Formula for the Volume of a Sphere Formula for the Volume of a Sphere 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, Volume of a cylinder = Area of the base x Height of the cylinder. Know More About Geometry Calculator
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= 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 ExamplesBack to Top 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 = 43 x π x (8.2)3 =43 x 3.14 x 551.368 Volume of the sphere (v) = 2308.39 cm3 Learn More Vector Calculator Tutorvista.com
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Define Perpendicular Lines Define Perpendicular Lines Two intersecting lines will have four angles formed at the intersection points. If all the four angles are equal, then the two lines are said to be perpendicular to each other. We already know by linear postulate theorem that the two vertically opposite angles are equal. Hence if these two lines are perpendicular, then all four angles are 90 degrees. Examples of perpendicular lines: – In the graph paper, The X-axis and Y-axis are perpendicular. – In an ellipse two axes, minor axis, and major axis are perpendicular. – For a line segment, any shortest line from a point outside the circle is perpendicular. – Tangent and normal to any curve are perpendicular lines. Slopes of two perpendicular lines: In coordinate Geometry, when two lines are perpendicular, the product of the slopes of the lines is -1. This property has a lot of applications in finding the equation of perpendicular lines, length of perpendicular segment from a point to a given line, etc. For any curve in a graph with equation y = f(x), the slope of the tangent is defined as the rate of change of y with respect to x at that point. The normal to this curve at this point is perpendicular to the tangent line.
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Example: In a circle, with centre at the origin and radius 3, the equation will be of the form (x)²+(y)² = 3². Take any point say (0,3). To find the tangent, we have to find dy/dx. Differentiating, 2x+2y =0 Hence, the slope of the normal is perpendicular to x axis or parallel to y axis. Example for Perpendicular Lines from a Point to a Line Let AB be a line with coordinates (1,2) and (3,4). Measure the length of perpendicular line from (-1,1) to this line segment. We know that the perpendicular line from (-1,1) has a slope of -1/slope of AB. Equation of AB is (x-1)/(3-1) = (y-2)/(4-2) Or x-1 = y-2 Or y = x+1 Slope of AB passing through (1,2) and (3,4) is 4 - 2/3 -1 =1. Slope of perpendicular line to AB is -1. Since the perpendicular line passes through (-1,1) equation of the perpendicular is y-1 = -1(x+1) or y =-x -1 +1 or y = -x. To get the foot of the perpendicular line on AB, we solve the two equations by substitution method. y = x+1 = -x This on simplification gives 2x = -1 or x = -1/2. Since y = -x , we have y = +1/2, So, foot of the altitude from the point (-1,1) is (-1/2,1/2). The length of the perpendicular segment is between (-1,1) and (-1/2,1/2) is √[ (-1/2+1)²+(1/2-1)²] = √(1/4+1/4) = √(1/2) = 1/1.414 = 0.707 approximately. Read More About Formula for Volume of Sphere Tutorvista.com
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