## GEOMETRY

### Introduction

• Sum of the angles of a triangle is 180°.
• Sum of any two sides of a triangle is greater than the third side.
• Area of a rectangle = length * breadth.
• Area of a square = side^2 = 1/2(diagonal)^2
• Area of a circle = π * r*r ; r = radius and π / PI = 22/7 or 3.14
• Area of a kite: 1/2 * (sum of the length of two diagonals)
• Area of a semi circle = π * r * r / 2 ; Circumference = π * r
• To find the area of a parallelogram, just multiply the base length (b) times the height (h): Area = b * h

### Volume and Surface Area

Cuboid: If length = L, breadth = B, height = H then Volume = L * B * H.

volume of a cube = a, Surface area = 6 *a * a sq m, diagonal = sq-root(3) * a

cylinder = b h = π *  r2 h

pyramid = (1/3) b h

volume of a cone = (1/3) b h = 1/3 * π *r2 h

volume of a sphere = (4/3) * π *  r3

1 m^3 = 1000 liters ; 1 cm^3 = 1 liter

Basic Conversions

• 1 m = 100 cm = 1000 mm

• 1 km = 1000 m = 5/8 miles

• 1 inch = 2.54 cm

• 100 kg = 1 quintal

• 10 quintal = 1 tonne = 1000 kg

• 1 kg = 2.2 pounds

• 1 m = 39.37 inches

• 1 mile = 1760 yd = 5280 ft; 1 nautical mile (knot) = 6080 ft

• 1 litre = 1000 cc

• 1 acre = 100 sq m; 1 hectare = 10000 sq m

• Parallel Lines : Two straight lines are parallel if they lie on the same plane and do not intersect however far produced.

• Transversal: It is a straight line that intersects two parallel lines. When a transversal intersects two parallel lines then

1. Corresponding angles are equal, (that is: For the following figure) 1 = 5; 2 = 6; 4 = 8; 3 = 7

1. Alternate interior angles are equal, that is (Refer following figure) 4 = 6; 5 = 3

1. Alternate exterior angles are equal, that is 2 = 8; 1 = 7

1. Interior angles on the same side of transversal add up to 180°, that is 4 + 5 = 3 + 6 = 180°

• Polygons are plane figures formed by a closed series of rectilinear (straight) segments. Example: Triangle, Rectangle

• Regular polygons: Polygons with all the sides and angles equal.

• Irregular polygons: Polygons in which all the sides or angles are not of the same measure.

• Polygon can also be divided as concave or convex poly-gons. Convex polygons are the polygons

• Sum of all the angles of a polygon with n sides = (2n – 4)p/2 or (n – 2)p Radians = (n – 2) 180° degrees

• Sum of all exterior angles = 360°, i.e. In the figure below: θ1 + θ2 + … + θ6 = 360°, In general, θ1 + θ2 + … + θn = 360°

• No. of sides = 360°/exterior angle

• Area = (ns2/4) × cot (180/n); where s = length of side, n = no. of sides.

• Perimeter = n × s.

• A triangle is a polygon having three sides. Sum of all the angles of a triangle = 180°.

• Acute angle triangle: Triangles with all three angles acute (less than 90°)

• Obtuse angle triangle: Triangles with one of the angles obtuse (more than 90°). we cannot have more than one obtuse angle in a triangle.

• Right angle triangle: Triangle with one of the angles equal to 90°.

• Equilateral triangle: Triangle with all sides equal. All the angles in such a triangle measure 60°

• Isosceles triangle: Triangle with two of its sides equal and consequently the angles opposite the equal sides are also equal.

• Scalene Triangle: Triangle with none of the sides equal to any other side.

Properties (General)

• Sum of the length of any two sides of a triangle has to be always greater than the third side.

• Difference between the lengths of any two sides of a triangle has to be always lesser than the third side

• Side opposite to the greatest angle will be the greatest and the side opposite to the smallest angle the smallest

• The sine rule: a/sin A = b/sin B = c/sin C = 2R (where R = circum radius.)

• The cosine rule: a2 = b2 + c2 – 2bc cos A This is true for all sides and respective angles

• In case of a right angle, the formula reduces to a2 = b2 + c2, Since cos 90 = 0

• The exterior angle is equal to the sum of two interior angles not adjacent to it. –ACD = –BCE = –A + –B

• Area = 1/2 base × height or 1/2 bh. Height = Perpendicular distance between the base and vertex opposite to it

• Area = rs (where r is in radius)

• Area = 1/2 × product of two sides × sine of the included angle

• Area = s(sa)(sb)(sc)−−−−−−−−−−−−−−−−−√ where s=a+b+c2

• = 1/2 ac sin B

• = 1/2 ab sin C

• = 1/2 bc sin A

• Area = abc/4R; where R = circum radius

• Area = 1/2 (product of diagonals) × (sine of the angle between them)

• If θ1 and θ2 are the two angles made between themselves by the two diagonals, we have by the property of intersecting lines : θ1 + θ2 = 180°

• Area = 1/2 × diagonal × sum of the perpendiculars to it from opposite vertices = d(h1+h2)2

• Area of a circumscribed quadrilateral = A=(Sa)(Sb)(Sc)(Sd)−−−−−−−−−−−−−−−−−−−−−−−√ where S = a+b+c+d2 (where a, b, c and d are the lengths of the sides.)

• A parallelogram is a quadrilateral with opposite sides parallel (as shown in the figure below.) Area = Base (b) × Height (h)

• Area = product of any two adjacent sides × sine of the included angle = ab sin Q

• Perimeter = 2 (a + b) where a and b are any two adjacent sides

• Diagonals of a parallelogram bisect each other.

• Bisectors of the angles of a parallelogram form a rectangle.

• A parallelogram inscribed in a circle is a rectangle.

• A parallelogram circumscribed about a circle is a rhombus.

• The opposite angles in a parallelogram are equal.

• The sum of the squares of the diagonals is equal to the sum of the squares of the four sides in the figure: AC2+BD2=2(AB2+BC2).

Cuboid

• A cuboid is a three dimensional box. It is defined by the virtue of it’s length l, breadth b and height h. It can be visualised as a room which has its length, breadth and height different from each other.

• Total surface area of a cuboid = 2 (lb + bh + lh)

• Volume of the cuboid = lbh

Cube of side s

• A cube is a cuboid which has all its edges equal i.e. length = breadth = height = s

• Total surface area of a cube = 6s2.

• Volume of the cube = s3.

Prism

• A prism is a solid which can have any polygon at both its ends. It’s dimensions are defined by the dimensions of the polygon at it’s ends and its height.

• Lateral surface area of a right prism = Perimeter of base * height

• Volume of a right prism = area of base* height

• Whole surface of a right prism = Lateral surface of the prism + the area of the two plane ends

Cylinder

• A cylinder is a solid which has both its ends in the form of a circle. Its dimensions are defined in the form of the radius of the base (r) and the height h. A gas cylinder is a close approximation of a cylinder.

• Curved surface of a right cylinder = 2prh where r is the radius of the base and h the height.

• Whole surface of a right circular cylinder = 2prh + 2pr2

• Volume of a right circular cylinder = pr2h

Pyramid

• A pyramid is a solid which can have any polygon as its base and its edges converge to a single apex. Its dimensions are defined by the dimensions of the polygon at its base and the length of its lateral edges which lead to the apex. The Egyptian pyramids are examples of pyramids.

• Slant surface of a pyramid = 1/2 * Perimeter of the base* slant height

• Whole surface of a pyramid = Slant surface + area of the base

• Volume of a pyramid = area of the base * 3 / heights

Cone

• A cone is a solid which has a circle at its base and a slanting lateral surface that converges at the apex. Its dimensions are defined by the radius of the base (r), the height (h) and the slant height (l). A structure similar to a cone is used in ice cream cones

• Curved surface of a cone = prl where l is the slant height

• Whole surface of a cone = prl + pr2

• Volume of a cone = πr2h3

Sphere

• A sphere is a solid in the form of a ball with radius r.

• Surface Area of a sphere = 4 * π * r

• Volume of a sphere = 4/3 * π r3

Frustum of a pyramid

• When a pyramid is cut the left over part is called the frustum of the pyramid.

• Slant surface of the frustum of a pyramid = 1/2 * sum of perimeters of end * slant height.

• Volume of the frustum of a pyramid = h3[B1+(B1B2)−−−−−−−−√+B2] where h is the thickness and B1, B2 the areas of the ends

### Solved Question Papers

Q.
A vertical stick 20 m long casts a shadow 10 m long on the ground. At the same time, a tower casts the shadow 50 m long on the ground. Find the height of the tower

1.100 m
2.120 m
3.25 m
4.200 m

Ans.1

Explanation :
When the length of stick = 20 m, then length of shadow = 10 m i.e. in this case length = 2 ×
shadow with the same angle of inclination of the sun, the length of tower that casts a shadow of 50
m fi 2 × 50 m = 100 m
i.e. height of tower = 100 m

Q.
The area of similar triangles, ABC and DEF are 144 cm2 and 81 cm2 respectively. If the longest side of larger △ ABC be 36 cm, then the longest side of smaller △ DEF is

1.20 cm
2.26 cm
3.27 cm
4.30 cm

ANS.3

Explanation :
For similar triangles we have (Ratio of sides)2 = Ratio of areas
Then as per question = (36/x)2 = 144/81. {Let the longest side of △ DEF = x}. So. 36/x = 12/9 so x = 27 cm

Q.
Two isosceles △’s have equal angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding heights

1.4/5
2.5/4
3.3/2
4.5/7

ANS.1

Explanation :
(Ratio of corresponding sides)2 = Ratio of area of similar triangle \ Ratio of corresponding sides in this question. sqrt1625=4/5

Q.
The areas of two similar △s are respectively 9 cm2 and 16 cm2 . Find the ratio of their corresponding sides.

1.3 : 4
2.4 : 3
3.2 : 3
4.4 : 5

ANS.1

Explanation :
Ratio of corresponding sides = sqrt916=3/4

### Quiz

Score more than 80% marks and move ahead else stay back and read again!

Q1: Area of a kite with diagonals 10,8cm is
1.9
2.40
3.80
4.18

ANS.1

Q2: area of a parallelogram with base length of 10cm and height of 5 cm is
1.50
2.25
3.10
4.25

ANS.1

Q3: volume of a sphere with radius 7 cm is
1.1437.33
2.1500
3.1466
4.1343

ANS.1

Q4: volume of a cone with radius 7cm and height 3 cm is
1.160
2.154
3.144
4.130

ANS.2

Q5:Volume of a cube is
1.a^3
2.6a^2
3.PI*R^3
4.PI*R^2*H

ANS.1