Geometry Tutorial Part - 7

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Geometry Tutorial Part - 7

Author: Sureshbala

QUADRILATERALS

(i) For any quadrilaterals

$\text{ Area of the quadrilateral} =\cfrac {1}{2}* \text{One diagonal*Sun of the offsets drawn to that diagonal}$

Hence, for the quadrilateral ABCD shown in Fig. 1.23,

$\text {Area of quadrilateral ABCD } =\cfrac {1}{2}*AC*(BE+DF)$

(ii) For a cyclic quadrilateral where the four sides measure a, b, c and d respectively,

$Area= \sqrt{(s-a)(s-b)(s-c)(s-d)}$ where s is the semi-perimeter, i.e., (s = a + b + c + d)/2

(iii) For a trapezium

$\text{ Area of a trapezium } =\cfrac {1}{2}* \text{Sum of parallel sides *Distance between them}$ $=\cfrac {1}{2}*(AD+BC)*AE \text{ (refer to fig 5.25)}$

(iv) For a parallelogram

$\text{(a)Area = Base Height}$ $\text{ (b)Area = Product of two sides Since of included angle}$

(v) For a rhombus

$Area=\cfrac {1}{2}* \text{ product of diagonals}$ $Perimeter =4* \text{ side of rhombus}$

(vi) For a rectangle

Area = Length * Breadth

Perimeter = 2(l + b), where l and b are the length and the breadth of the rectangle respectively

(vii) For a square

(a) Area=Side^2

(b) $Area=\cfrac {1}{2}Diagonal^2$

Where the diagonal $=\sqrt{2}*side$

(viii) For a polygon

(a) $\text{Area of a regular polygon =1/2 * perimeter * Perpendicular distance from the centre of the polygon to any side}$

(Please note that the centre of a regular polygon is equidistant from all its sides)

(b) For a polygon which is not regular, the area has to be found out by dividing the polygon into suitable number of quadrilaterals and triangles and adding up the areas of all such figures present in the polygon.

CIRCLE

(i) $\text {Area of the circle } =\pi r^2 \text {where r is the radius of the circle }$

$Circumference = 2 \pi r$

(ii) Sector of a circle

$\text {Length of arc}=\cfrac {\theta}{360^{\circ}}* 2\pi r$ $Area = \cfrac{\theta}{360^{\circ}} * \pi r^2\; where$ $\theta$ $\text{the angle of the sector in degrees and r is the radius of the circle.}$ $Area=\Big(\cfrac {1}{2}\Big)$ $lr \text{l is the length of arc and r is radius}$

(iii) Ring : Ring is the space enclosed by two concentric circles

$Area=\pi R^2=\pi R^2=\pi(R+r)(R-r)$ where R is the radius of the outer circle and r is the radius of the inner circle.

ELLIPSE

$Area = \pi ab$ is semi-major axis and "b" is semi-minor axis $Perimeter =\pi (a+b)$

AREAS AND VOLUMES OF SOLIDS

Solids are three-dimensional objects which, in addition to area, have volume also. For solids two different types of areas are defined

(a) Lateral surface area or curved surface area and

(b) Total surface area

As the name itself indicates, lateral surface area is the area the LATERAL surface of the solid. Total surface area includes the areas of the top and the bottom surfaces also of the solid. Hence, Total surface area = Lateral surface area + Area of the top face + Area of the bottom face

In solids (like cylinder, cone, sphere) where the lateral surface is curved, the lateral surface area is usually referred to as the "curved surface area

For any solid, whose faces are regular polygons, there is a definite relationship between the number of vertices, the number of sides and the number of edges of the solid. This relationship is given by "Euler's Rule".

Number of faces + Number of vertices = Number of edges + 2 (Euler's Rule)

PRISM

A right prism is a solid whose top and bottom faces (bottom face is called base) are parallel to each other and are identical polygons (of any number of sides) that are parallel. The faces joining the top and bottom faces are rectangles and are called lateral faces. There are as many lateral faces as there are sides in the base. The distance between the base and the top is called height or length of the right prism.

In a right prism, if a perpendicular is drawn from the center of the top face, it passes through the centre of the base,

$\text {Lateral Surface Area = Perimeter of base x Height of the prism}$ $\text {Total Surface Area = Lateral Surface Area + 2 x Area of base}$ $\text {Volume = Area of base x Height of the prism}$

CUBOID OR RECTANGULAR SOLIDI

A right prism whose base is a rectangle is called a rectangular solid or cuboid. If l and b are respectively the length and breadth of the base and h, the height, then

$\text{Lateral Surface Area=2(l+b).h}$
$\text{Total Surface Area=2(l+b)h+2lb=2(lb+lh+bh)}$
$\text{Longest diagonal of the cuboid} =\sqrt{1^2+b^2+h^2}$

CUBE

A right prism whose base is a square and height is equal to the side of the base is called a cube.

$Volume=a^3 \text{ where a is the edge of the cube}$
$\text{Lateral Surface Area }=4a^2$
$\text{Total Surface Area }=6a^2$

The longest diagonal of the cube (i.e., the line joining one vertex on the top face to the diagonally opposite vertex on the bottom face) is called the diagonal of the cube. The length of the diagonal of the cube is $\sqrt{3}$

CYLINDER

A cylinder is equivalent to a right prism whose base is a circle. A cylinder curved surface as its lateral faces. If r is the radius of the base and h is the height of the cylinder

$Volume = \pi r^2 h$
$\text{Curved Surface Area } =2\pi rh$
$\text{Total Surface Area } =2\pi rh+2 \pi r^2 =2 \pi r(h+r)$

A hollow cylinder has a cross-section of a ring. Volume of the material contained in a hollow cylindrical $ring=\pi(R^2-r^2)h$ where R is the outer radius, r is the inner radius and h, the height.

PYRAMID

A solid whose base is a polygon and whose faces are triangles is called a pyramid. The triangular faces meet at a common point called vertex. The perpendicular from the vertex to the base is called vertex to base is called the height of the pyramid.

A pyramid whose base is a regular polygon and the foot of the perpendicular from the vertex to the base coincides with the center of the base, is called a right pyramid

The length of the perpendicular from the vertex to any side of the base (please note that this side will be the base of one of the triangular lateral faces of the prism) along the slant lateral surface is called a right pyramid.

$\text{Volume of a pyramid} =\cfrac {1}{3}* \text{area of base*height}$ $\text{Lateral Surface area } =\cfrac {1}{2}* \text{Perimeter of the base * Slant height}$

Total Surface Area = Lateral Surface Area + Area of the base.

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