About this video
The video finds the shaded area between a sector (radius 11 cm, central angle 120 degrees) and an isosceles triangle with two 8 cm sides inside it. The sector is 1/3 of 121 pi, giving 121 pi/3; the triangle is split into two 30-60-90 triangles to get height 4 and base 8 sqrt(3), so its area is 16 sqrt(3). The shaded area is 121 pi/3 - 16 sqrt(3) square centimetres.
You'll be able to
- find the area of a sector as (central angle / 360) x pi r^2.
- use 30-60-90 triangle ratios to find the height and base of an isosceles triangle with a 120 degree angle.
- find a shaded area by subtracting one area from another.
Key words
- sector
- A 'pizza slice' of a circle bounded by two radii and an arc.
- central angle
- The angle at the centre of the circle that tells you what fraction of the circle the sector is.
- isosceles triangle
- A triangle with two equal sides; a line from the top vertex perpendicular to the base splits it into two matching halves.
- 30-60-90 triangle
- A right triangle whose sides are in the ratio 1 : sqrt(3) : 2 (short side : long side : hypotenuse).
- exact answer
- An answer left with pi and square roots instead of rounded.
Transcript
- [Instructor] We're told an isosceles triangle with two of its sides as eight centimeters sits inside a sector with a radius of 11 centimeters and a central angle of 120 degrees. What is the area of the shaded region? Either enter an exact answer in terms of pi or round to the nearest integer.
So pause this video, have a go at it before we do this together. All right, now to do this, we just have to figure out the area of this sector of the circle and then subtract out the area of this triangle. So let's first figure out the area of this sector of the circle.
So we know that the area of the entire circle, I could draw that better, the area of the entire circle if we were to go all the way around is pi r squared. So that would be pi times 11 centimeters squared. So that would be equal to 121 pi. This would be the entire, that would be the area of the entire circle.
Now, if we just wanted this sector, and that would be even including this, we're gonna subtract out the area of that triangle in a second. Well, we just have to figure out what fraction of the total area is it going to be? Well, we have a central angle of 120 degrees.
So the fraction of the entire circle, it's 120 degrees over 360 degrees. If we had the whole circle, we would go all the way 360. So 120 over 360 is essentially 1/3. And so this is going to be 121, over 3, pi, or we could even say 121 pi over 3 square centimeters, is the area of this whole sector.
Now we have to figure out the area of this triangle, and let me draw it a little bit differently here just to make it hopefully a little bit clearer what's going on. So we know that it's isosceles, so that length is eight, that length is eight. We know that this angle right over here is 120 degrees, 120.
But if we were to draw a perpendicular right over here, that would bisect this angle, and it would also bisect the other side there. Because we're dealing with an isosceles triangle, it's now symmetric. And so this would be 60 degrees. This would be 60 degrees.
And how would we figure out this length? Well, you might be familiar with 30-60-90 triangles and 30-60-90 triangles, and that's what's here. We have 60, 90. This angle right over here is going to be 30 degrees. And so in 30-60-90 triangles, you have the opposite side of the 30 degree is going to be half the hypotenuse.
So this length over here is going to be four, that's half of eight, that's four right over here, that height. And then this side over here is square root of three times the short side. So this over here is going to be square root of three times four. Now if you want the area of the entire triangle, you just have to multiply 1/2 times base times height.
So that's going to be 1/2 times the base is going to be two times this, 'cause we have this whole thing right over here for the entire base. So two times four square roots of three, I should have written that like that here, four square roots of three, it's a more natural way of writing it, times the height, times four.
Well, these two cancel out, and we end up with 16 square roots of three. 16 square root of three is the area of this triangle right over here. And so if I want the area of just a shaded region, it would be 121 pi over three minus 16, minus 16 square roots of three, and this would be in centimeters.
And we're done.
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