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.

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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

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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

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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

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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

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

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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

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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

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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

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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

What is the surface area of a composite figure of a cone and a sphere, both with a radius of 5 cm, if the height of the cone is 12 cm? Consider an ice cream cone as an example of the composite figure, where half of the sphere is above the edge of the cone.

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Answer

Calculate the slant height height of the cone using the Pythagorean Theorem. The height will be the height of the cone, the base will be the radius, and the hypotenuse will be the slant height.

$s=$\sqrt{169}$=13$

The surface area of the cone (excluding the base) is given by the formula $A=\pi rs$ . Plug in our values to solve.

The surface area of a sphere is given by $A=4\pi $r^2$$ but we only need half of the sphere, so the area of a hemisphere is $A=2\pi $r^2$$ .

$A_${hemisphere}=2\pi(5)^2$=50\pi\ $cm^2$$

So the total surface area of the composite figure is $65\pi\ $cm^2$+50\pi\ $cm^2$=115\pi\ $cm^2$$ .

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Question

What is the surface area of a hemisphere with a diameter of 4 cm?

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Answer

A hemisphere is half of a sphere. The surface area is broken into two parts: the spherical part and the circular base.

The surface area of a sphere is given by SA = 4pi $r^{2}$.

So the surface area of the spherical part of a hemisphere is SA = 2pi $r^{2}$.

The area of the circular base is given by A = pi $r^{2}$. The radius to use is half the diameter, or 2 cm.

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Question

What is the surface area of a sphere with a radius of 15?

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Answer

To solve for the surface area of a sphere you must use the equation

First, plug in 15 for and square it

Multiply by 4 and to get

$900\pi=225(4)(\pi)$

The answer is $900\pi$ .

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Question

What is the surface area of a sphere whise radius is .

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Answer

The surface area of a sphere is found by the formula $SA=4\pi $r^{2}$$ using the given radius of .

$SA=4\pi $(1.5cm)^{2}$$

$\rightarrow 9\pi $cm^{2}$$

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Question

Find the surface area of a sphere whose diameter is $D=\pi$ .

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Answer

The surface area of a sphere is found by the formula $SA=4\pi $r^{2}$$ . We need to first convert the given diameter of $D=\pi$ to the sphere's radius.

$\pi =2r$

$r=\pi/2$

Now, we can solve for surface area.

$\dpi{100} \dpi{100} SA=4\pi \left ( $\frac{\pi $}{2}\right)^{2}$$

$=4\pi \cdot $$\frac{\pi^{2}$$ }{4}$

$\dpi{100} \rightarrow $\pi^{3}$$

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Question

What is the surface area of a sphere with a radius of ?

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Answer

To solve for the surface area of a sphere you must remember the formula:

First, plug the radius into the equation for :

Since , the surface area is $(4)(144)(\pi)=576\pi$ .

The answer is therefore $576\pi$ .

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Question

To the nearest tenth of a square centimeter, give the surface area of a sphere with volume 1,000 cubic centimeters.

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Answer

The volume of a sphere in terms of its radius is

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

Substitute and solve for :

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

$$\frac{4\pi $r^3$}{3}$ = 1,000$

$\frac{4\pi $r^3$}{3}$\cdot $\frac{3}{4\pi }$= 1,000\cdot $\frac{3}{4\pi }$

$r=\sqrt[3]{ $\frac{3,000}{4\pi }$} \approx 6.20 \textrm{ cm }$

Substitute for in the formula for the surface area of a sphere:

$A = 4\pi $r^2$ \approx 4 \cdot \pi \cdot 6.20 ^2 \approx 483.6 \textrm{ $cm}^{2}$$

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Question

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

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Answer

The standard equation to find the area of a sphere is $4\Pi $r^2$$ .

Substitute the given radius into the standard equation to get the answer:

$4\Pi $(4)^2$=4\Pi (16)=64\Pi$

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Question

Given that the radius of a sphere is 3, find the surface area.

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Answer

The standard equation to find the area of a sphere is

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

where denotes the radius. Plug in the given radius to find the surface area.

$SA=4\pi $(3)^2$=4\cdot 9\cdot \pi =36\pi$

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Question

Find the surface area of the following sphere.

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Answer

The formula for the surface area of a sphere is:

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

where is the radius of the sphere.

Plugging in our values, we get:

$SA = 4 \pi $(6m)^2$$

$SA = 4 \pi (36m) = 144 \pi $m^2$$

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Question

Find the surface area of the following sphere.

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Answer

The formula for the surface area of a sphere is:

$SA = 4 \pi $(r^2$)$

Where is the radius of the sphere

Plugging in our values, we get:

$SA = 4 \pi $(9m)^2$$

$SA = 324 \pi $m^2$$

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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.

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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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