The surface area of a sphere is the total area of the curved surface of the sphere since the shape of a sphere is completely round.

Given the radius, the total curved surface area of a sphere can be found by using the following formula:

S.A. = 4πr^{2}

pi or π is a special mathematical constant, and it is approximately equal to 22/7 or 3.14.

If r or the radius of the sphere is known, the surface area is four times the product of pi and the square of the radius of the sphere.

If you must use the diameter of the sphere, S.A. = 4π(d/2)^{2} = 4π(d^{2}/4) = πd^{2}

The surface area is expressed in square units.

- If r is measured in feet, then the surface area is measured in square feet or ft
^{2}.

- If r is measured in centimeters, then the surface area is measured in square centimeters or cm
^{2}.

- If r is measured in inches, then the surface area is measured in square inches or in.
^{2}

Then, do a ratio of the **area of the pyramid** to the **volume of the pyramid**.

The area of the pyramid is A_{pyramid}.

The volume of the pyramid is V_{pyramid} = (1/3) × A_{pyramid} × r = (A_{pyramid} × r) / 3

So, the ratio of the area of the pyramid to the volume of the pyramid is the following:

A_{pyramid} / V_{pyramid} = A_{pyramid} ÷ (A_{pyramid} × r) / 3

A_{pyramid} / V_{pyramid} = (3 × A_{pyramid}) / (A_{pyramid} × r )

A_{pyramid} / V_{pyramid}= 3 / r

Now pay careful attention to the following important stuff!

**Observation # 1:**

For a large number of pyramids, let's say that n is such a large number, the ratio of the **surface area of the sphere** to the **volume of the sphere** is the same as 3 / r.

Why is that? That cannot be true! Well, here is the reason:

For n pyramids, the total surface area is n × A_{pyramid}

Also for n pyramids, the total volume of the sphere is n × V_{pyramid}

Therefore, ratio of total surface area of the sphere to total volume of the sphere is

(n × A_{pyramid}) / (n × V_{pyramid}) = A_{pyramid} / V_{pyramid}

**We have already shown above that A _{pyramid} / V_{pyramid} = **3 / r

Therefore, S.A._{sphere} / V_{sphere} is also equal to 3 / r.

**Observation # 2:**

Furthermore, n × A_{pyramid} = S.A._{sphere} (The total area of the bases of all pyramids or n pyramids is approximately equal to the surface area of the sphere)

n × V_{pyramid} = V_{sphere} ( The total volume of all pyramids or n pyramids is approximately equal to the volume of the sphere.

Using observation #2, do a ratio of S.A._{sphere} to V_{sphere}

S.A._{sphere} / V_{sphere} = n(A_{pyramid}) / n(V_{pyramid})

Cancel n

S.A._{sphere} / V_{sphere} = (A_{pyramid}) / (V_{pyramid})

S.A._{sphere} / V_{sphere} = 3 / r

Therefore, observation #1 and observation #2 help us to make the following important observation:

S.A.sphere / Vsphere = 3 / r

Therefore, the total surface area of a sphere, call it SA is:

SA = 4 × pi × r

Find the surface area of a sphere with a radius of 6 cm

SA = 4 × pi × r

SA = 4 × 3.14 × 6

SA = 12.56 × 36

SA = 452.16

Surface area = 452.16 cm

Find the surface area of a sphere with a radius of 2 cm

SA = 4 × pi × r

SA = 4 × 3.14 × 2

SA = 12.56 × 4

SA = 50.24

Surface area = 50.24 cm

The surface area of a hemisphere is the total area of the surface of the hemisphere. The surface of a hemisphere consists of a circular base and the curved surface of the hemisphere.

For a hemisphere, the area of the curved surface is half the surface area of the sphere.

Area of the curved surface = (1/2)4πr^{2}

Area of the curved surface = (1/2)4πr^{2}

Area of the **curved surface** = 2πr^{2}

The area of the **circular base** is πr^{2}

Surface area of hemisphere = 2πr^{2} + πr^{2}

Surface area of hemisphere = 3πr^{2}

**Example #3:**

The diameter of a sphere is 8 cm. Find the surface area of the hemisphere.

r = d/2 = 8/2 = 4

Surface area of hemisphere = 3πr^{2} = 3π(4)^{2} = (3)(3.14)(16) = 150.72 cm^{2}

**Example #4**

The volume of a sphere is 33.5103 cubic units. Find the surface area of the sphere.

**Step 1**

Use the volume to find the radius of the sphere.

V = (4/3)πr^{3}

33.5103 = (4/3)πr^{3}

33.5103 = (1.3333)(3.14)r^{3}

33.5103 = (4.186562)r^{3}

Divide both sides by 4.186562

33.5103 / 4.186562 = r^{3}

8.004 = r^{3}

r = cube root of 8.004 = 2

**Step 2**

Use the radius to find the surface area.

S.A. = 4πr^{2}

S.A. = 4(3.14)(2)^{2}

S.A. = (12.56)(4)

S.A. = 50.24 square units