Surface areas and Volumes

Topics covered:-

FormulasInformation on:-oCuboidoCubeoCylinderoConeoSphere

Shape Total surface area

Curved (Lateral)surface area

Volume Longest Diagonal

Cuboid 2(lb+bh+hl) 2h(l+b) lbh

Cube 6a2 4a2  a3 a√3

Cylinder 2πr(r + h)  2πrh  πr2h

Hollow Cylinder

π(R2- r2) 2πh(R+r) πh(R2-r2)

Cone πr[r + √(r2 + h2)]

 πr√(r2 + h2) 1/3πr2h

Sphere 4πr2 4πr2 4/3 πr3

Hemisphere 3πr2 2πr2 (2/3)πr3

Frustum π[ r12 + r2

2 + (r1 * r2) * √[(r1 - r2)2 + h2) ]

π(r1 + r2)√[(r1 - r2)2 + h2]

(1/3)πh [r12 +

r22 + (r1 * r2)]

Formulae

Cuboid

 A cuboid is a convex polyhedron bounded by six quadrilateral faces.

Cube

 A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.

CylinderA cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the

points at a fixed distance from a given line segment,

the axis of the cylinder. Thesolid enclosed by this surface and by two planes

perpendicular to the axis is also called a cylinder. The surface area and

the volume of a cylinder have been known since deep antiquity. A cylinder whose cross section is an ellipse, 

parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively.

Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

More precisely, it is the solid figure bounded by a base in a plane and by a surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base, such that there is a circular cross section. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.

SphereA sphere is a perfectly round geometrical object in three-

dimensional space.Like a circle a sphere is defined ,mathematically, as the

set of points that are all the same distance r from a given point in three-dimensional space.

This distance r is the radius of the sphere, and the given point is the center of the sphere.

The maximum straight distance through the sphere passes through the center and is thus twice the radius; it is the diameter.

Areas related to circles

Let us recall some formulae related to circles

Circumference= 2πr Area= πr2

Area of semi-circle= πr2/2 Area of quadrant of a circle= πr2/4 Area of a sector of a circle= πr2θ/360

Quadrant

A quarter of a circle made by two radiuses at right angles and the connecting arc is known as Quadrant.

Sector of a circle

A circular sector or sector of a circle(symbol: ⌔), is the portion of a disk enclosed by two radii and an arc.

Sector

Minor sector

Major sector

Major sector

Minor sector

Example of a

sector

Difference between a minor sector and a major sectorMinor sector

The smaller portion of a circle that is cut off by 2 radii of a circle, is known as Minor Sector.

Major sectorThe larger

portion of a circle that is cut off by 2 radii of a circle, is known as Major Sector.

SEGMENT

In geometry, a circular segment (symbol: ⌓) is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant or a chord.

Segment

Minor segment

Major segment

Difference between a minor segment and a major segment

Minor segment The smaller portion of

a circle that is cut off by a chord is known as minor segment.

Major segment The larger portion of a

circle that is cut off by a chord is known as major segment.

Difference between a quadrant and a sector

Quadrant SectorA quarter of a circle

made by two radiuses at right angles and the connecting arc is known as Quadrant.

A circular sector or sector of a circle(symbol: ⌔), is the portion of a disk enclosed by two radii and an arc.

Efforts by- Arpan Goyal(X th a)

SUBMITTED TO- RAJENDER SIR