Here is a circle drawn on a centimetre grid. Part (a) On the grid, draw a tangent to the circle. Part (b) Harold says, “The area of the circle must be more than four centimetres squared.” Explain why Harold must be correct.
So for part a, first of all, we need to know what is a tangent. Well, a tangent is a straight line, which touches the edge or circumference of a circle once. And the key here is that it’s straight and it only touches the circle once. We’re gonna be careful here because often when you draw these lines, by mistake, you can touch the circle more than once. And therefore, it would not be a tangent. I’ve drawn a horizontal tangent here. But it wouldn’t need to be horizontal. It could be diagonal or an angle. It could be vertical. The key is, is it a straight line? and does it touch the edge of the circle only once? So we’ve drawn our tangent and have answered part a. So now, let’s move on to part b.
Harold says, “The area of the circle must be more than four centimetres squared.” Explain why Harold must be correct.
There’re a couple of ways that we could do this. And they both use the fact that the circle is drawn on a centimetre grid, and we’re told that. So therefore, because it’s a centimetre grid, we know that one square is worth one centimetre squared because it’s one centimetre by one centimetre. So therefore, if we colored in four squares, this will be worth four centimetres squared. Well because our four centimetres squared square can fit inside the circle, and we’ve shown this by coloring in four squares. But also we could have worked out because we can see that the square is two centimetres by two centimetres. Well, if we multiply these together, we get four centimetres squared. Therefore, the circle must be bigger than four centimetres squared. And that’s because there’s extra area within the circle, which is outside of this two-by-two square.
So now I’ve answered the question because I’ve explained why the area of the circle must be more than four centimetres squared. So I’ve explained why Harold must be correct. However, I said there are other methods we could use and I’m gonna demonstrate one of them now. And that’s to use the area of the circle. Well, we know that the area of a circle is equal to 𝜋𝑟 squared, where 𝑟 is the radius. So therefore, if we look at a circle with an area four centimetres squared, we can rewrite this as 𝜋𝑟 squared equals four. So then, if we divide by 𝜋, then what we’re gonna have is 𝑟 squared on the left-hand side of the equation and four divided by 𝜋 on the right-hand side.
So therefore, if we want to work out the radius, what we can do is take the square root because it’s the inverse operation of squaring. And we’ll take the square root of both sides. Well, if you take the square root of 𝑟 squared which is left with 𝑟, and then we take the square root of four over 𝜋. And we could do this on our calculator by typing this in. And if we do this, we can see that the radius is approximately 1.13 centimetres. So therefore, for four-centimetre squared circle, that would be the radius.
Well, if we have a look at our circle, we can clearly see that the radius is, in fact, greater than 1.5 centimetres. Because I’ve marked here where 1.5 centimetres would be if it was the radius. So therefore, it cannot be 1.13 centimetres because as I said it’s greater than 1.5. So therefore, the circle must be greater than four centimetres squared. So that was the other method you could have used.