What is 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, Know More About Types of Quadrilaterals
Tutorvista.com
Page No. :- 1/7
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 the Midpoint Formula
Tutorvista.com
Page No. :- 2/7
= 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
Tutorvista.com
Page No. :- 3/7
ARC Of A Circle Arc of a circle can be taken as a part of the circle formed by two radii making an angle less than 180 degrees. Arc of a Circle Definition states that an arc is a segment of the total circumference of a circle. We know that a circle has different parts of arc, whenever two points are marked on the circle. Thus, any two points on a circle forms a minor arc and a major arc. Minor arc of a circle is an arc containing less than 12 the perimeter of the circle. The minor arc is also called as small arc. Major arc of a circle is an arc containing more than 12 the perimeter of the circle. It is also called as a great arc. An arc made with 210 degrees with centre is the major arc of a circle. An arc made with 90 degrees at the center is the minor arc of a circle. An arc made with 180 degrees is the semi circle of the circle. Arc of a Circle Formula In the above figure, the part of the circle L is called the arc of the circle. The length of an arc of a circle with radius r and subtending an angle θ in the center of the circle is defined as,
Tutorvista.com
Page No. :- 4/7
If θ is in radians, L = θ *r If θ is in degrees, L = (θ/3600) * 2 r This is the formula for finding the arc of a circle. This is based on the principle that the total angle at the centre is 360 degrees. So, the arc subtending an angle θ will be (θ/360)* 2. This can be used to find the length of the arc of a circle. The length of the arc to the perimeter is in the same proportion as the subtended angle to the centre to 2pi radians. Arc area is given by the following formula, Arc area = θ/2 Or Arc area = (θ/3600) ( r²). This is also based on the proportion principle. The area of the segment is the area of isosceles triangle formed by two radii and chord. The formula for area of a segment is given by, Area of the segment = r²/2( - ) Arc of a Circle Example ProblemsBack to Top Below are the examples based on arc of a circle Example 1: An arc of a circle subtends i. 120 degrees ii. 270 degrees. iii. 180 degrees iv. 40 degrees v. 320 degrees. Determine whether it is minor arc or major arc. Solution: Read More About Adjacent Angles Definition
Tutorvista.com
Page No. :- 5/7
120, Minor arc since 120<180 270, Major arc since 270>180 180, semi circle as 180 = 3602 40, minor arc as 40 <180 320, major arc as 320 > 180. Example 2: Find the length of arc with angle 60 at the centre for a circle of radius 7. Solution: Length of the arc = (60/360) = 2*227*7*16 = 22/3.
Tutorvista.com
Page No. :- 6/7
Thank You
TutorVista.com