11.2. Surface Area of a Sphere

What is a sphere? Is it the same as a circle? Can you draw a circle on a paper? Yes, you can, because a circle is a plane closed figure whose every point lies at a constant distance (called radius) from a fixed point, which is called the centre of the circle.

A sphere is a three-dimensional object that can be thought of as a collection of points all equidistant from a center point. Unlike a circle, which is a flat shape, a sphere has depth. The distance from the center to any point on the surface is the radius. This fundamental understanding distinguishes spheres from two-dimensional shapes like circles.

Think of a soccer ball or a basketball, which are perfect spheres. Just like if you draw a circle on paper, but then imagine that you've inflated that circle into a three-dimensional ball.

Now if you paste a string along a diameter of a circular disc and rotate it as you had rotated the triangle in the previous section, you see a new solid. What does it resemble? A ball? Yes. It is called a sphere.

By rotating a circle around its diameter, you create a three-dimensional figure called a sphere. The rotation causes all points on the circle to sweep out a space, forming the surface of the sphere. The center point remains fixed during this rotation, which becomes the center of the sphere.

Imagine spinning a pizza dough in the air. As it spins, it expands outward into a flat circle, but if you could take that dough and form it into a ball shape, it would create a spherical object, just like a basketball.

Can you guess what happens to the centre of the circle, when it forms a sphere on rotation? Of course, it becomes the centre of the sphere.

When a circle rotates around its diameter, the center of the circle also moves, but it remains fixed at the center of the sphere. This point serves as the pivotal location from which all points on the surface of the sphere are equidistant. Therefore, this center is crucial for defining the sphere and its properties.

If you've ever used a very precise compass to draw circles, the point of the compass is the center of the circle, and when it's made into a sphere, that point is now the center of the entire ball, just like the way Earth rotates around its axis.

This suggests that the surface area of a sphere of radius r = 4 times the area of a circle of radius r = 4 × ( π r2) So, Surface Area of a Sphere = 4 π r2.

The surface area of a sphere can be calculated by recognizing that its surface is equivalent to four times the area of a circle with the same radius. This is mathematically represented as 4πr², where r is the radius of the sphere. This formula helps us quantify how much space the surface of the sphere occupies, which can be important in real-world applications, such as determining materials needed to cover a ball.

If you have a globe (a spherical model of the Earth), think about how much paint you'd need to cover it entirely. The formula 4πr² gives you a way to figure out exactly how much paint to buy based on the radius of the globe.

Now, let us take a solid sphere, and slice it exactly ‘through the middle’ with a plane that passes through its centre. What happens to the sphere? Yes, it gets divided into two equal parts. Each half is called a hemisphere.

When a sphere is halved through its center, it results in two equal portions known as hemispheres. Each hemisphere has a curved surface (the outer part of the half-sphere) and a flat circular base. This division is essential in various applications such as architectural designs or when considering how to measure sections of a sphere.

Think of a watermelon. When you cut it in half, each half is a hemisphere. You can easily see the curved side and the flat cut side where it was sliced. If you wanted to know how much fruit is in each half, you would need to calculate the surface area of the hemisphere.

The curved surface area of a hemisphere is half the surface area of the sphere, which is 1/2 of 4 πr². Therefore, Curved Surface Area of a Hemisphere = 2 πr².

The curved surface area of a hemisphere is derived from the total surface area of a sphere. Since a hemisphere is half of a sphere, the curved part alone is 2πr². This becomes important when calculating how much material is needed to cover just the curved part of a hemisphere, such as a dome.

Imagine a half-dome architecture in a park. If you're tasked with covering just the top curved section with tiles, you would use the formula for the curved surface area of a hemisphere to determine how many tiles are needed.

Now taking the two faces of a hemisphere, its surface area is 2 πr² + πr². So, Total Surface Area of a Hemisphere = 3 πr².

The total surface area of a hemisphere includes both the curved outer surface and the flat circular base. This yields a total surface area of 3πr², which is useful for understanding how much surface needs to be painted or covered, not just the curved part but also the base.

Returning to our half-watermelon example, if you were to paint both the green outer curve and the flat inner cut surface, you'd need to know the total surface area—this formula helps you figure out that need accurately.

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