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Home / Drawing a Circle on the Surface of a Sphere

Drawing a Circle on the Surface of a Sphere

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

Definition: Surface Surface area of a Sphere

A sphere is the three-dimensional analog of a circle. It tin can exist defined as a perfectly rounded object that has no edges or vertices.

All points located on the surface of a sphere are at an equal distance from the center. This distance is chosen the radius, which is unremarkably denoted by 𝑟 .

An interesting property of a sphere is that amidst all iii-dimensional shapes with the aforementioned book, information technology is the one with the minimum surface area. For this reason, spheres arise in a multifariousness of concrete systems where surface area is minimized, such as h2o aerosol and lather bubbles!

The surface surface area of a sphere tin can be calculated by means of the following formula.

Formula: Surface Expanse of a Sphere

The area, 𝐴 , of a sphere of radius 𝑟 is given by the formula 𝐴 = iv 𝜋 𝑟 .

Since 𝜋 = 3 . ane 4 1 five 9 … is just a number, this means that as long as nosotros know the radius of a sphere, we can always apply this formula to find its surface area.

Let's starting time with a simple case.

Example 1: Finding the Surface Surface area of a Sphere given Its Radius

Find the surface area of the given sphere to the nearest tenth.

Answer

Recall that the surface area, 𝐴 , of a sphere of radius 𝑟 is given by the formula 𝐴 = iv 𝜋 𝑟 .

From the diagram, this sphere has radius 𝑟 = 6 , so nosotros can substitute this value into the formula and rearrange to become 𝐴 = iv × 𝜋 × 𝑟 = 4 × 𝜋 × 6 = 4 × 𝜋 × three six = 4 × 3 6 × 𝜋 = one 4 four × 𝜋 = 4 v 2 . three eight ix … .

Nosotros were asked to round our answer to the nearest 10th. Remember that the tenths digit is the first digit after the decimal point, which in this example is a 3. The digit following this (the hundredths digit) is an 8, then the answer rounds up to 452.4 to the nearest tenth.

Since the radius of the sphere was given in centimetres, the surface area must be in square centimetres. The expanse of the sphere, rounded to the nearest tenth, is 452.4 cmtwo .

Next, we will piece of work through an example where we are given the diameter of the sphere, non the radius. Our approach is very like but with one boosted step. Ever make sure that you lot check whether you are given a radius or a diameter in the question.

Instance 2: Finding the Surface Surface area of a Sphere given Its Diameter Using an Approximation of Pi

Find the surface surface area of a sphere whose diameter is 12.half dozen cm. Employ 𝜋 = two 2 7 .

Answer

Call back that the surface expanse, 𝐴 , of a sphere of radius 𝑟 is given by the formula 𝐴 = 4 𝜋 𝑟 .

Hither, nosotros are given the bore of the sphere, 12.6 cm, which is twice its radius. To utilize the formula to piece of work out the surface area, we get-go need to summate the radius, and then nosotros halve the diameter to get 𝑟 = 1 2 . half dozen ÷ 2 = 6 . 3 . And so, substituting for 𝑟 in the formula, we have 𝐴 = iv × 𝜋 × 𝑟 = four × 𝜋 × ( half dozen . 3 ) = 4 × 𝜋 × 3 nine . vi 9 = four × three 9 . 6 9 × 𝜋 = 1 v 8 . 7 6 × 𝜋 .

Annotation that in the question, we are given the approximation 2 2 vii for 𝜋 , so substituting this value gives 𝐴 = i 5 8 . 7 six × 𝜋 = i 5 eight . 7 6 × ii two 7 = ane v eight . seven half-dozen × two two vii = iii 4 9 2 . 7 two 7 = 4 9 8 . 9 half dozen .

As the diameter was given in centimetres, the surface area must exist in square centimetres; the surface area of the sphere is 498.96 cm2 .

The formula for the expanse of a sphere contains but two variables, 𝐴 and 𝑟 . This means that if nosotros are given the expanse of a sphere, and then we tin can always piece of work astern to find its radius. Once we have worked out the radius, if needed, we can double this value to obtain the bore. The next example shows how to rearrange the formula to solve this type of trouble.

Example 3: Finding the Bore of a Sphere given Its Surface area

What is the diameter of a sphere whose surface expanse is iii six 𝜋 cm2 ?

Answer

Kickoff, we recall the formula to calculate the surface surface area, 𝐴 , of a sphere of radius 𝑟 : 𝐴 = 4 𝜋 𝑟 .

We take been given the expanse, 𝐴 = 3 6 𝜋 , and then substituting into the formula, we have 3 half-dozen × 𝜋 = 4 × 𝜋 × 𝑟 .

Conveniently, we have been given the area in terms of 𝜋 , which allows united states of america to neatly divide both sides of our equation by 𝜋 to get iii half-dozen = 4 × 𝑟 .

So, we carve up both sides by 4, giving 9 = 𝑟 .

We can now notice the radius of our sphere by taking the square roots of both sides of this equation: √ ix = √ 𝑟 , and then 3 = 𝑟 , which is the same as 𝑟 = three . It is worth noting that there are actually two solutions to this equation. Although we could say that 𝑟 = ± 3 , in this case, 𝑟 represents a length and we can therefore ignore the negative solution.

The units of the surface area are square centimetres, and so the radius is in centimetres. Doubling the value of the radius, we notice that the diameter of the sphere is 2 × 3 = 6 c m .

Observe that in the in a higher place example, we substituted the value for 𝐴 , the surface area of the sphere, into the formula 𝐴 = 4 𝜋 𝑟 and then rearranged to detect the value of the radius, 𝑟 . An culling approach to this strategy is to rearrange the formula to make 𝑟 the subject so substitute for 𝐴 directly, as follows.

Starting with the original formula 𝐴 = 4 𝜋 𝑟 and rewriting the right-hand side to include multiplication signs, we have 𝐴 = 4 × 𝜋 × 𝑟 .

Dividing both sides by iv and and so by 𝜋 , we get 𝐴 4 𝜋 = 𝑟 .

Finally, nosotros accept the square roots of both sides: 𝐴 4 𝜋 = √ 𝑟 , so 𝑟 = 𝐴 four 𝜋 .

Again, we tin can safely ignore the negative solution when defining this human relationship, so we take obtained our formula for the radius. If we were to substitute 𝐴 = 3 six 𝜋 straight into this formula, we would get 𝑟 = 3 . As expected, this is the same value for the radius that nosotros worked out in the previous example.

Formula: Radius of a Sphere given Its Surface Area

The radius, 𝑟 , of a sphere of surface area 𝐴 is given by the formula 𝑟 = 𝐴 four 𝜋 .

Now, nosotros examine a very important concept when studying spheres: the great circle.

Definition: Slap-up Circle

A smashing circle is the largest circle that can be fatigued on whatever given sphere. It can be formed on the surface of a sphere by the intersection of a aeroplane that passes through the sphere'due south eye. Since a peachy circle shares its center with the parent sphere, it will also share its radius, 𝑟 .

A great circle will always cut a sphere into two equal hemispheres, every bit shown in Figure 1.

Information technology is possible to draw other circles on the surface of a sphere that do not pass through the heart of the sphere. These circles will not be nifty circles and will have a smaller radius than a great circle (and the parent sphere).

Figure 2 shows a keen circumvolve of radius 𝑟 and another circumvolve that lies on the surface of the sphere and has a radius 𝑟 . The smaller circle divides the sphere into two unequal parts.

The centers of both circles share a mutual axis, which, by the definition of a bang-up circle, as well passes through the center of the sphere.

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

By taking this view, we can clearly see that the radius of a great circle, 𝑟 , 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 𝑟 in this example) will be smaller than the radius of the sphere, and so 𝑟 < 𝑟 .

Finally, we know that all circles on a given sphere that are classified as a great circle will have the same radius, 𝑟 . We can therefore conclude that all great circles on a sphere will be identical to each other, even if they occupy a dissimilar prepare of points along the surface of the sphere.

Note that we discover another interesting fact by recalling the formula for the area of a circle, 𝐴 , in terms of its radius: 𝐴 = 𝜋 𝑟 .

By comparison this to the formula for the surface area of a sphere, we see that the sphere will accept exactly four times the surface area of a circle with the same radius, which we now know is a cracking circle; that is, 𝐴 = four 𝜋 𝑟 = iv × 𝜋 𝑟 = 4 𝐴 .

Formula: Surface Area of a Sphere given Its Great Circle

The surface surface area of a sphere, 𝐴 , is exactly four times the surface area of its great circle, 𝐴 : 𝐴 = 4 𝐴 .

Since a slap-up circle shares some of its properties with its parent sphere, you lot may be asked to solve questions using the relationship betwixt these 2 shapes. Let's look at an example.

Example 4: Finding the Surface Surface area of a Sphere given Information about Its Great Circle

Find the surface area of a sphere to the nearest tenth if the expanse of the bang-up circle is four 4 1 𝜋 inii .

Answer

Call up that the surface surface area, 𝐴 , of a sphere of radius 𝑟 is given past the formula 𝐴 = 4 𝜋 𝑟 .

Our strategy will exist to work out the value of 𝐴 from the expanse of the great circle given in the question.

Whatever great circumvolve has the same radius as its parent sphere. Therefore, the area of this cracking circumvolve, 𝐴 , will be given by the formula 𝐴 = 𝜋 𝑟 .

Using this concluding formula, we tin substitute for 𝜋 𝑟 in the surface area formula as follows: 𝐴 = four 𝜋 𝑟 = 4 × 𝜋 𝑟 = 4 𝐴 .

Now that we have expressed the surface area of the sphere in terms of the surface area of its corking circle, we can use the fact that 𝐴 = iv 4 one 𝜋 to get 𝐴 = iv 𝐴 = iv × four four 1 𝜋 = iv × 4 4 i × 𝜋 = 1 7 6 4 × 𝜋 = five five iv 1 . 7 6 nine … .

From the question, nosotros need to round our respond to the nearest tenth. The tenths digit is the first digit afterwards the decimal point, which here is a seven. The digit following this (the hundredths digit) is a 6, and then our answer must round up to 5‎ ‎541.8 to the nearest 10th.

The area of the great circle was given in square inches, so the area of the sphere will besides exist in square inches. Nosotros conclude that the surface area of the sphere is v‎ ‎541.viii intwo , rounded to the nearest tenth of a square inch.

Next, we have another example where we must use the backdrop of a great circle to work out the surface area of its parent sphere; this time we are given the circumference.

Case 5: Finding the Surface Area of a Sphere given the Bang-up Circle'south Circumference

Find, to the nearest tenth, the surface expanse of a sphere whose great circumvolve has a circumference of 1 four 0 𝜋 ft.

Answer

Recall the formula for the surface area, 𝐴 , of a sphere of radius 𝑟 : 𝐴 = 4 𝜋 𝑟 .

Hither, we are given the circumference of a great circumvolve. Since we know that the radii of a great circle and its parent sphere are the same, then our first pace will be to use this data to calculate the value of 𝑟 . We know that the human relationship between circumference and radius is c i r c u thou f e r e northward c e = 2 𝜋 𝑟 .

Now, nosotros substitute the value ane 4 0 𝜋 for the circumference to get i 4 0 𝜋 = two 𝜋 𝑟 .

To solve this equation for 𝑟 , nosotros divide both sides by 2 𝜋 , so 1 4 0 𝜋 two 𝜋 = 𝑟 , which implies 𝑟 = vii 0 .

We now accept a familiar situation, where the surface area of the sphere, 𝐴 , tin exist constitute using our formula. Substituting for 𝑟 , nosotros have 𝐴 = iv × 𝜋 × 𝑟 = 4 × 𝜋 × ( seven 0 ) = iv × 𝜋 × 4 9 0 0 = 4 × 4 nine 0 0 × 𝜋 = 1 9 6 0 0 × 𝜋 = 6 1 5 7 five . 2 1 half dozen … .

The question states that nosotros should provide our answer to an accuracy of the nearest tenth. The tenths digit is the first digit afterward the decimal bespeak, which here is a two. The digit following this (the hundredths digit) is a ane, and so our answer must round downwards to 61‎ ‎575.two to the nearest tenth.

The circumference of the great circle was given in feet, so the surface area of the sphere will exist in square feet. We conclude that the surface area of the sphere is 61‎ ‎575.2 fttwo , rounded to the nearest tenth of a square foot.

Recall that a great circumvolve will always cut a sphere into ii equal hemispheres. Consequently, we can use data about great circles to help united states of america calculate the surface area of corresponding hemispheres or other fractions of a sphere. Here is an example.

Example half-dozen: Finding the Full Surface Surface area of a Hemisphere given Its Radius

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

Reply

The diagram includes a great circle with a radius of eighteen cm. As we know that the radius of a slap-up circle is too the radius, 𝑟 , of its parent sphere (or hemisphere), then 𝑟 = 1 8 .

Now, retrieve the formula for the surface area, 𝐴 , of a sphere of radius 𝑟 : 𝐴 = four 𝜋 𝑟 .

Detect that the total surface of a hemisphere is made up of a curved surface and a apartment, circular surface. The area of the curved part, which nosotros shall telephone call 𝐴 , is half the surface area of the corresponding sphere; that is, 𝐴 = ane 2 × 𝐴 = 1 2 × 4 𝜋 𝑟 = 2 𝜋 𝑟 .

Furthermore, the at surface is just a circle of radius 𝑟 , so writing 𝐴 for its area, we take 𝐴 = 𝜋 𝑟 .

Therefore, the total surface surface area of the hemisphere, which we tin write as 𝐴 , must satisfy 𝐴 = 𝐴 + 𝐴 .

Substituting for 𝐴 and 𝐴 from above, we get 𝐴 = ii 𝜋 𝑟 + 𝜋 𝑟 = iii 𝜋 𝑟 .

Finally, nosotros have 𝑟 = one eight , so making this substitution gives 𝐴 = 3 × 𝜋 × 𝑟 = 3 × 𝜋 × ( 1 eight ) = 3 × 𝜋 × 3 two 4 = iii × iii 2 four × 𝜋 = 9 7 ii × 𝜋 = 3 0 5 three . 6 two 8 … .

We were asked to give our answer to the nearest 10th. The tenths digit is the first digit later the decimal point, which here is a six. The digit following this (the hundredths digit) is a 2, so our answer must round down to 3‎ ‎053.6 to the nearest tenth.

The radius of the dandy circle was given in centimetres, so the total surface area of the hemisphere will be in square centimetres.

The full surface surface area of the hemisphere is 3‎ ‎053.vi cmtwo , rounded to the nearest tenth of a square centimetre.

For our terminal case, we have a question with a existent-world context that is given as a give-and-take problem and without a diagram. In cases similar this, it is always important to read the question advisedly and work out exactly what we are being asked to find.

Example seven: Solving a Word Trouble Involving a Hemisphere

A water characteristic can exist modeled every bit a hemisphere with its base set up onto a square patio. If the bore of the hemisphere is 4 feet and the patio has a side of length 6 anxiety, what would the visible surface area of the patio be? Give your answer accurate to two decimal places.

Answer

Recall that a keen circle will always cutting a sphere into two equal hemispheres. Therefore, the base of the hemisphere in the h2o feature will be a great circumvolve. Moreover, a swell circle and its parent hemisphere must have the aforementioned radius, 𝑟 . Our strategy volition exist to use information about the hemisphere to summate the surface area of this nifty circle. We can then subtract it from the area of the whole patio to find the visible area of the patio.

We are told that the patio is square with a side of length six anxiety, then the expanse of the whole patio is half dozen × six = three six .

We also know that the bore of the hemisphere is four feet, so, to discover its radius 𝑟 , we divide by 2 to get 𝑟 = 4 ÷ 2 = 2 . Thus, its base of operations will be a swell circle with a radius of two.

Moreover, the expanse of this nifty circle will exist 𝐴 = 𝜋 × 𝑟 = 𝜋 × 2 = 𝜋 × 4 = 4 𝜋 .

For at present, we go along this answer in its exact form because we need to apply information technology in a further calculation.

Our last step is to obtain the visible area of the patio by subtracting the area of the circle from the expanse of the square, giving 3 6 − 4 𝜋 = 3 6 − 1 2 . 5 six six … = 2 3 . four iii iii … , which is 23.43, accurate to ii decimal places.

The lengths in the question were given in feet, and so this area must be in square feet. The visible expanse of the patio, accurate to two decimal places, is 23.43 ft2 .

Let's terminate past recapping some key concepts from this explainer.

Key Points

  • The surface area, 𝐴 , of a sphere of radius 𝑟 is given by the formula 𝐴 = four 𝜋 𝑟 .
  • The formula can be rearranged in order to more hands notice the radius (or diameter) of a sphere, given its surface area: 𝑟 = 𝐴 4 𝜋 .
  • Always make sure that you check whether you are given the radius or the diameter of the sphere in the question.
  • A peachy circle is the intersection of a sphere with a plane that passes through the sphere's center. It cuts the sphere exactly in half, forming two hemispheres.
  • A great circumvolve is the largest circle that can be formed on the surface of its parent sphere, and both shapes share the aforementioned radius.
  • The surface area of a sphere, 𝐴 , is exactly four times the surface area of its great circle, 𝐴 .
  • We can use the formula to observe the surface area of a hemisphere or other fractions of a sphere, in particular, in existent-globe questions presented equally give-and-take problems.

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Source: https://www.nagwa.com/en/explainers/632175148743/