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In this section, the formulas for the curved surface area, total surface area, and volume of cones, spheres, and hemispheres are presented. Key concepts such as the relationships between dimensions and the derived formulas are highlighted.
In this section, the following important formulas are summarized: 1. Curved Surface Area of a Cone: - Formula: C.S.A = πrl - Where r is the radius and l is the slant height of the cone.
Contextual Importance: Understanding these formulas is crucial as they are fundamental for solving various practical problems in geometry involving three-dimensional shapes.
Curved Surface Area of Cone: πrl, where l is the slant height and r is the base radius.
Total Surface Area of Cone: πr(l + r), integrating both curved and base.
Volume of Cone: (1/3)πr²h, showing how height and radius affect volume.
Surface Area of Sphere: 4πr², covering the entirety of the sphere's surface.
Volume of Sphere: (4/3)πr³, representing how space is filled in three dimensions.
For cone's curved surface, πrl is the key, the height will help, you will see.
Imagine a baker shaping a cone with frosting. To know how much icing to use, they need to calculate the curved surface area and the size of the base.
To remember sphere formulas: '4S: Sphere's Surface, 4V: Volume has 4 parts of 3'.
Example: Calculate the total surface area of a cone with radius 5 cm and slant height 10 cm. Solution: Using the formula T.S.A = πr(l + r), we find T.S.A = π × 5 × (10 + 5) = 75π cm².
Example: Find the volume of a hemisphere with a radius of 7 cm using V = (2/3)πr³. Solution: V = (2/3)π × 7³ = approximately 143.6 cm³.
Term: Curved Surface Area
Definition: The area of the surface of a three-dimensional object, excluding its base.
The area of the surface of a three-dimensional object, excluding its base.
Term: Total Surface Area
Definition: The total area that the surface of an object occupies, including all faces and edges.
The total area that the surface of an object occupies, including all faces and edges.
Term: Slant Height
Definition: The distance measured from the base to the apex of a cone along the lateral surface.
The distance measured from the base to the apex of a cone along the lateral surface.
Term: Volume
Definition: The amount of three-dimensional space an object occupies.
The amount of three-dimensional space an object occupies.
Term: Sphere
Definition: A perfectly round three-dimensional shape, every point of which is equidistant from the center.
A perfectly round three-dimensional shape, every point of which is equidistant from the center.
Term: Hemisphere
Definition: Half of a sphere, divided by a plane passing through its center.
Half of a sphere, divided by a plane passing through its center.