How to find the surface area of a sphere - Math

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Question

In terms of $\pi$ , give the surface area, in square inches, of a spherical water tank with a diameter of 20 feet.

Answer

feet = $20 \times 12 = 240$ inches, the diameter of the tank; half of this, or 120 inches, is the radius. Set , substitute in the surface area formula, and solve for :

$A =4\pi $r^{2}$$

$A =4\pi \cdot $120^{2}$$

$A =4\pi \cdot 14,400$

$A = 57,600 \pi$

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Question

Which is the greater quantity?

(a) The surface area of a sphere with radius 1

(b) 12

Answer

The surface area of a sphere can be found using the formula

$A = 4\pi $r^{2}$$ .

The surface area of the given sphere can be found by substituting :

$A = 4\pi \cdot $1^{2}$ = 4\pi$

$\pi > 3$ so $4 \times \pi >4 \times 3$ , or $4\pi > 12$

This makes (a) greater.

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Question

Sphere A has volume $972 \pi \textrm{ $in}^{3}$$ . Sphere B has surface area $400 \pi \textrm{ $in}^{2}$$ . Which is the greater quantity?

(a) The radius of Sphere A

(b) The radius of Sphere B

Answer

(a) Substitute $V = 972 \pi$ in the formula for the volume of a sphere:

$$\frac{4}{3}$\pi $r^{3}$= V$

$$\frac{4}{3}$\pi $r^{3}$= 972 \pi$

$$\frac{4}{3}$\pi $r^{3}$ \div $\frac{4}{3}$\pi = 972 \pi \div $\frac{4}{3}$\pi$

$\frac{4}{3}$\pi $r^{3}$ \cdot $\frac{3}{4\pi}$ = 972 \pi\cdot $\frac{3}{4\pi}$

$r =\sqrt[3]{ 729} = 9$ inches

(b) Substitute $A = 400 \pi$ in the formula for the surface area of a sphere:

$4\pi $r^{2}$= A$

$4\pi $r^{2}$= 400 \pi$

$4\pi $r^{2}$ \div 4\pi = 400 \pi \div 4\pi$

$r = $\sqrt{100}$ = 10$ inches

(b) is greater.

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Question

is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius

(B) The surface area of a cube with edges of length

Answer

The surface area of a sphere is $4 \pi$ times the square of its radius, which here is ; the surface area of the sphere in (A) is $4 \pi $t^{2}$$ .

The area of one face of a cube is the square of the length of an edge, which here is , so the area of one face of the cube in (B) is $(2t) ^{2} = $4t^{2}$$ . The cube has six faces so the total surface area is $6 \cdot $4t^{2}$ = $24t^{2}$$ .

$\pi < 4$ , so $4 \pi $t^{2}$ < 16 $t^{2}$ < 24 $t^{2}$$ , giving the sphere less surface area. (B) is greater.

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Question

The diameter of a sphere is . What is its surface area?

Answer

The surface area of a sphere is given by the equation: $SA=4\pi $r^2$$

The only given information is the diameter: . In order to solve for the surface area, the only necessary information is the radius. The missing variable (radius) can be calculated through the given diameter because diameter is twice the length of the radius. That is:

Using that information, radius can be solved for by

$($\frac{18:cm}{2}$)=r$

Given that the radius is , this value can be substituted in to solve for the final surface area:

$SA= 4 \pi $(9:cm)^2$$

$SA= $4(81:cm^2$)\pi$

$SA= $324\pi:cm^2$$

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Question

The diameter of a sphere is . What is the sphere's surface area?

Answer

The equation to find the surface area of a sphere is: $Area = 4 \cdot \pi \cdot $r^2$$

The only information required to solve for the area is the radius. This information is given to us in a sense. If the diameter of the sphere is , the radius must be . This is because the radius is equivalent to half of the diameter.

Now that we know the radius, we can solve for the area by substituting in for the value of , the radius.

$Area = 4 \cdot \pi \cdot $(4^2$)$

$A = 4 \cdot \pi \cdot 16$

$A = 64 \cdot \pi$

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Question

If the radius of a sphere is , what is the sphere's surface area?

Answer

Write the formula for surface area of a sphere:

$A=4\pi $r^2$$

Plug in the value of radius and solve for the surface area:

$A=4\pi $(1)^2$ = 4\pi$

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Question

Find the surface area of a hemisphere that has a radius of .

Answer

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}$=4\pi $r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}$=\pi $r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}$=2\pi $r^2$+\pi $r^2$=3 \pi $r^2$

Plug in the given radius to find the surface area.

$$\text{Surface Area of Hemisphere}$=3\pi $(2)^2$=37.70$

Make sure to round to places after the decimal.

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Question

Find the surface area of a sphere with a circumference of $\small \small 10\pi,cm$ .

Answer

The formula for the surface area of a sphere is simply

$\small A=4\pi $r^2$$

However, the problem is that we are given the circumference rather than the radius. Nonetheless, this isn't too big of an obstacle, as the formula for circumference in terms of radius is simply.

$\small C=2\pi r$

Substituting or value gives

$\small 10\pi=2\pi r$

Solving for the radius gives

$\small r=5$

We can then substitute this value into our formula for surface area.

$\small $A=4\pi(5^2$)$

$\small A=100\pi$

Therefore, our surface area is $\small $100\pi,cm^2$$

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Question

If the radius of a sphere is , what is the surface area of the sphere?

Answer

Write the equation for the surface area of a sphere.

$A=4\pi $r^2$$

Substitute the radius and find the area.

$A=4\pi $(15)^2$ = 4\pi(225)=900\pi$

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Question

If the radius of the sphere is one half, what is the surface area?

Answer

Write the formula for the surface area of a sphere, plug in the radius, and find the area.

$A=4\pi $r^2$ = 4\pi $\left($\frac{1}{2}\right)^2$ = 4\pi\left($\frac{1}{4}\right)= \pi$

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Question

If the diameter of a sphere is , what is the surface area?

Answer

The radius is half the length of a diameter.

$r=\frac{d}{2}$=\frac{15}{2}$=7.5$

Write the formula for the surface area of a sphere and plug in the radius.

$A=4\pi $r^2$ = $4\pi(7.5)^2$= 225\pi$

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Question

Find the surface area of a sphere if the volume is .

Answer

Write the formula for the volume of a sphere and substitute volume to find radius.

$V=\frac{4}{3}$\pi $r^3$$

$1=\frac{4}{3}$\pi $r^3$$

$\frac{3}{4\pi}$= $r^3$

$r=\sqrt[3]{$\frac{3}{4\pi}$}= ($\frac{3}{4\pi}$)^$\frac{1}{3}$$

Write the formula for the surface area of a sphere and plug in the radius.

$A=4\pi $r^2$ = 4\pi \left[\left($\frac{3}{4\pi}$\right)^$$\frac{1}{3}\right]^2$ = 4\pi\left($\frac{3}{4\pi}$\right)^$\frac{2}{3}$$

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Question

Find the surface area of a hemisphere that has a radius of .

Answer

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}$=4\pi $r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}$=\pi $r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}$=2\pi $r^2$+\pi $r^2$=3 \pi $r^2$

Plug in the given radius to find the surface area.

$$\text{Surface Area of Hemisphere}$=3\pi $(3)^2$=84.82$

Make sure to round to places after the decimal.

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Question

Find the surface area of a hemisphere that has a radius of .

Answer

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}$=4\pi $r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}$=\pi $r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}$=2\pi $r^2$+\pi $r^2$=3 \pi $r^2$

Plug in the given radius to find the surface area.

$$\text{Surface Area of Hemisphere}$=3\pi $(4)^2$=150.80$

Make sure to round to places after the decimal.

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Question

Find the surface area for a hemisphere that has a radius of $\frac{1}{2}$ .

Answer

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}$=4\pi $r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}$=\pi $r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}$=2\pi $r^2$+\pi $r^2$=3 \pi $r^2$

Plug in the given radius to find the surface area.

$$\text{Surface Area of Hemisphere}$=3\pi $($\frac{1}{2}$)^2$=2.36$

Make sure to round to places after the decimal.

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Question

Find the surface area of a hemisphere that has a radius of .

Answer

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}$=4\pi $r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}$=\pi $r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}$=2\pi $r^2$+\pi $r^2$=3 \pi $r^2$

Plug in the given radius to find the surface area.

$$\text{Surface Area of Hemisphere}$=3\pi $(5)^2$=235.62$

Make sure to round to places after the decimal.

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Question

Find the surface area of a hemisphere that has a radius of .

Answer

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}$=4\pi $r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}$=\pi $r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}$=2\pi $r^2$+\pi $r^2$=3 \pi $r^2$

Plug in the given radius to find the surface area.

$$\text{Surface Area of Hemisphere}$=3\pi $(6)^2$=339.29$

Make sure to round to places after the decimal.

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Question

Find the surface area of a hemisphere if it has a radius of .

Answer

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}$=4\pi $r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}$=\pi $r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}$=2\pi $r^2$+\pi $r^2$=3 \pi $r^2$

Plug in the given radius to find the surface area.

$$\text{Surface Area of Hemisphere}$=3\pi $(7)^2$=461.81$

Make sure to round to places after the decimal.

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Question

Find the surface area of a hemisphere with a radius of .

Answer

In order to find the area of a hemisphere, which is just half a sphere, you will need to first find half the surface area of the sphere.

Recall how to find the surface area of a sphere:

$\text{Surface Area of Sphere}$=4\pi $r^2$ , where is the radius.

Now, since the sphere is cut in half, it also has a circle as its base.

Recall how to find the area of a circle:

$\text{Area of Circle}$=\pi $r^2$

Now add together half the surface area of a sphere with the area of the circular base to find the surface area of a hemisphere.

$\text{Surface Area of Hemisphere}$=2\pi $r^2$+\pi $r^2$=3 \pi $r^2$

Plug in the given radius to find the surface area.

$$\text{Surface Area of Hemisphere}$=3\pi $(8)^2$=603.19$

Make sure to round to places after the decimal.

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