calculation of the total area by adding the rectangle and trapezium areas.

Encouragement to visit a live past paper website for further resources.

Conclusion of the video with a summary of the key points covered.

Transcripts

[Music]

in today's video we're going to cover

how you can find the area of strange

looking shapes like these three

the problem with shapes like these is

they don't have their own formulas like

rectangles or triangles do

and so to find their areas we instead

have to split them up into smaller

shapes that we do know the formulas for

to quickly recap the main formulas you

need to know are these four here

which are the ones for rectangles

parallelograms

triangles and trapeziums

so when you're given a weird shape the

aim is to break it up into some

combination of these four shapes that we

know the formulas for

and one thing to point out is that we

sometimes call these four easy shapes

simple shapes because we have formulas

for them

whereas the more complicated shapes that

we're focusing on in this video are

often called compound or composite

shapes

because they're made up of two or more

simple shapes

let's start with this one in the middle

and add some measurements so that we can

work through it

the first thing to spot here is that if

we draw a horizontal line across the top

here

then we can split this shape into a

rectangle at the bottom and a triangle

at the top

so then all we need to do is find the

area of each of them and add the two

together

so to find the area of this rectangle we

need to use this formula in the top left

as do length times width

which in our case would be the length of

5 centimeters

times the width of four centimeters

which gives us 20 square centimeters

then to find the area of the triangle at

the top we need to use this formula of

one half times base times height

the base will just be five centimeters

because it's the same length as this

base of the overall shape

however we haven't actually been told

the height of the triangle so we're

gonna have to work that out for

ourselves

what we do know is that the height of

the entire shape is seven centimeters

and the rectangle makes up four

centimeters of that

so the height of the triangle must just

be the difference between four

centimeters and seven centimeters

which we can find by doing seven minus

four to get three centimeters

and now that we have our dimensions we

can find the area by just doing one half

times the base of five times the height

of 3 and we get 7.5 square centimeters

and then to finish the question we work

out the total area of the shape by

adding together the areas of our

rectangle and our triangle so 20 plus

7.5

which is 27.5 square centimeters

and that's our answer

if you look at this next one

you can see that this bit at the top is

sort of sticking out a bit

and so to make it easier for ourselves

we can just cut that bit off with a

horizontal line along here

and we're now left with a big rectangle

at the bottom

and a small trapezium on top

to find the area of the rectangle we

just do a length times width again

so 15 times 12

which is 180 centimeters squared

next if you look at the formula for

trapeziums in the bottom left

we first have to find the average length

by doing a plus b over two and then

multiply that by the height

and remember a and b in this formula are

just the top and bottom lengths of the

trapezium

so in our case we'll do four plus six

all over two

and then times that by the height of

five

and if we simplify that the four plus

six is ten

and then the ten divided by two is just

five so we have five times five which is

25 centimeters squared

then to finish we just add together the

areas of the rectangle and the trapezium

so 180 plus 25

to get a total area of 205 centimeters

squared

anyway that's everything for this video

so hope it was helpful

if you haven't seen it already we now

have a live past paper website up so you

can just click this button in the top

right corner of this screen and you can

check out our new website

hope you enjoy