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Discovering the Surface Area of Spheres and Hemispheres: A Hands-On Adventure

Another day of Math. Five of us sitting in front of the 1.8 kg, thick Math textbook looking at an endless bunch of questions. Typical Math class, except for the fresh air blowing and the mewing of kittens. Finally Aunty came, armed with a tennis ball.

Now I don't know whether it's me or everyone, but there is something about a tennis ball that makes you energetic or alert, especially in a Math class. Is it the bright colour or that fact that it is a ball or just being out of the dull atmosphere of a textbook? Whatever it was, it made us eager to attend the class and intrigued by why she got it? After the excitement died down a little, aunty announced we were going to find the surface area of the ball without destroying it.

All of us were excited with the idea. Soon we were thinking up ideas. We thought we could cover it with strips of paper and measure the strips. But, it wouldn’t be the accurate surface area. What could we do?

We sat there wondering and we began to brainstorm ideas. Some of the ideas were like find the circumference and multiply it by the height and so on. Until we were struck upon an idea of wrapping the ball with a string and measuring it out. Don’t get it? Let me explain.

We were to wrap the ball in woollen yarn, then take the same yarn and find out how many circles, with the same radius as the sphere, could it make. Sounds easy right? It definitely is not easy as it sounds, at least for us. Maybe it was the short spans of chatting that we did. But the yarn kept slipping of the ball and pretty soon Aunty was fed up because we weren’t making much progress. Finally, she gave in and told us to push a thumb tack through the exact centre of the ball. Before doing that there were a few squeals from pricks.

Pretty soon we were tying the yarn from the tack all around the ball. Yet the ball slipped a few times but we finally managed to cover half the ball which we decided was enough. So we unraveled the yarn and tried to fill it in a circle of the same radius. But it wouldn’t stay in, so we put a thumbtack through the middle. But then the tack kept spinning when we tried to put the yard around it. Finally after some help from Aunty, we then saw we could fit the yarn in two of those circles. But we had measured only half of the sphere so we had to multiply it by two.

So we finally concluded that the surface area of the sphere was 4 times the circle’s area, which was π r^2. So the Total Surface Area of sphere is 4π r^2. For a hemisphere, the curved surface area was 2π r^2 and the Total Surface Area is 3π r^2 because there was an extra surface under it.

And in this way, we had figured out the surface area of a sphere and hemisphere without a board or without a teacher telling us the answer and making us memorise it. We had done it hands on in a fun way again, as usual. Like we do it always at Sunnyside.

Abdul Muiz Syed is a nature and sports enthusiast. He likes writing and creating pencil illustrations. He plays sports when he isn’t at his beloved school, Sunnyside.


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