Volume of a Sphere Formula Volume of a Sphere is a measurement of the occupied units of a Sphere. The volume of a Sphere is represented by cubic units like cubic centimeter, cubic millimeter and so on. Volume of a Sphere is the number of units used to fill a Sphere. Generally the volume of a solid is calculated as the area of the base times its height as long the area is constant throughout the height of the solid. But this concept can not be directly applied to find the volume of a sphere because the area changes with every cross section of the sphere. Volume of a Sphere Formula Formula for Volume of a Sphere was found by Archimedes. Archimedes found after several experiments that the volume of a sphere and also its surface area is exactly rd of the volume and the surface area of a cylinder with the same outer dimensions In the above diagram, let r be the radius of the sphere. Since the over all dimensions of both the sphere and the cylinder are the same, the height of the cylinder is 2r. Under this condition, Know More About Equation of Line
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Volume of a cylinder = Area of the base x Height of the cylinder. = πr2 x 2r = 2πr3 Therefore, as per Archimedes formula the volume of the sphere is, ( )( 2πr3) = ( )πr3 So much happy about this result by himself, Archimedes wished a cylinder and globe be placed on his tomb! (This wish was fulfilled) Volume of a Sphere Examples Given below are some examples to find the volume of a sphere Example 1: The sphere has a radius of 8.2 cm. Solve for volume of sphere. Solution: Given: Radius (r) = 8.2 cm Formula: Volume of the sphere (v) = 43 π r3 cubic unit Learn More What is a Parallelogram
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= 43 x π x (8.2)3 =43 x 3.14 x 551.368 Volume of the sphere (v) = 2308.39 cm3 Example 2: The sphere has radius of 8.3 m. Solve for volume of sphere. Solution: Given: Radius (r) = 8.3 m Formula: Volume of the sphere (v) = 43 π r3 cubic unit = 43 x π x (8.3)3 =43 x 3.14 x 571.78 Volume of the sphere (v) = 2393.88 m3
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Diameter of a Circle Circle is defined as the set of points that is at an equal distant from the centre of the circle. There are a number of terminologies involved in a Circle. Some of them are as follows: Centre: The predetermined point from which the surface of the circle is at an equidistant is called the centre of a circle. Radius: The constant distance from the centre to a point on the surface of the circle is called its radius . Circumference: The boundary of a circle is called its circumference. Chord: A line segment whose end points is present on the circumference of a circle is called a chord . Diameter: A chord crossing through the midpoint of a circle is called its diameter. Circle Formulas Diameter of a Circle: Diameter = 2 X Radius
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Radius of a Circle: Radius(R) = Diameter / 2 Area of a Circle: Area = pi X R2 Circumference of a Circle: Circumference = 2 X pi X R Circle TheoremBack to Top Theorem 1: A perpendicular from the centre of a circle to a chord bisects the chord. Given : AB is a chord in a circle with centre O. OC ⊥ AB. To prove: The point C bisects the chord AB. Construction: Join OA and OB Proof: In triangles OAC and OBC, m∠OCA = m∠OCB = 90 (Given) OA = OB (Radii) OC = OC (common side) ∠OAC = ∠OBC (RHS) CA = CB (corresponding sides) The point C bisects the chord AB. Hence the theorem is proved. Read More About Parallelogram Definition
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Theorem 2: AB and CD are equal chords of a circle whose centre is O. OM ⊥ AB and ON ⊥ CD. Prove that m∠OMN = m∠ONM. Given : In a circle with centre O chords AB and CD are equal OM ⊥ AB, ON ⊥ CD (Fig.6.11). To prove : ∠OMN = ∠ONM Proof : AB = CD (given) OM ⊥ AB (given); ON ⊥ CD (given) OM = ON (equal chords equidistant from the centre) In triangle OMN, m∠OMN = m∠ONM ( Δ OMN is isosceles) Hence Proved.
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Thank You
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