### Video Transcript

In the figure, a circle is
inscribed in an equilateral triangle 𝐴𝐵𝐶 of side length 12 centimetres. Find the radius of the circle and
the area of the shaded region. Take 𝜋 is equal to 3.14 and root
three is equal to 1.73.

Since the circle is inscribed
within the triangle, we know that the circumference of the circle meets each side of
the triangle exactly once. Let’s call these points of
intersection 𝑥, 𝑦, and 𝑧. We can add lines joining the centre
of the circle to each of these points. They are the radii of the
circle. Let’s call them 𝑟. We know that the radius and the
tangent meet at angle of 90 degrees. So the radii meet the sides of the
triangle at 90 degrees.

Next, we can split the triangle up
into three equal triangles. These are triangle 𝐴𝑂𝐵, triangle
𝐵𝑂𝐶, and triangle 𝐴𝑂𝐶. The area of these three triangles
makes up the area of the larger triangle 𝐴𝐵𝐶. The area of 𝐴𝑂𝐵 plus 𝐵𝑂𝐶 plus
𝐴𝑂𝐶 equals the area of 𝐴𝐵𝐶. Since each of these triangles are
equal, we can say that the area of 𝐴𝑂𝐵 is equal to one-third of the area of
𝐴𝐵𝐶.

Now, one of the formulae we know
for area of a triangle is a half multiplied by base multiplied by height. If we treat the base of triangle
𝐴𝑂𝐵 as the side 𝐴𝐵, we can write the area of 𝐴𝑂𝐵 as being a half multiplied
by 12 multiplied by 𝑟. Remember, the radius is
perpendicular to the tangent. So the radius is the height of this
triangle. That gives us that the area of
𝐴𝑂𝐵 is six 𝑟.

Now, we don’t know the vertical
height of triangle 𝐴𝐵𝐶. So we’ll use an alternative
formula. We’ll use the formula half
multiplied by 𝑎𝑏 multiplied by sin 𝐶. Since this is an equilateral
triangle, each of its interior angles are equal. They’re all 60 degrees. Each of the sides of this triangle
are 12 centimetres. So the area is a half multiplied by
12 multiplied by 12 multiplied by sin 60. Sin 60 is root three over two. So this becomes a half multiplied
by 12 multiplied by 12 multiplied by root three over two. We can cross cancel by two and by
two again. And that gives us that the area of
triangle 𝐴𝐵𝐶 is 36 root three.

Now remember, we said that the area
of 𝐴𝑂𝐵 is equal to a third of the area of 𝐴𝐵𝐶. So let’s substitute in the areas we
have. That gives us six 𝑟 is equal to a
third multiplied by 36 root three. A third of 36 is 12. So we get six 𝑟 equals 12 root
three. And if we divide through by six, we
get that 𝑟 is equal to two root three. We’re told to use root three as
1.73. So the radius is two multiplied by
1.73, which is 3.46. We’ve calculated the radius of this
circle to be 3.46 centimetres.

Now that we have the radius, we
need to find the area of the shaded region. The shaded area is given by the
area of the triangle minus the area of the circle. Remember, we said that the area of
triangle 𝐴𝐵𝐶 is given by a half 𝑎𝑏 sin 𝐶. And the area of the circle is 𝜋𝑟
squared. We already worked out that a half
𝑎𝑏 sin 𝐶 was 36 root three. 𝜋𝑟 squared is 𝜋 multiplied by
two root three squared. It’s actually easier here to use
two root three rather than 3.46 at this stage. Two root three all squared is the
same as having two squared multiplied by root three squared. Two squared is four. And root three squared is
three. So we have four multiplied by three
which is 12.

Using the values of 𝜋 and root
three that we were given in the question, and we get that the shaded area is 36
multiplied by 1.73 minus 3.14 multiplied by 12. We we’ll need to use some formal
written methods for multiplication to work out the value of 36 multiplied by 1.73
and 3.14 multiplied by 12. To work out 36 multiplied by 1.73,
let’s first calculate 36 multiplied by 173. Three multiplied by six is 18. So we put an eight in this first
column and carry the one. Seven multiplied by six is 42. And when we add the one, we get
43. One multiplied by six is six and
six plus four is 10.

Next, we’re going to multiply the
digits of 173 by three. But this three isn’t really a
three; it’s actually a 30. So we put a zero in this column to
show that when we multiply, our answer is going to be 10 times bigger than we
expect. Three multiplied by three is
nine. Seven multiplied by three is
21. And one multiplied by three is
three. Three plus two is five. Finally, we add these two
numbers. And we get 173 multiplied by 36 is
6228. 1.73 is 100 times smaller than
173. So 36 multiplied by 1.73 is going
to be 100 times smaller than 173 multiplied by 36. It’s 62.28.

We can use a similar method to
calculate 3.14 multiplied by 12. 314 multiplied by 12 is 3768. 3.14 is 100 times smaller than
314. So our answer is also going to be
100 times smaller. 3.14 multiplied by 12 is 37.68. 62.28 minus 37.68 is 24.6. We have that the radius is 3.46
centimetres. And the shaded area is 24.6
centimetres squared.