In this chapter we will study to find the Surface area and volume of solid figure , like parallelepiped , cube, cuboid, cylinder , cone , frustum of cone , sphere and hemisphere and we will study to find the surface area and volume of combination of solid figures combination of two or more different or similar solid figures.
SOLID FIGURES
The objects which occupy space (i.e. they have three dimensions ) are called solids.
The solid figures can be derived from the plane figures.
e.g. In figure (i) , we have a paper cut in the form as shown. It is a plane figure. But when we fold the paper along the dotted lines, a box can be made as shown in figure (ii) , which occupies some part of the space. It has more than two dimensions and therefore it fulfils the criteria of being a solid figure.
A solid figure has surface area and volume which are defined below :
SURFACE AREA OF SOLID FIGURE
Surface area of a solid body is the area of all of its surfaces together and it is always measured in square units.
VOLUME OF SOLID FIGURE
The measure of part of space occupied by a solid is called its volume.It is always measured in cubic units.
PARALLELOPIPED
A solid bounded by three pairs of parallel plane surfaces (or faces) is called a parallelepiped . A parallelepiped whose faces are rectangles is called a rectangular parallelepiped or a rectangular solid or a cuboid .
CUBOID
A figure which is surrounded by six rectangular surfaces, is called cuboid.
* A cuboid has 8 corners and 12 edges and four diagonals.
* Volume of cuboid = l * b * h
where, l = length , b = breadth , h = height
* Total surface area of cuboid = 2(lb + bh + hl)
* Diagonal of the cuboid = √(l² + b² + h²)
* Total length of cuboid = 4(l + b + h)
* Lateral surface area or Area of 4 walls = 2(l + b)h
CUBE
A cuboid whose length, breadth and height are same is called cube.
* A cube has six surfaces , twelve edges, eight corners, four diagonals.
Cube is a special case of cuboid which has 6 faces of equal length.
* Total length of cube = 12 * side
* Volume of a cube = (side)³
* Total surface area of a cube = 6 * (side)²
* Diagonal of a cube = √3 * Edge
RIGHT CIRCULAR CYLINDER
A right circular cylinder is a solid figure obtained by revolving the rectangle say ABCD about its one side, say BC.
Let base of right circular cylinder be r and its height be h.
Then,
(i) Volume of cylinder = (area of base) * height = 𝞹r²h cubic units
(ii) Curved surface area = Circumference of the base * Height = 2𝞹rh sq units
(iii) Total surface area = Curved surface + Area of two ends = 2𝞹rh + 2𝞹r²
Also, h = V / 𝞹r²
HOLLOW CYLINDER
The volume of material in a hollow cylinder is the difference between the volume of a cylinder having the external dimensions and the volume of a cylinder having the internal dimensions.
Let R and r be the external and internal radii of the hollow cylinder and b be its height . Then ,
(i) Volume of hollow cylinder = 𝞹(R² - r²)
(ii) Total surface area = 2𝞹(R + r)(h + R - r) sq units
(iii) Curved surface area = 2𝞹Rh + 2𝞹rh sq units
(iv) Total outer surface area = 2𝞹rh + 𝞹R² + 𝞹 (R² - r²) sq units
RIGHT CIRCULAR CONE
A right circular cone is a solid , generated by the revolution of a right angled triangle about one of its sides containing the right angle as axis . Let height of a right circular cone be h, slant height be l and its radius be r , Then,
(i) slant height of the cone = l = AC = √(r² + h²) units
(ii) Volume of cone = 𝞹r²h/3 cubic units
(iii) Curved surface area of cone = 𝞹rl sq units
(iv) Total surface area of a cone = Curved surface area + Base area = 𝞹r(l + r) sq units
FRUSTUM OF A CONE
If a right circular cone is cutoff by a plane parallel to the base of the cone, then the portion of the cone between the cutting plane and base of the cone is called the frustum of a cone . Let h be the height then,
(i) Volume of frustum of right circular cone = 𝞹h (R² + r² + Rr) cubic units
(ii) Curved (lateral) surface area of frustum for right circular cone = 𝞹l (R + r) sq units
where, slant height , l² = h² + (R - r)² units
(iii) Total surface area of frustum of right circular cone = area of base + area of top + lateral surface area
= 𝞹(R² + r² + l(R + r)) sq units
(iv) Total surface area of bucket = 𝞹[(R + r)l + r²] sq units [it is open at bigger end]
SQHERE
A sphere is a solid generated by the revolution of a semi circle about its diameter .
Let the radius of a semi circle about its diameter.
Let radius of sphere be r, then
(i) Volume of sphere = 4𝞹r³ / 3 cubic units
(ii) Surface area of sphere = 4𝞹r² sq units
(iii) Volume of a hollow sphere = 4𝞹(R³ - r³) / 3 cubic units
where, r = inner radius and R = outer radius
HEMISPHERE
A plane passing through the centre cuts the sphere in tow equal parts , each part is called a hemisphere.
Let the radius of hemisphere r then
(i) Volume of hemisphere = 2𝞹r³ / 3 cubic units
(ii) Curved surface area of hemisphere = 2𝞹r² sq units
(iii) Total surface area = 2𝞹r² + 𝞹r² sq units
EXERCISES
THANKS FOR READING THIS AT Math Capsule .
