Explainer: Surface Areas of Spheres

In this explainer, we will learn how to use the surface area formula of a sphere in terms of its radius or diameter to find the sphere’s area, radius, or diameter.

Definition: Surface Area of the Sphere

A sphere is the 3-dimensional analogue of a circle. It can be defined as a perfectly rounded object that has no edges or vertices.

All points located on the surface of a sphere are an equal distance from the center. This distance is called the radius (π‘Ÿ).

The surface area (𝐴) of a sphere can be calculated using the following formula: 𝐴=4πœ‹π‘Ÿ.

An interesting property of a sphere is that of all 3-dimensional shapes with the same volume, a sphere has the minimum surface area. For this reason, spheres arise in a variety of physical systems where surface area is minimized, such as water droplets and soap bubbles!

Example 1: Finding the Surface Area of a Sphere given Its Diameter

Find the surface area of a sphere whose diameter is 12.6 cm. Use πœ‹=227.

Answer

The first thing we can do is to find the radius (π‘Ÿ) of our sphere. We know that the radius of a sphere or circle is half the diameter (𝑑): π‘Ÿ=𝑑2.

Using the diameter given in the question, we can therefore calculate the radius: π‘Ÿ=12.62,π‘Ÿ=6.3.cm

Now that we have the radius, we can use the formula for the surface area (𝐴) of our sphere, which is 𝐴=4πœ‹π‘Ÿ.

Let us substitute in both the radius and the approximate value for πœ‹ given in the question: 𝐴=4ο€Ό227(6.3).

Squaring our radius and tidying up the fraction, we find the following expression: 𝐴=4Γ—22Γ—39.697.

Finally, we can reduce this fraction and find the surface area of our sphere: 𝐴=489.96.cm

Note

Remember to use the correct units for area when answering this type of question.

The formula can also be used in the reverse situation, in order to find the radius or diameter of a sphere, given its surface area.

Example 2: Finding the Diameter of a Sphere given Its Surface Area

What is the diameter of a sphere whose surface area is 36πœ‹ cm2?

Answer

The question gives us the surface area of a sphere, and hence we can write out our equation and substitute in the known value: 𝐴=4πœ‹π‘Ÿ,36πœ‹=4πœ‹π‘Ÿ.

We have conveniently been given the surface area in terms of πœ‹, which allows us to neatly divide both sides of our equation by 4πœ‹ and cancel the terms: 36πœ‹4πœ‹=4πœ‹π‘Ÿ4πœ‹,9=π‘Ÿ.

We can now find the radius of our sphere by square rooting both sides of the equation: π‘Ÿ=√9=3.cm

It is worth noting that there are actually two solutions to this equation. Although we could say that (π‘Ÿ=Β±3), in our case π‘Ÿ represents a length and we can therefore ignore the negative solution.

Finally, we can observe that the question asks for the diameter of the sphere. We can hence multiply the radius by 2, to find the answer: 𝑑=2π‘Ÿ=2(3)=6.cm

This method can be generalized by rearranging the formula to find the radius in terms of surface area: 𝐴=4πœ‹π‘Ÿ,𝐴4πœ‹=π‘Ÿ.

Again we can safely ignore the negative solution when defining this relationship: π‘Ÿ=ο„žπ΄4πœ‹.

Definition: Great Circles

A great circle is the largest circle that can be drawn on any given sphere. It can be formed on the surface of a sphere by the intersection of a plane that passes through the spheres center. Since a great circle shares its center with the parent sphere, it will also share its radius (π‘Ÿ).

A great circle will always bisect a sphere into two equal hemispheres, as shown in figure 1.

It is possible to draw other circles on the surface of a sphere that do not pass through the center of the sphere.

These circles will not be great spheres and will have a smaller radius than a great circle (and the parent sphere).

Figure 2 shows a great circle of radius π‘ŸοŠ§ and another circle that lies on the surface of the sphere and has a radius π‘ŸοŠ¨. The smaller circle divides the sphere into two unequal parts.

The centers of both circles share a common axis, which, by the definition of a great circle, also passes through the center of the sphere.

Figure 3 shows a top-down view of the sphere shown in figure 2.

By taking this view, we can clearly see that the radius of a great circle (π‘Ÿ) is the radius of the sphere. We can also see by comparison that the radius of any other circle on the surface of the sphere (represented by (π‘Ÿ) in this example) will be smaller than the radius of the sphere: π‘Ÿ>π‘Ÿ.

Finally, we know that all circles on a given sphere that are classified as a great circle will have the same radius, π‘ŸοŠ§. We can therefore conclude that all great circles on a sphere will be identical to each other, even if they occupy a different set of points along the surface of the sphere.

We can notice another interesting fact by recalling the formula for the area of a circle (𝐴) in terms of its radius: 𝐴=πœ‹π‘Ÿ.

By comparing this to the formula for the surface area of a sphere, we can see that the sphere will have exactly four times the area of a circle with the same radius, which we now know is a great sphere: 𝐴=4πœ‹π‘Ÿ,𝐴=4𝐴.

Since a great circle shares some of its properties with a sphere, you may be asked to solve questions using the relationship between these two shapes. Let’s look at a couple of examples.

Example 3: Finding the Total Surface Area of a Sphere given the Area of Its Great Circle

Find the surface area of a sphere to the nearest tenth if the area of the great circle is 441πœ‹ in2.

Answer

First, we can show that the area of a sphere (𝐴) is four times the area of a circle (𝐴) with a common radius: 𝐴=πœ‹π‘Ÿ,𝐴=4πœ‹π‘Ÿπ΄=4𝐴.

Since we know that the radii of a great circle and its parent sphere are the same (π‘Ÿ), we can conclude that this relationship applies to a great circle.

We can therefore substitute the area of the great circle given into the equation and solve: 𝐴=4(441πœ‹)=1,764πœ‹.in

Finally, the question states that we should provide our answer to an accuracy of the nearest tenth, or one decimal place. We therefore round our answer to find the following result: π΄β‰ˆ5,541.8.in

Example 4: Finding the Surface Area of a Sphere Given the Great Circle’s Circumference

Find, to the nearest tenth, the surface area of a sphere whose great circle has a circumference of 140πœ‹ ft.

Answer

Since we know that the radius (π‘Ÿ) of a great circle and its parent sphere are the same, finding this value can be our first step. We know that the relationship between circumference (𝐢) and radius is 𝐢=2πœ‹π‘Ÿ.

We can now substitute in the given value for the circumference of the great circle: 140πœ‹=2πœ‹π‘Ÿ.

Solving for π‘Ÿ, we obtain the following result: 140πœ‹2πœ‹=π‘Ÿ,π‘Ÿ=70.cm

We now have a familiar situation, in which the surface area (𝐴) of the sphere can be found using our formula 𝐴=4πœ‹π‘Ÿ.

Let us substitute in the radius for the great circle (and hence the sphere) into the equation: 𝐴=4πœ‹(70).

Multiplying our terms, we find 𝐴=4πœ‹(4,900)=19,600πœ‹.ft

The question states that we should provide our answer to an accuracy of the nearest tenth; hence, the surface area of the sphere should be given in the following form: π΄β‰ˆ61,575.2.ft

Example 5: Finding the Total Surface Area of a Hemisphere Given Its Radius

Find the total surface area of the hemisphere. Round your answer to the nearest tenth.

Answer

Here, we have a shape formed of two distinct surfaces: the curved surface of the hemisphere, and the flat circle which forms its base. As we can see from the diagram, these two surfaces share a common radius (π‘Ÿ).

We can say that the total surface area of the hemisphere (𝐴) can be found by adding the areas of these two surfaces: 𝐴=𝐴+𝐴.ο‚οŠ§οŠ¨

Since we know that the hemisphere is half of a sphere, we can see that the area of the curved surface (𝐴) will also be half that of a sphere. Therefore, we have 𝐴=𝐴2,=4πœ‹π‘Ÿ2,=2πœ‹π‘Ÿ.

When cutting any 3-dimensional shape, new surfaces may be created. In this case, the base of our hemisphere is a great circle. We can see this by recognizing that a sphere bisected by a great circle would give rise to two hemispheres.

The area of the great circle (𝐴) can be written as π΄πœ‹π‘Ÿ.

The total area of our hemisphere can therefore be found by substituting the area of our two surfaces into our original equation for 𝐴: 𝐴=𝐴+𝐴=2πœ‹π‘Ÿ+πœ‹π‘Ÿ=3πœ‹π‘Ÿ.ο‚οŠ§οŠ¨οŠ¨οŠ¨οŠ¨

We can now substitute in the radius given in the question and solve for 𝐴: 𝐴=3πœ‹(18)=3πœ‹(324)=972πœ‹.ο‚οŠ¨οŠ¨cm

Finally, we round our answer to one decimal place (or one-tenth), to satisfy the conditions of our question: π΄β‰ˆ3,053.6.ο‚οŠ¨cm

Key points

  1. The surface area (𝐴) of a sphere can be calculated using the following formula: 𝐴=4πœ‹π‘Ÿ.
  2. The formula can be rearranged in order to more easily find the radius (or diameter) of a sphere, given its surface area: π‘Ÿ=ο„žπ΄4πœ‹.
  3. A great circle is the intersection of a plane that passes through a sphere’s center. It will always bisect a sphere into two hemispheres.
  4. A great circle is the largest circle that can be formed on the surface of its parent sphere and both shapes share the same radius.
  5. A sphere will have exactly four times the area of its great circle (𝐴): 𝐴=4𝐴.

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