Mathematik Übungen

Volume and surface of cylinders, spheres, and cones

Sphere

A sphere is a perfectly round solid figure with every point on its surface equidistant from its centre. The radius (r) of a sphere is the distance from the exact center of the sphere (C) to any point on the outside edge of that sphere. We calculate volume and surface area of a sphere using these formulae:

Cr
$$S=4πr^2$$

Surface area is actually equal to areas of four circles of the same radius.

$$V = 4/3 πr^3$$

Of all the solids, a sphere has the smallest surface area for a volume.

Cylinder

A cylinder is a solid that has two parallel (usually circular) bases connected by a curved surface.

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The height (altitude, h) of a cylinder is the perpendicular distance between its bases.

A cylinder can be right or oblique. A right cylinder has bases aligned one directly above the other. In an oblique cylinder, the bases remain parallel to each other, but the sides lean over at an angle that is not 90°. If they have equal height and base, they will have the same volume.

$$V=base×height=πr^2h$$

The surface area is the sum of two bases and the lateral area. We need to know the side length to calculate the lateral area. Only for a right cylinder the side length is equal to the height (a=s1). For oblique cylinder it is s2. S=2×bases+side length×base circumference=2πr2+2πr×side length

Cone

A cone is the solid figure that is generated by a right triangle turning around one of its cathetus. It can be right (the vertex is over the center of its base) or oblique (vertex is not over the center). The base can be a circle or an ellipse.

rhsV

A cone is closely related to a pyramid and the formulae for their volume are similar (volume of a pyramid is one third of a prism with the same width, length and height and a cone is one third of a cylinder with the same base and height).

$$V=1/3×base×h$$

The slant height (s) of a right cone is the length of the segment from the vertex of the cone to the circle of the base. We don't define this for oblique cones.

The total surface area of a cone is the sum of the area of its base and the lateral (side) surface. The lateral surface of a right cone can be calculated as:


Al=$1/2$×circumference×s

Total area of right circular cone is:

$$A=A_b+A_l=πr^2+1/2×2πr×s=πr^2+πrs$$

   
   

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