Volume Bola

Volume

The volume of a solid body is the amount of "space" it occupies. Volume has units of length cubed (i.e., , , , etc.) For example, the volume of a box (rectangular parallelepiped) of length , width , and height is given by

The volume can also be computed for irregularly-shaped and curved solids such as the cylinder and cone. The volume of a surface of revolution is particularly simple to compute due to its symmetry.

The following table gives volumes for some common surfaces. Here denotes the radius, the height, and the base area, and, in the case of the torus, the distance from the torus center to the center of the tube (Beyer 1987).

surface Volume

cone

conical frustum

cubecylinder

ellipsoid

oblate spheroid

prolate spheroid

pyramid

pyramidal frustum

sphere

spherical cap

spherical sector

spherical segment

torus

Even simple surfaces can display surprisingly counterintuitive properties. For instance, the surface of revolution of around the x -axis for is called Gabriel's horn, and has finite volume, but infinite surface area.

The generalization of volume to dimensions for is known as content.

For many symmetrical solids, the interesting relationship

holds between the surface area , volume , and inradius . This relationship can be generalized for an arbitrary convex polytope by defining the harmonic parameter in place of the inradius (Fjelstad and Ginchev 2003).

SEE ALSO: Arc Length, Area, Bellows Conjecture, Cavalieri's Principle, Content, Harmonic Parameter, Height, Length, Surface Area, Surface of Revolution, Volume Element, Volume

Theorem, Width

Sphere

A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius") from a given point (the "center"). Twice the radius is called the diameter, and pairs of points on the sphere on opposite sides of a diameter are called antipodes.

Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of " -sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of

all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). As a result, geometers call the surface of the usual sphere the 3-sphere, while topologists refer to it as the 2-sphere and denote it .

Regardless of the choice of convention for indexing the number of dimensions of a sphere, the term "sphere" refers to the surface only, so the usual sphere is a two-dimensional surface. The colloquial practice of using the term "sphere" to refer to the interior of a sphere is therefore discouraged, with the interior of the sphere (i.e., the "solid sphere") being more properly termed a "ball."

The sphere is implemented in Mathematica as Sphere[ x, y, z , r].

The surface area of a sphere and volume of the ball of radius are given by

(1)

(2)

(Beyer 1987, p. 130). In On the Sphere and Cylinder (ca. 225 BC), Archimedes became the first to derive these equations (although he expressed in terms of the sphere's circular cross section). The fact that

(3)

was also known to Archimedes (Steinhaus 1999, p. 223; Wells 1991, pp. 236-237).

Any cross section through a sphere is a circle (or, in the degenerate case where the slicing plane is tangent to the sphere, a point). The size of the circle is maximized when the plane defining the cross section passes through a diameter.

The equation of a sphere of radius centered at the origin is given in Cartesian coordinates by

(4)

which is a special case of the ellipsoid

(5)

and spheroid

(6)

The Cartesian equation of a sphere centered at the point with radius is given by

(7)

A sphere with center at the origin may also be specified in spherical coordinates by

(8) (9) (10)

where is an azimuthal coordinate running from 0 to (longitude), is a polar coordinate running from 0 to (colatitude), and is the radius. Note that there are several other notations sometimes used in which the symbols for and are interchanged or where is used instead of . If is allowed to run from 0 to a given radius , then a solid ball is obtained.

A sphere with center at the origin may also be represented parametrically by letting , so

(11) (12) (13)

where runs from 0 to and runs from to .

The generalization of a sphere in dimensions is called a hypersphere. An -dimensional hypersphere, also known as an -sphere (in a geometer's convention), that is centered at the origin can therefore be specified by the equation

(14)

Of course, topologists would regard this equation as instead describing an -sphere.

The volume of the sphere, , can be found in Cartesian, cylindrical, and spherical coordinates, respectively, using the integrals

(15)

(16)

(17)

The interior of the sphere of radius and mass has moment of inertia tensor

(18)

Converting to "standard" parametric variables , , and gives the coefficients of the first fundamental form

(19) (20) (21)

second fundamental form coefficients

(22) (23) (24)

area element

(25)

Gaussian curvature

(26)

and mean curvature

(27)

Given two points on a sphere, the shortest path on the surface of the sphere which connects them (the sphere geodesic) is an arc of a circle known as a great circle. The equation of the sphere with points and lying on a diameter is given by

(28)

Four points are sufficient to uniquely define a sphere. Given the points with , 2, 3, and 4, the sphere containing them is given by the beautiful determinant equation

(29)

(Beyer 1987, p. 210).

SEE ALSO: Ball, Bing's Theorem, Bowl of Integers, Bubble, Circle, Cone-Sphere Intersection, Cylinder-Sphere Intersection, Dandelin Spheres, Diameter, Double Sphere, Ellipsoid, Exotic Sphere, Fejes Tóth's Problem, Geodesic Dome, Glome, Hypersphere,

Liebmann's Theorem, Liouville's Sphere-Preserving Theorem, Mikusiński's Problem, Noise Sphere, Oblate Spheroid, Osculating Sphere, Parallelizable, Prolate Spheroid, Radius, Space Division by Spheres, Sphere Packing, Sphere Line Picking, Sphere Point Picking, Sphere-

Sphere Intersection, Spherical Lune, Spherical Wedge, Tangent Spheres, Tennis Ball Theorem

Volume BoIa

Gambar diatas merupakan gambar setengah bola dengan,jari-jari r. dan menunjukkan dua buah kerucut dengan jari-jari r dan tinggi r. Jika dilakukan percobaan dengan menuangkan cairan pada kedua kerucut sampai penuh, kemudian cairan dari kedua kerucut tersebut dituangkan dalam setengah bola maka cairan tersebut tepat memenuhi bentuk setengah bola. Dari percobaan tersebut dapat dituliskan sebagai berikut.

Volume bola =4/3πr3 dengan r = jari-jari bolaKarena r = 1/2 d maka bentuk lain rumus volume bola adalah sebagai berikut.

3. BOLA3.1. Pengertian BolaBola adalah bangun ruang yang dibatasi oleh sebuah sisi lengkung/kulit bola.

3.2. Unsur-unsur BolaBola memiliki satu sisi.

3.3. Luas dan volume Bola•Luas bola :L = 4 x luas lingkaran   = 4 x π r2

   = 4 π r2

•Volume bola :V = 4 x volume kerucut

  = 4 x 1/3 π r2 tkarena pada bola, t = r maka  = 4 x 1/3 π r2 r

  = 4 x 1/3π r3

  = 4/3 π r3

Mencari rumus volume bola

4 November 2010 msihabudin Tinggalkan komentar Go to comments

Mungkin kita bertanya-tanya. Sebenarnya dari mana asal rumus bola? Mengapa rumus untuk volume bola sama dengan

Lalu bagaimana mendapatkan nilai . Sebenarnya rumus bola ini didapatkan dari mana? Tentunya, bagi siswa atau mahasiswa yang sudah tahu mengenai integral, sudah pasti akan mengetahui tentang hal ini. Tentu, hal yang bukan lagi menarik bagi mereka yang sudah mengetahuinya. Dengan menggunakan konsep volume benda putar yang dicari dengan menggunakan integral, akan didapatkan suatu rumus volume bola seperti yang disebutkan tadi.

Perhatikan persamaan lingkaran bagian atas yang diberikan di bawah ini

Suatu persamaan lingkaran bisa kita tuliskan menjadi dua fungsi. Tentunya masih ingat mengenai persamaan lingkaran. Persamaan umum lingkaran dengan pusat di (0,0) adalah

dengan r adalah jari-jari lingkaran.

Persamaan itu bisa kita tuliskan dan bisa kita bagi menjadi 2. Yaitu lingkaran bagian atas dan lingkaran bagian bawah. Untuk persamaan lingkaran bagian atas, perhatikan persamaan berikut ini

Untuk persamaan lingkaran yang bagian bawah yaitu

Untuk gambar di atas, adalah persamaan lingkaran bagian atas. Persamaannya yaitu

Untuk mencari volume bola, kita akan memutar setengah lingkaran tersebut. Coba bayangkan, jika setengah bola tersebut diputar dengan sumbu porosnya yaitu sumbu x. Benda apa yang akan terbentuk? Benda yang akan terbentuk adalah sebuah bola dengan jari-jari sama dengan jari-jari lingkaran.

Lalu bagaimana mencari volume benda putarnya? Masih ingat kan mengenai mencari volume benda putar untuk fungsi dan diputar terhadap sumbu x. Bagi yang belum pernah belajar hal ini. Bisa dipelajari pelan-pelan di sini.

Volume benda putar untuk adalah

Untuk mencari volume bola. Setengah lingkaran tersebut kita pandang sebagai dan untuk nilai a dan b,

Perhitungannya sebagai berikut:

Bentuk terakhir adalah rumus untuk mencari volume bola yang sudah kita kenal sejak SMP. Bahkan SD juga ada yang sudah mendapatkannya.

Jadi, dengan menggunakan integral. (volume benda putar), kita bisa dengan mudah menemukan rumus-rumus untuk menghitung volume bola, volume kerucut, volume tabung atau volume benda-benda ruang lengkung yang lain.

Volume BolaDari percobaan tersebut terbukti bahwa volum setengah bola sama dengan volum tabung sama dengan volum kerucut.

Formula untuk Menemukan Volume sebuah Sphere By Mark Kennan , eHow Contributor Oleh Mark Kennan , eHow Kontributor

A sphere is a three-dimensional object that is formed by connecting all points that are a specified distance away from a single center point. bola adalah objek tiga-dimensi yang dibentuk dengan menghubungkan semua titik yang jarak tertentu dari titik pusat tunggal. The volume of a sphere is the amount of space enclosed by it. Volume bola adalah jumlah ruang tertutup olehnya. The volume is measured in three dimensions, so the answer must be reported in cubic units, whether it be cubic centimeters, cubic inches or cubic feet. Volume diukur dalam tiga dimensi, maka jawabannya harus dilaporkan dalam unit kubik, apakah itu sentimeter kubik, inci kubik atau kaki kubik. Examples of spheres include marbles and soccer balls. Contoh bola termasuk kelereng dan bola sepak bola.

Formula Rumus

1. The formula for finding the volume of a sphere is 4/3 * π * R^3, where R is the radius. Rumus untuk mencari volume bola adalah 4 / 3 * π * R ^ 3, dimana R adalah jari-jari. The radius is calculated by measuring the distance from one side of the sphere, through the center point to the other side of the sphere and dividing that measurement by 2. radius dihitung dengan mengukur jarak dari satu sisi bola, melalui titik pusat ke sisi lain dari bola dan membagi pengukuran bahwa dengan 2. You can also calculate the radius from the circumference by measuring the distance around the sphere at its widest point and dividing that by 2π. Anda juga dapat menghitung jari-jari dari keliling dengan mengukur jarak sekitar sphere pada titik terlebar dan membagi bahwa dengan 2π. For example, if the circumference was 31.4 inches, the radius would be 5 inches. Sebagai contoh, jika lingkar itu 31,4 inci, jari-jari akan menjadi 5 inci. If the radius of a sphere is 5 inches, the volume of the sphere is 523.3 inches cubed. Jika jari-jari bola adalah 5 inci, volume bola adalah 523,3 inci potong dadu.

Reasoning for the Formula Penalaran untuk Formula yang

2. Since the formula is for volume, it must involve three-dimensional measurements like length, width and depth. Karena rumus untuk volume, hal itu harus melibatkan pengukuran tiga dimensi seperti panjang, lebar dan kedalaman. Though a sphere has these dimensions, they are not constant because they depend on where you measure the distance across. Meskipun bola memiliki dimensi ini, mereka tidak konstan karena mereka bergantung pada tempat Anda mengukur jarak di seluruh. The radius is the average of all measurements across the sphere from top to bottom, so it represents length, width and height. radius adalah rata-rata semua pengukuran di lingkup dari atas ke bawah, sehingga mewakili panjang, lebar dan tinggi. Since it represents three dimensions, the radius is cubed. Karena mewakili tiga dimensi, radius adalah pangkat. The "4/3" has been derived, because even though a sphere may have the same radius as a cube's height, the volumes will not be equivalent. The "4 / 3" telah diturunkan, karena meskipun bola mungkin memiliki jari-jari sama tinggi kubus itu, volume tidak akan setara.

Applications for Formula Aplikasi untuk Formula

3. The volume of a sphere can be applied to a number of real-life situations. Volume bola dapat diterapkan ke sejumlah situasi kehidupan nyata. For example, when manufacturers design a sports ball, such as a basketball or soccer ball, they must make sure that it meets the specifications of that sport. Sebagai contoh, ketika produsen merancang olahraga bola, seperti bola basket atau sepak bola, mereka harus memastikan bahwa itu memenuhi spesifikasi olahraga itu. To do this, they determine how large the ball needs to be and then use the formula for the volume of the sphere to determine the dimensions. Untuk melakukan hal ini, mereka menentukan seberapa besar bola perlu dan kemudian menggunakan rumus untuk volume bola untuk menentukan dimensi.

Read more: Formula untuk Menemukan Volume sebuah Sphere | eHow.com http://www.ehow.com/way_5268924_formula-finding-volume-sphere.html#ixzz16rYbG6NZ

Bola

Sebuah bola (dari Yunani σφαῖρα - sphaira, "dunia, bola") adalah sempurna bulat geometris objek dalam ruang tiga dimensi , seperti bentuk putaran bola . Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. Seperti sebuah lingkaran dalam dua dimensi, lingkup yang sempurna benar-benar simetris di sekitar pusat, dengan semua titik pada permukaan berbaring r jarak yang sama dari titik pusat. This distance r is known as the radius of the sphere. Jarak r ini dikenal sebagai jari-jari dari bola. The maximum straight distance through the sphere is known as the diameter of the sphere. Jarak lurus maksimal melalui bola dikenal sebagai diameter dari bola. It passes through the center and is thus twice the radius. Melewati pusat dan dengan demikian dua kali jari-jari.

In higher mathematics, a careful distinction is made between the sphere (a two-dimensional spherical surface embedded in three-dimensional Euclidean space ) and the ball (the three-dimensional shape consisting of a sphere and its interior). Dalam matematika lebih tinggi, perbedaan hati-hati dibuat antara lingkungan (suatu dimensi bulat-dua permukaan yang tertanam dalam tiga-dimensi ruang Euclidean ) dan bola (dalam bentuk tiga dimensi yang terdiri dari sebuah bola dan interior).

Eleven properties of the sphere

In their book Geometry and the imagination[3] David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane which can be thought of as a sphere with infinite radius. These properties are:

1. The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.

The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar result of Apollonius of Perga for the circle. This second part also holds for the plane.

2. The contours and plane sections of the sphere are circles.

This property defines the sphere uniquely.

3. The sphere has constant width and constant girth.

The width of a surface is the distance between pairs of parallel tangent planes. There are numerous other closed convex surfaces which have constant width, for example the Meissner body. The girth of a surface is the circumference of the boundary of its orthogonal projection on to a plane. It can be proved that each of these properties implies the other.

A normal vector to a sphere, a normal plane and its normal section. The curvature of the curve of intersection is the sectional curvature. For the sphere each normal section through a given point will be a circle of the same radius, the radius of the sphere. This means that every point on the sphere will be an umbilical point.

4. All points of a sphere are umbilics.

At any point on a surface we can find a normal direction which is at right angles to the surface, for the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane containing the normal with the surface will form a curve called a normal section and the curvature of this curve is the sectional curvature. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures. It can be proved that any closed surface will have at least four points called umbilical points. At an umbilic all the sectional curvatures are equal; in particular the principal curvatures are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.

5. The sphere does not have a surface of centers.

For a given normal section there is a circle whose curvature is the same as the sectional curvature, is tangent to the surface and whose center lines along on the normal line. Take the two centers corresponding to the maximum and minimum sectional curvatures: these are called the focal points, and the set of all such centers forms the focal surface.For most surfaces the focal surface forms two sheets each of which is a surface and which come together at umbilical points. There are a number of special cases. For channel surfaces one sheet forms a curve and the other sheet is a surface; For cones, cylinders, toruses and cyclides both sheets form curves. For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This is a unique property of the sphere.

6. All geodesics of the sphere are closed curves.

Geodesics are curves on a surface which give the shortest distance between two points. They are a generalisation of the concept of a straight line in the plane. For the sphere the geodesics are great circles. There are many other surfaces with this property.

7. Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.

These properties define the sphere uniquely. These properties can be seen by observing soap bubbles. A soap bubble will enclose a fixed volume and due to surface tension its surface area is minimal for that volume. This is why a free floating soap bubble approximates a sphere (though external forces such as gravity will distort the bubble's shape slightly).

8. The sphere has the smallest total mean curvature among all convex solids with a given surface area.

The mean curvature is the average of the two principal curvatures and as these are constant at all points of the sphere then so is the mean curvature.

9. The sphere has constant positive mean curvature.

The sphere is the only imbedded surface without boundary or singularities with constant positive mean curvature. There are other immersed surfaces with constant mean curvature. The minimal surfaces have zero mean curvature.

10. The sphere has constant positive Gaussian curvature.

Gaussian curvature is the product of the two principle curvatures. It is an intrinsic property which can be determined by measuring length and angles and does not depend on the way the surface is embedded in space. Hence, bending a surface will not alter the Gaussian curvature and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries and the sphere is the only surface without boundary with constant positive Gaussian curvature. The pseudosphere is an example of a surface with constant negative Gaussian curvature.

11. The sphere is transformed into itself by a three-parameter family of rigid motions.

Consider a unit sphere placed at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three coordinate axis, see Euler angles. Thus there is a three-parameter family of rotations which transform the sphere onto itself, this is the rotation group, SO(3). The plane is the only other surface with a three-parameter family of transformations (translations along the x and y axis and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the surfaces of revolution and helicoids are the only surfaces with a one-parameter family.