Lesson video

Hello everybody, I'm Mr. Gratton and thank you so much for joining me for another math lesson.

In today's lesson, we'll be looking at the missing length in a composite shape that involves arcs of circles when we are given the area.

Here are some key words that we'll be using during this lesson.

Pause here to familiarise yourself with some of these words.

Before we jump ahead and look at missing lengths from composite shapes, let's first of all have a look at how to find the missing lengths from circles and sectors of a circle.

The area of this quarter-circle is 9 pi centimeter-squared.

How many of these quarter-circles fit into a full circle? Well, four of them can.

Each of these quarter-circles are congruent to the first and therefore, each of them will have the same area as the first.

So if this first quarter-circle has an area of 9 pi, then all of them will have an area of 9 pi and the total area of this whole circle will be 9 pi four times, which is 36 pi centimeter-squared.

As you can see here, you need four quarter-circles to create one full circle.

This time, we've got an area given as a decimal as opposed to in terms of pi.

The area of this third of a circle is 103.

2 centimeter-squared.

What is the area of the full circle that this sector comes from? There are 1, 2, 3 of these sectors that create one full circle and therefore, the total area of this full circle is 103.

2 three times, which is 309.

6 centimeter-squared.

In general, if you're given the area of a sector of a circle, you can find out the area of the full circle that the sector comes from, no matter if it's written in terms of pi or as a decimal, but what if we are given a more challenging sector of a circle than 1/4 or 1/3? The area of this three-quarter-circle is 27 pi centimetre squared.

The reason why this sector is more challenging than the previous two is because you cannot fit a multiple of these sectors to create one full circle.

Before we looked at four-quarter-circles made one full circle, three-thirds of a circle made one full circle, but you cannot put together 2, 3, 4, 5, or however many integer number of these sectors to create one full circle and so there must be an extra step involved to make it possible.

We can split this three-quarter-circle into three one-quarter-circles and therefore, the area of this one-quarter-circle is going to be 1/3 of the area of the total three quarter-circle and 1/3 of 27 pi is 9 pi and therefore, each of the quarters that make up this three-quarter-circle is going to be 9 pi each.

As with before, if a quarter-circle is going to be 9 pi, then a full circle is going to be four lots of that at 9 pi four times for a total area of the full circle of 36 pi.

We can represent this as dividing by three to find a one-quarter sector and then multiplying by four defined four-quarter-circles, which represent one full circle.

Okay, onto a check.

Each of these four sectors has the same area of 18 pi.

Match the sector to the calculations below, define the area of the full circle that each sector comes from, and hence, find the value of the area of the full circle found by options E, F, G and H.

Pause now to have a look at all four sectors, all four calculations and match them with the four answers.

For sector number one, the answers are B and H.

Sector two is C and E.

Sector three is A and F, and sector four is D and G.

Notice how for sector number four, the three quarters of a full circle that the calculation is more complicated than the 1/3, the 1/4 and 1/2 sectors of a full circle.

This demonstration will be finding the radius when given the area.

It is very helpful to write down the formula, area equals pi times r-squared for any question involving the area.

What values we are able to substitute in will depend on the information given and the question.

In this particular question, we are given the area is 254.

5 millimeter-squared.

We therefore take the 254.

5 and substitute it in where the area part of the formula is.

Therefore, we now have 254.

5 equals pi times r-squared.

Now that I've substituted 254.

5, I've got an equation that I can solve to find the value of r.

I can divide both sides of the equation by pi, this cancels out the value of pi on the right hand side of the equation, leaving only r-squared and by typing in on my calculator, 254.

5 divided by pi gives me approximately 81, therefore, r-squared equals 81.

The radius, a number, when squared, gives you 81.

By square rooting the value of 81, I can find r my radius for this question.

What number when multiplied by itself gives you 81? The radius of this circle is approximately nine millimetres.

Pause now to give the same question a go for the circle on the right.

And your calculations should have looked like this.

And so the radius of this circle is five inches when rounded to the nearest integer.

We can bring together the two bits of information from this lesson in order to find the radius of a sector given the area of a sector.

Step one is to find the area of the full circle that a sector comes from.

With this example, we have a quarter-circle and four-quarter-circles make up a full circle and so if one-quarter-circle has an area of 25 pi, then four-quarter-circles have a total area of 100 pi, so the full circle that this quarter circular sector comes from has an area of 100 pi.

Step two, by solving an equation, we can find the radius using the area of the full circle that you found in step one.

As I said before, it is always sensible to write out the formula first, area of a full circle equals pi times r-squared and I can substitute 100 pi into the area of a full circle part of the calculation.

So 100 pi equals pi times r-squared.

I can divide both sides by pi leaving 100 equals r-squared.

What number when multiplied by itself gives you a hundred? That number is 10 and so the radius of this quarter circular sector is 10 centimetres.

Okay, onto a check.

A semicircle has an area of 72 pi meter-squared.

What is the area of the full circle that this sector is a part of? Pause now to give this question a go.

And the answer is 144 pi meters-squared.

This is double the area of the original semicircle.

Now we want to find the diameter of that same semicircle.

What value will go in this blank, in order to help calculate the radius and diameter of the semicircle? Pause here to think what would go in that blank.

The value that goes in the blank is 144 pi.

You never put the area of the sector in this formula, you must always put the area of the full circle that the sector belongs to.

Next up, I've given you the second step of calculations.

I divide both sides of the equation by pi, what would go in both the left and right hand blanks after I've divided both sides of the equation by pi? Pause now to give that a go.

After dividing both sides by pi, I get 144 equals r-squared.

Knowing that r-squared equals 144, what is the radius of this semicircle? Pause now to have a think.

What number multiplied by itself gives you 144? The answer to that is 12.

However, notice the original question, find the diameter of the semicircle.

Currently we have the radius, so what value is the diameter of this semicircle? Pause now to figure out how I can use the radius to find the diameter.

The diameter of a sector or of any circle is twice the radius and twice 12 is 24, in this case, 24 metres.

We can also find the radius or diameter of any sector by using algebra throughout the entire process.

Now the area of a full circle is pi times r-squared and so the area of a quarter-circle is gonna be pi times r-squared with an extra multiplier.

And that multiplier for a quarter-circle is times by a quarter.

We multiply by a quarter because the area of a quarter-circle is a quarter of the area of a full circle.

As with before, I take some information given in the question and substitute it into my modified formula.

The area of a quarter-circle is 25 pi and so substituting that into the formula gives me the equation, 25 pi equals pi times r-squared times by a quarter.

I can multiply both sides by four as this is the reciprocal of one quarter.

For the left hand side of the equation, I can simplify 4 times 25 pi to give me 100 pi and I can simplify a quarter times by four because they are reciprocals, they multiply to make one and anything multiplied by one is just the same value and so the right hand side of this equation becomes pi times r-squared.

We are now at the same stage of the algebraic process as with our previous method, so we can divide both sides by pi then square root to find the value of our radius, in this case, 10 centimetres.

Onto our next check.

A third of a circle has an area of 75 pi.

What would go in this blank in order to create a formula to find the area of a third-circle? Pause now to think what will go in that blank? Because we are dealing with a third of a circle, the value that would go in there is the fraction, 1/3.

Okay, we can substitute 75 pi in place of the area of a third of a circle to give you the equation, 75 pi equals pi times r-squared times by 1/3.

I can multiply both sides by.

Oh, what number? Pause now to think what number I can find the reciprocal of to go into these two blanks.

And the answer is three.

I always will look at the fraction of 1/3 and think what is the reciprocal of 1/3? The answer to that is three.

For the left hand side of this equation, 3 times 75 is 225 and so the left hand side of the equation is 225 pi.

The 1/3 and the three cancel each other out because they are reciprocals of each other and so have a total value of one.

My next step is to divide both sides of the equation by pi to give me r-squared equals what? Pause now to think what would go in this blank.

And the answer is 225.

The pi on the numerator and denominator of the step above cancel out.

I can then square root both sides of the equation to give me r equals what? Pause now to think what would go in that final blank.

The radius of this third of a circle is 15 metres.

For this demonstration, I'll go through the steps to find the radius of this three-quarter sector of a circle.

After each set of steps, try the same method for the question on the right.

So step number one, find the area of the one-quarter-circle.

To do that, I take my three quarters of circle and split it into three one quarter sectors.

I then take my current area of 461.

8 which represents the three quarters of the full circle divided by 3 to get 153.

9 which is the area of each one quarter part of this sector.

Try the exact same method with your question by also spitting your three-quarter-circle up into three one quarter sectors.

Pause now to give that a go.

And the area of each quarter-circle for your sector is going to be 50.

3 centimeter-squared.

Step number two is to write an equation for one of the quarter-circles that you found, the area of.

The area is going to be pi times r-squared times by a quarter, since we are dealing with a quarter of a full circle.

Substituting the area of one of those quarters in will give you the equation, 153.

9 equals pi times r-squared times by a quarter.

Pause now to look at the formula on my side and adapt it to create one for your sector.

For your question, the area of each quarter-circle is 50.

3 centimeter-squared and therefore, the 50.

3 would go on the left hand side of the equation with pi times r-squared times by a quarter still on the right because we are still dealing with a one quarter of a circle.

Step three is to solve the equation that we have written to find the radius.

To do this, we multiply both sides by four, giving us for the left hand side 615.

6 and for the right hand side, the quarter and the times by 4 cancel out because they are reciprocals leaving us only with the pi times r-squared and then we can divide both sides by pi.

I can use a calculator to figure out that the left hand side equals 195.

951.

The final step to solving our equation is to square root both sides and so the radius of this three quarter circular sector is 14 centimetres.

Give this a go for your r-squared value.

Pause now to square root and see what value you get at the end.

Your answer should be a radius of eight centimetres.

Onto the practise.

Pause here to fully fill in this table.

For question number two, find the area of the full circle that this quarter circular sector comes from and then use that value in part B to fill in the blanks.

Pause the video now to do this.

For question number three, fill in all of the blanks for each stage of working to find the diameter of this semi-circular sector.

And for question number four, without any guidance, find the radius of this sector.

Pause now to do both of these questions.

Onto the answers.

Pause here to match the answers on screen with your answers.

For question number two, because we have a quarter-circle, the area of a full circle is gonna be four times that amount and four times 900 pi is 3,600 pi.

We can use the area of a full circle and substitute that into the area of a full circle formula, so 3,600 pi equals pi r-squared and therefore, 3,600 equals r-squared.

We know this is the right answer because the square root of 3,600 is the 60 given.

For question number three, the missing values are 78.

6, 25.

01, 5.

001 and then the diameter of this semicircle is 10 centimetres.

For question number four, the radius is approximately 24 centimetres.

Now that we found the value of the radius or diameter of a sector in isolation, we can use those algebraic strategies to find the radius, the diameter, or any other missing values of a composite shape involving sectors of a circle.

You can break a composite shape up into its component parts if you can spot that it's made from many different rectangles or circular sectors.

In this diagram, we have two component parts, a quarter of a circle and a square.

We know it is a square and not a rectangle because the hash marks mean both the base and the height of that rectangle are the same, meaning it is a square.

We do not know any measurements on this composite shape and so we can label the radius of this quarter-circle as r and work from there.

Because this horizontal radius is shared with the square, we know that the base and the height of the square are also this r value.

Because any given radius of a circle or sector is equal to any other radius of that same shape, the vertical radius can also be called r.

If this is a quarter-circle with a radius of r, then the formula for the area is gonna be pi times r-squared times by a quarter.

Because this is a square with side length of r, then the area of this square is going to be r-squared.

We can use the expressions for the areas of the quarter-circle and the square and use it as part of an equation for the area of the whole composite shape, which we know is 1606.

9 centimeters-squared.

In order to solve this equation, we first of all need to factorised out the r-squared from both terms of the expression on the left.

And to help with that, we can treat the r-squared as one r-squared.

Note the r-squared in each of the two terms and we can factorised that out so the r-squared is outside a pair of brackets with the quarter pi and the plus one inside the brackets.

The expression inside the bracket is just a number, so I can type that into the calculator to get 1.

785 and then it continues.

I can divide both sides of my equation by 1.

785 to give me on the left hand side just r-squared and on the right hand side using a calculator I get 900.

02.

Now that I've got an equation in r-squared, I can square root both sides to give me r equals 30 centimetres and so the radius of this composite shape is 30 centimetres round to the nearest integer.

This composite shape has an area of 200.

5 centimeter-squared.

If I were to break up this composite shape, I would end up with these two component parts.

R is the radius of the semicircular component part of this composite shape.

Which of these expressions shows the height of this square component part? Pause now to look through all of those options and choose the correct one.

And the answer is 2r.

The width and the height of the square at the bottom are both twice the radius of the semicircle.

This is because they're both the same as the diameter of that semicircle.

Sticking with the same composite shape, which of these expressions represents the area of the semicircle and which represents the area of the square? Pause now to have a look through all five of those options.

The area of the square is 4r-squared or 2r times by 2r and the area of the semicircle is 1/2 of pi r-squared, pi r-squared being a full circle, so half pi r-squared being the area of a half circle or a semicircle.

I now know that the area of the full composite shape is 1/2 pi times r-squared plus 4r-squared.

Which of these three fully factorised forms is correct for the area of this composite shape? Pause now to have a look to see which one is the correct fully factorised form.

And C is the correct answer.

A is almost correct, but they factorised out r and not r-squared.

And below, we'll start to appear the method to calculate the value of r.

Fill in any blanks as they appear.

Pause now to fill in this blank.

I can find this value by typing in 1/2 pi plus 4 on my calculator to get about 5.

571.

I can divide both sides through by 5.

571 to get r, what would go in this blank? Pause now to give it a go.

And again, I can use my calculator to divide 200.

5 divided by 5.

571 to get r-squared equals 35.

991.

Therefore, what is the value of r round to the nearest integer? Pause now to give that a go.

And the radius of this composite shape is six centimetres round to the nearest centimetre By rearranging the component parts of this composite shape, which two shapes can you create? Pause now to have a look.

I can rearrange the semicircle at the top and on the bottom to create a circle along with that square in the middle.

If the radius of this circle is r, what is the area of this component square? Pause now to have a think.

And so if the radius of that circle is r, so is the base of the square, therefore, the area of the component square is r-squared.

Here is a mostly complete method to calculate the value of the radius.

Fill in both of these blanks.

Pause now to do so.

I can divide through by the value of pi plus 1 to get r-squared equals 360.

996 and then I can square root the value of 360.

996 to get an answer of 19 centimetres when rounded to the nearest centimetre.

And onto the practise.

For question number one, pause twice, once for parts A and B and then once for part C, pause now for parts A and B.

Complete the calculation to find the radius of the quarter-circle in this composite shape.

Pause now to do that.

And for question number two, fill in every blank to complete the method to find the radius of this composite shape.

Pause now to fill in all of those blanks.

And for question number three, by rearranging the component parts of this composite shape, find the value of r, which is the radius of that sector component part of the composite shape.

Pause now to go to the method to do this.

For question number one, the area of the two squares are r-squared and the area of the quarter-circle is going to be one quarter pi r-squared.

For part B, I can fact raise out the r-squared which gives me inside of the brackets one quarter pi plus 2.

When typed in on a calculator, this gives me a value of 2.

785.

For part C, the radius of the component quarter-circle is 13 metres.

And for question number two, pause here to match all of your answers to the blanks that been filled in on screen.

For question number three, the radius is 28 centimetres when rounded to the nearest in integer.

Whilst it is always possible to find a missing length by solving algebra equation, sometimes we can skip a few steps to find that missing length by comparing light terms instead, let's have a look.

If an area is given to you in terms of pi and not in decimal form, and you know that the radius is rational, then it may be possible to very much simplify the process to find the lengths in the composite shape.

In all examples of questions that you'll see during this learning cycle or side lengths that we will will find the value of typically the radius or some length k, they will all be rational numbers.

We can represent the area of this composite shape by this algebraic expression, one quarter r-squared times pi for the quarter circular part of it, and r-squared for the square part of it.

We are also given that the area is 20.

25 pi plus 81.

Sometimes we can compare the non-pi terms of the algebraic and numeric representations of the area and create a more simple equation from these two values.

These two non-pay values both represent the square part of the composite shape and therefore have equal value.

Because they have equal value, we can write down the equation, r-squared equals 81, which can be solved to get the radius of 9.

This method can also be used when there are multiple missing lengths in a composite shape.

This is especially helpful when the rectangles are more general and not just squares.

Algebraically, the area of the semicircle can be written as a half pi times r-squared and the area of the rectangle can be represented by 2r for the length times by k for the height.

And so the total area of this composite shape can be seen in two terms, the half pi times r-squared term and the 2r times k term.

Numerically, the area is also given to us as the number 8 pi plus 160.

Let's have a look at the non-pi term, just like the previous example, 2r times k equals 160.

I don't think we can solve that.

This is because 2r times k equals 160 has two variables in it and we can only solve equations with one variable in it.

However, we can compare the pi terms. 1/2 pi times r-squared equals 8 pi.

We can solve this equation just like before.

We times both sides by two because two is the reciprocal of 1/2 that gives us pi times r-squared equals 16 pi.

We can divide both sides by pi to get r-squared equals 16 and so the radius is four centimetres.

And now that we know the value of r, we can use that to find the value of k.

The equation that we can make for the non-pi terms is gonna be 2r times k equals 160 and we can substitute r equals 4 into this to get 2 times 4 times k equals 160.

We can simplify that 2 times 4 to become 8 and then we can divide through by eight to get k equals 20 centimetres.

Okay, onto a check.

The area of this composite shape can be represented algebraically as 1/2 times pi times r-squared plus 4r-squared where 4r-squared represent the square part of the composite shape.

Furthermore, numerically, the area can be expressed as 2 pi plus 16 by comparing the non-pi terms and creating an equation from the two representations of the non-pi terms, find the value of r.

Pause now to create that equation and then solve it.

And so the equation is 4r-squared equals 16, dividing through by 4 gives you r-squared equals 4.

Then square rooting the four gives you a radius of two.

Okay, onto the final set of practise questions.

For question one A, explain why that rectangle component part of the composite shape is actually a square? And for part B, find the value of r by comparing the non-pi terms both algebraically and numerically.

Pause now to give that question a go.

And for question number two, write down the algebraic representation of this composite shape in its simplest form and then use this algebraic representation and compare the non-pi term of the algebraic representation to the numerical representation to find the value of r.

Pause now to do this.

Question three is similar, but notice how we have a general rectangle and not a square for this question.

By first of all, writing down the algebraic representation of this composite shape and by comparing both the pi terms and the non-pi terms, find the values of r and k.

Pause now to do this.

For question number four, by possibly doing some rearrangement of the component parts, find the values of r and k.

Pause now to do this.

Okay, onto the answers.

For question one A, both the length and the width of this rectangle are the same at 2r each because the length and the width of a rectangle are the same, it is a square.

And by comparing the 4r-squared to the 400 the non-pi terms both numerically and algebraically, you can figure out that the radius is 10 centimetres.

For question number two, this is the algebraic representation that gives us a value for the radius of 25 centimetres.

This is the algebraic representation for this composite shape.

You would first of all have to compare the pi terms to find the value of the radius at 20 centimetres and then you can use the value of 20 centimetres to find that k is 18 centimetres.

And for question number four, the radius is six centimetres and the value of k is 4.

5 centimetres.

And that is some absolutely amazing work everyone on a really challenging lesson where we have found the radius or diameter of a sector of a circle by first of all finding the area of the full circle that the sector came from.

We have applied algebraic methods to find the radius and diameter of sectors of circles or composite shapes that contain sectors in them.

We've also found a little bit of a cheat way of finding the radius or diameter by comparing pi terms and non-pi terms of both algebraic and numeric representations of a composite shape.

That is all for today's lesson.

Thank you so much for taking part.

I hope to see you soon for some more maths.

Have a good day.