The diagram shows two tangential circles inscribed in a rectangle. The length of the rectangle is 10 centimetres. The two circles are cut out from the rectangle. Calculate the area of paper left inside the rectangle. Give your answer in terms of 𝜋.
To be tangent to each other means to touch once in one place. So that would be here. And they are inscribed in a rectangle. So they’re inside of this rectangle. We’re told the length of the rectangle is 10 centimetres and the two circles are cut out from it. We wanna find the area of the paper left inside the rectangle. So we need to find the area of the circles and take them away from the area of the rectangle.
So again, to find the area of the paper that would be left, we will need to find the area of the rectangle, this one, which would be found using length times width, and then subtracting the area of the two circles, so two times the area of a circle, 𝜋 times the radius squared. So we already know the length of the rectangle. It’s 10 centimetres. But we don’t know the width. And we don’t know the radius of the circles.
Well, in order for two circles to fit inside of this rectangle, this length, this length, this length, and this length would all have to be the exact same because these are the diameters going through the circles’ centre. So if the length is 10, these blue lengths would need to be exactly half of that, five.
So if the diameters are five centimetres, the radius is exactly half of the diameter. So here’s a radius. And we want it to be half of five, so five-halves or 2.5 as a decimal. We will go ahead and use the fraction form. So we have the length, we have the radius, but we need the width of the rectangle.
Well, as we said, if this length is five centimetres and it’s the diameter, then this length will also be five centimetres, which is the width of the rectangle. So starting over, as we said, so starting over, we said the length of the rectangle is 10 centimetres and the width was five centimetres. And then, for our circles, we said that the radius was five-halves.
Now it’s okay to use the brackets or the multiplication symbol here, just to represent multiplying by two. Either form is fine. So before we begin multiplying, we need to take five-halves and square it. So we need to take five squared, which is 25, and two squared, which is four. So after we square the radius, it becomes twenty-five fourths.
So let’s simplify the area of the rectangle and the area of the circles and then subtract. So looking at the area of the circles, we have two on the numerator and a four on the denominator. Two goes into itself once and goes into four twice. So what we’re left with is one times 𝜋 times 25 halves. Well, one times 𝜋 is just gonna be 𝜋, and then times 25 halves, so we have 25 halves 𝜋.
Now for the area of the rectangle, 10 times five is 50. So the area that would be left after we cut out the circles from the rectangle would be 15 minus 25 halves.
Now we need units. Since our lengths were in centimetres, our area should be square centimetres or centimetres squared. And we’re asked to give our answer in terms of 𝜋, meaning don’t multiply by 𝜋; leave 𝜋 in our answer. So once again, the area would be 50 minus 25 halves 𝜋 square centimetres.