SURFACE AREAS

AND VOLUMES

contentsSurface areas and volumes of :-

Cube CuboidCyinderCone Sphere and hemisphere

Cube

Cube DEFINATION:A cuboid whose length ,breadth, height is

called a cube. SOLID CUBE:A solid cube is the part of the space enclosed

by six faces of the cube.

SURFACE AREA OF A CUBE SURFACE AREA OF CUBE:Since all six faces of a cube are

squares of the same size i.e. For a cube we have l=b=h. Thus, if l cm is the length of edge or side or a cube,then

Therefore, surface area of a cube=6lsq

LATERAL SURFACE AREA OF A CUBE and VOLUME and TOTAL SURFACE AREA Lateral surface area of cube=4(edge)sq

I. Volume of cube=(edge) cubic

II. L.S.A of a cuboid =2 (l + b) hT.S.A of a cuboid =2(lb+bh+lh)Volume of the cuboid =lbh

III. T.S.A of a cube =6a2< Total surface area of a cube,  sum areas of all the faces of a cube >

Parts of cube FaceAlso called facets or sides. A cube has six faces which are all

squares, so each face has four equal sides and all four interior angles are right angles. See Definition of a square. In the figure above, drag the 'explode' slider to see the faces separated for clarity.

EdgeA line segment formed where two edges meet. A cube has 12 edges. Because all faces are squares and congruent to each other, all 12 edges are the same length.

VertexA point formed where three edges meet. A cube has

8 vertices.

Cuboid

What is a cuboidA solid which has six rectangular faces at right angles to each other.

Curved surface area of the cuboid

Surface of the cuboid without the top = 2 (bh + hl) + lb

SURFACE AREA OF CUBOID WITHOUT THE TOP AND THE BOTTOM

= 2 (bh + hl)

Total surface area of the cuboidAREA OF RECTANGLE 1 = (l x h) +AREA OF RECTANGLE 2 = (l x b) + AREA OF RECTANGLE 3 = (l x h) + AREA OF RECTANGLE 4 = (l x b) +AREA OF RECTANGLE 5 = (b x h) +AREA OF RECTANGLE 6 = (b x h) = 2 (l x b) + 2 (b x h) + 2(l x

h)

= 2 (lb + bh + hl)

Volume

Volume is the space occupied by an object. Volume is also referred to capacity of an

object.

THUS, VOLUME OF CUBOID = BASE AREA x

HEIGHT = (l x b) x h

= l x b x hVOLUME OF CUBOID = l x b x h

Cylinder

Introduction to Cylinder A 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. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder.

Surface area of Cylinder

TSA of a cylinder = area of the base + area of top + CSA of the cylinder

= ∏r2 + ∏r2 + 2∏rh

= 2∏r2 + 2∏rh = 2∏r(r + h) Where, r is the radius h is the height of the

cylinder

Volume of the cylinder Volume of the cylinder = area of the

base x height = r2 x h = ∏r2h Volume of hollow cylinder = ∏(R2 - r2)

h Where, r is the radius and h is the

height

cone

Introduction to 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.

Curved surface area of the cone

when we cut a cone from its slant height curved surface area of cone =area of sector

=1/2 *l *(2

∏r) = ∏r l

Total surface area of a cone

TOTAL SURFACE AREA =curved surface area

+area of the base

= ∏rl+ ∏r2

+ = ∏r(r+l)

VOLUMEWHEN WE TAKE A CONE AND A CYLINDER OF SAME HEIGHT AND RADIUS WE GET

Sphere

and

Hemi-

sphere

Introduction to Sphere & Hemi-sphere1. A sphere is a perfectly round geometrical object in three-

dimensional space. Like a circle, which is in two dimensions in a mathematical sense, a sphere is 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 centre and is thus twice the radius; it is the diameter.

2. Hemisphere refers to the equal halves of the sphere and can also be called the 3d design for a semi-circle.

Surface Area of Sphere

When we talk about painting or polishing the surface it is related to the surface area. Surface-Area (TSA) = 4∏r2

Where, ‘r’ is the radius from the center to surface.

Surface Areas of Hemisphere

TSA of hemisphere = 3∏r2 CSA of hemisphere = 2∏r2

Where, ‘r’ is the radius.

Volume of Sphere When we talk about the air in the

solid or want to count the no. of small object from the bigger one then it is related to the volume.

Volume of the sphere = 4/3∏r3

Where, r is the radius.

Volume of Hemisphere Volume of the Hemisphere = 2/3∏r3

Where, r is the radius.

Credits

Cube – by Stuti SomaniCuboid –by Niriksha Mogaveera Cylinder – by Aditya WarriorCone – by Shreyans MaliwalSphere and Hemisphere-by

Pakshal Shah Animation– by Shreyans Maliwal