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# Volume of a right circular cone

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### Volume of a right circular cone

1. 1. MATHS PROJECT VOLUME OF A RIGHT CIRCULAR CONE Presented By- SHRABANTI IX-C
2. 2. CONE <ul><li>Q. What is circular cone ? </li></ul><ul><li>Answer : A circular cone is a surface generated by </li></ul><ul><li>a straight line passing through a fixed </li></ul><ul><li>point and moving on a circle. </li></ul><ul><li>Q. What is a right circular cone ? </li></ul><ul><li>Answer : Right circular cone: A right circular </li></ul><ul><li>cone is a surface generated by revolving </li></ul><ul><li>a straight line, which passes through a </li></ul><ul><li>fixed point and makes a constant angle </li></ul><ul><li>with a fixed line. </li></ul><ul><li>In all the above cases, hollow cone is generated. </li></ul>
3. 3. TYPES OF CONES A B C D V e r t i c a l h e i g h t Radius Hereafter, we mean by cone a right circular cone. In the figure, D is the vertex of a cone, the vertical distance between the vertex and base of the cone is called its height. CONE CIRCULAR CONE RIGHT CIRCULAR CONE Slant height =l
4. 4. <ul><li>Height of the cone </li></ul><ul><li>The length of the segment is the height of the cone and is usually denoted by h. </li></ul><ul><li>Slant height of the cone </li></ul><ul><li>The distance between the vertex and any point on the circumference of the base circle is called its slant height. The length of the segment is called the slant height of the cone and is generally denoted by l. </li></ul><ul><li>Radius of the cone </li></ul><ul><li>The radius of the base circle is called the radius of the cone and is usually denoted by r. </li></ul><ul><li>In a rt. angled triangle, </li></ul><ul><li>l x l =(h x h) + (r x r) </li></ul><ul><li>l= h + r </li></ul>2 2
5. 5. DERIVATION <ul><li>VOLUME OF A RIGHT CIRCULAR CONE </li></ul><ul><li>Volume of a cone </li></ul><ul><li>= 1/3 x π (r x r) h </li></ul><ul><li>The above formula can be verified </li></ul><ul><li>experimentally. Take a cylindrical </li></ul><ul><li>jar of height h and radius r whose </li></ul><ul><li>volume is π x (r x r) h. </li></ul>Take a hollow cone which has the same height h and same radius r. Fill the cone with water and pour it into the jar. The jar will be fully filled, when three cones full of water is poured into it. It shows that volume of the cone is 1/3 x π (r x r) h.
6. 6. <ul><li>A sector of a circle of radius 15cm has </li></ul><ul><li>the angle </li></ul>120 0 <ul><li>It is rolled up so that two bonded radii </li></ul><ul><li>are joined together to form a cone. </li></ul>120 0 15cm A B O O h 15cm <ul><li>Here the radii are joined together. Clearly the radii </li></ul><ul><li>of circle is converted to the slant height of cone and </li></ul><ul><li>the arc AB is converted to circumference of the cone. </li></ul>A B
7. 7. <ul><li>To determine the volume of the cone so formed. We first </li></ul><ul><li>determine the length of arc AB. </li></ul><ul><li>Length of arc AB = 120 / 360 x 2π x 15 </li></ul><ul><li>= 10π </li></ul><ul><li>Suppose the radius of cone be r, </li></ul><ul><li>So, 2πr = 10π </li></ul><ul><li>or, r = 5cm </li></ul><ul><li>Slant height of the cone is 15cm. </li></ul><ul><li>Height of the cone = (15) - (5) </li></ul><ul><li>= (15+5) (15-5) </li></ul><ul><li>= 20 x 10 </li></ul><ul><li>= 10 2 cm </li></ul><ul><li>Volume of the cone = r h </li></ul>2 2 1 3 π 2 1 3 2 7 10 2 22 = x x x = 5 370.33 cm 3
8. 8. SOLVED EXAMPLES <ul><li>EXAMPLE 31 : Find the volume of the largest right circular cone that </li></ul><ul><li>can be cut out of a cube whose edge is 7 cm. </li></ul><ul><li>SOLUTION : For largest right circular cone to be cut, clearly the circle </li></ul><ul><li>will be inscribed in a face of the cube and its height will </li></ul><ul><li>be equal to an edge of the cube. </li></ul><ul><li>Radius of the base of cone, r = cm </li></ul><ul><li>So. Volume of the cone = π r h </li></ul><ul><li>= x x x cm </li></ul><ul><li>= </li></ul>7 2 1 3 2 3 1 22 7 7 2 2 7 3 88.83 cm 3
9. 9. SOLVED EXAMPLES <ul><li>EXAMPLE 3 : The circumference of the base of a 16 m high solid cone is 33 m. </li></ul><ul><li>Find the volume of the cone. </li></ul>SOLUTION : Radius of the base = r m, Height of the cone, h = 16 m Circumference of the base = 2 π r = 33 . r = 33 2 π = π 33 2 X 22 /7 = 21 4 m Volume of the cone = 1 3 r 2 h = 1 3 x 22 7 x 21 4 2 x 16 = 22 x 21 = 462 m 3
10. 10. THE END