How to locate the quantity of an octahedron

We can compute their quantity making use of the complying with formula:

$ latex V= frac {sqrt {2}} {3} {{} ^ 3} $

where, a is the size of among the sides of the octahedron.

Evidence of the formula for the quantity of an octahedron

An octahedron can be developed by signing up with 2 pyramids at their bases. This indicates that we can obtain the quantity of an octahedron if we include the quantities of both pyramids.

Considering that both pyramids coincide, this indicates we need to locate the quantity of one pyramid as well as just increase by 2 to obtain the quantity of the octahedron.

Currently, to compute the quantity of any kind of pyramid, we can make use of the complying with formula:

$ latex V _ {p} =frac {A _ {b} times h} {3} $

where, $latex A _ {b} $ is the location of the base as well as h is the elevation of the pyramid.

In this instance, the base of the pyramid is square, so its location amounts to $latex {{} ^ 2} $.

The elevation of the pyramid can be computed making use of the Pythagorean thesis as well as the complying with layout:

Pyramid with height to find the volume of octahedron

Considering that the faces of an octahedron are equilateral triangulars, every one of its sides have a size of a. As a result, the hypotenuse that we will certainly make use of amounts to a.

Likewise, considering that the diagonal of a square amounts to $latex asqrt {2} $, half the angled, which amounts to among the legs, amounts to $latex frac {asqrt {2}} {2} $.

As a result, the elevation of the pyramid is:

$ latex h= sqrt {{{} ^ 2} – {{(frac {asqrt {2}} {2})} ^ 2}} $

$ latex h= sqrt {{{} ^ 2} -frac {{{} ^ 2}} {2}} $

$ latex h= sqrt {frac {{{} ^ 2}} {2}} $

$ latex h= frac {} {sqrt {2}} $

$ latex h= frac {a sqrt {2}} {2} $

Multiplying the elevation by the location of the base as well as separating by 3, we obtain the quantity of the pyramid:

$ latex V _ {p} =frac {1} {3} times {{} ^ 2} times frac {a sqrt {2}} {2} $

$ latex V _ {p} =frac {{{} ^ 3} sqrt {2}} {6} $

Multiplying the quantity of the pyramid by 2, we obtain the quantity of the octahedron:

$ latex V= 2times frac {{{} ^ 3} sqrt {2}} {6} $

$ latex V= frac {{{} ^ 3} sqrt {2}} {3} $


How to locate the area of an octahedron

To compute the area of an octahedron, we make use of the complying with formula:

$ latex A _ {s} =2sqrt {3}|{{} ^ 2} $

where, a is the size of among the sides of the octahedron.

Evidence of the formula for the area of an octahedron

We can acquire the formula for the area of an octahedron by thinking about that the area of any kind of three-dimensional number amounts to the amount of the locations of all its faces.

When it comes to octahedrons, we have 8 consistent triangular faces. That is, we have 8 confront with the very same form as well as the very same measurements, so the area is:

$ latex A _ {s} =8A _ {t} $

where, $latex A _ {t} $ is the location of each triangular face.

Likewise, when we discuss an octahedron, we typically indicate a routine octahedron. If this holds true, each triangular face is an equilateral triangular.

As a result, keeping in mind that the formula for the area of an equilateral triangle is:

formula for the area of an equilateral triangle

we can replace that worth right into the area formula:

$ latex A _ {s} =8times frac {sqrt {3}} {4} {{} ^ 2} $

$ latex A _ {s} =2sqrt {3}|{{} ^ 2} $


Volume as well as location of an octahedron– Instances with answers

EXAMPLE 1

If an octahedron has sides 4 m long, what is its quantity?

To fix this trouble, we need to make use of the formula for the quantity of an octahedron with the size a= 4. So, we have:

$ latex V= frac {{{} ^ 3} sqrt {2}} {3} $

$ latex V= frac {{{4} ^ 3} sqrt {2}} {3} $

$ latex V= frac {64 sqrt {2}} {3} $

$ latex V= 30.17$

The quantity of the octahedron is $latex 30.17|{{m} ^ 3} $.

instance 2

What is the area of an octahedron that has sides with a size of 2 m?

We can fix this trouble by utilizing the formula for the area of an octahedron with the worth a= 2. As a result, we have:

$ latex A _ {s} =2sqrt {3}|{{} ^ 2} $

$ latex A _ {s} =2sqrt {3} times {{2} ^ 2} $

$ latex A _ {s} =2sqrt {3} times 4$

$ latex A _ {s} =13.86$

The area of the offered octahedron is $latex 13.86|{{m} ^ 2} $.

instance 3

Calculate the quantity of an octahedron that has sides with a size of 5 m.

Again, we make use of the formula for the quantity of a tetrahedron. In this instance, we make use of the worth a= 5:

$ latex V= frac {{{} ^ 3} sqrt {2}} {3} $

$ latex V= frac {{{5} ^ 3} sqrt {2}} {3} $

$ latex V= frac {125 sqrt {2}} {3} $

$ latex V= 58.93$

The quantity of the octahedron is $latex 58.93|{{m} ^ 3} $.

instance 4

Find the area of an octahedron that has sides with a size of 5 centimeters.

We make use of the area formula, replacing the worth a= 5. So, we have:

$ latex A _ {s} =2sqrt {3}|{{} ^ 2} $

$ latex A _ {s} =2sqrt {3} times {{5} ^ 2} $

$ latex A _ {s} =2sqrt {3} times 25$

$ latex A _ {s} =86.6$

Therefore, the area is $latex 86.6|{{centimeters} ^ 2} $.

instance 5

If an octahedron has sides 10 centimeters long, what is its quantity?

We are mosting likely to make use of the formula for the quantity of an octahedron making use of the worth a= 10. As a result, we have:

$ latex V= frac {{{} ^ 3} sqrt {2}} {3} $

$ latex V= frac {{{10} ^ 3} sqrt {2}} {3} $

$ latex V= frac {1000 sqrt {2}} {3} $

$ latex V= 471.4$

Then, the quantity of the offered tetrahedron is $latex 471.4|{{centimeters} ^ 3} $.

instance 6

If an octahedron has sides with a size of 8 centimeters, what is its area?

We use the area formula with the worth a= 8:

$ latex A _ {s} =2sqrt {3}|{{} ^ 2} $

$ latex A _ {s} =2sqrt {3} times {{8} ^ 2} $

$ latex A _ {s} =2sqrt {3} times 64$

$ latex A _ {s} =221.7$

The area of the octahedron is $latex 221.7|{{centimeters} ^ 2} $.

instance 7

If the quantity of an octahedron amounts to $latex 11.5|{{m} ^ 3} $, what is the size of among its sides?

In this instance, we understand the quantity of the octahedron as well as we wish to compute the size of the sides. As a result, we make use of the quantity formula as well as fix for a:

$ latex V= frac {{{} ^ 3} sqrt {2}} {3} $

$ latex 11.5= frac {{{} ^ 3} sqrt {2}} {3} $

$ latex 34.5= {{} ^ 3} sqrt {2} $

$ latex 24.4= {{} ^ 3} $

$ latex a= 2.9$

The size of among the sides of the octahedron is 2.9 m.

EXAMPLE 8

If the area of an octahedron is $latex 50|{{m} ^ 2} $, what is the size of its sides?

In this instance, we have the area of the octahedron as well as we wish to compute the size of among its sides. As a result, we need to make use of the area formula as well as fix for a:

$ latex A _ {s} =2sqrt {3}|{{} ^ 2} $

$ latex 50= 2sqrt {3}|{{} ^ 2} $

$ latex 14.43= {{} ^ 2} $

$ latex a= 3.8$

The sides of the octahedron have a size of 3.8 m.

EXAMPLE 9

Find the size of the sides of an octahedron that has a quantity of $latex 22|{{centimeters} ^ 3} $.

To fix this trouble, we need to make use of the formula for the quantity of an octahedron as well as fix for a. As a result, we have:

$ latex V= frac {{{} ^ 3} sqrt {2}} {3} $

$ latex 22= frac {{{} ^ 3} sqrt {2}} {3} $

$ latex 66= {{} ^ 3} sqrt {2} $

$ latex 46.67= {{} ^ 3} $

$ latex a= 3.6$

The octahedron has sides with a size of 3.6 centimeters.

instance 10

An octahedron has an area of $latex 73.3|{{m} ^ 2} $. Identify the size of among its sides.

Let’s usage the area formula with the offered area as well as fix for a:

$ latex A _ {s} =2sqrt {3}|{{} ^ 2} $

$ latex 73.3= 2sqrt {3}|{{} ^ 2} $

$ latex 21.16= {{} ^ 2} $

$ latex a= 4.6$

The octahedron has sides with a size of 4.6 m.


Volume as well as location of an octahedron– Technique problems

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You have actually finished the test!

If the area of an octahedron is 256.2 cm2, what is the size of its sides?

Write the solution to one decimal location.

$ latex l=$ m


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