Summary Advanced Problems in Mathematics, ISBN: 9781013293825 Engineering maths

Mensuration
Perimeter and Area of Plane Figure
Plane Figure A figure enclosed by three or more sides or by a
circular boundary.
Perimeter Total length of the sides of a plane figure.
Area Space covered by a plane figure.

Triangle
For any triangle having sides a, b and c, then A

Perimeter = a + b + c = 2 s b
c
1 h
Area = Base × Height = ( a × h )
2
B a C
or Area = s( s − a )( s − b)( s − c), it is called
Heron’s formula.
a+ b+ c
where, s = = semi-perimeter of the triangle.
2

Different Types of Triangles
(i) Right Angled Triangle A

Perimeter = b + d + h d
h
1
Area = ( b × h )
2
C
B b
Hypotenuse = d = h 2 + b2
(ii) Equilateral Triangle A
Perimeter = 3a
a a
3
Altitude = Height ( h ) = a h
2
3 2 B a C
Area = a
4

, (iii) Isosceles Triangle A
Perimeter = a + 2b
4b2 − a 2 b b
Altitude = Height (h) = h
2
a
Area = 4b2 − a 2 B
a
C
4
(iv) Isosceles Right Triangle A

Perimeter = 2a + 2a
b
Hypotenuse (b) = 2a a

1 2
Area = a
2 B C
a

(v) Triangle having Two Sides a
B C
and One Angle γ
β
Perimeter = a + b + c c
b
α
1 1 1
Area = ab sin γ = bc sin α = ac sin β A
2 2 2
B
(vi) Acute Angled Triangle
Perimeter = a + b + c c
a
h
2
bh b  a 2 + b2 − c2 
Area = = a2 −   A C
2 2  2b  b

B
(vii) Obtuse Angled Triangle
Perimeter = a + b + c
c h
2 a
bh h c − a − b
2 2 2
Area = = a2 −  
2 2  2b  A
b C

D
Quadrilateral
Perimeter = AB + BC + CD + DA A
h1 h2
1
Area = ( h1 + h2 )BD
2 B
C