Page 216 - Start Up Mathematics_6

Important Formulae in Geometry
Let’s learn how literals can be used to denote characteristics of certain geometric figures:

1. Perimeter of a Square
Perimeter = 4 times the length of a side of the square
= 4 × l = 4l
Thus, perimeter of a square having side l = 4l l

2. Area of a Rectangle
Area = length of the rectangle × breadth of the rectangle
Thus, area of a rectangle having length and breadth l and b respectively = l × b b
Here, l and b are variables. They can take any value independent of each other. l

Example 8: A cube is a three-dimensional figure as shown in the figure.
It has six faces and all of them are identical squares. The
length of an edge of the cube is given by l. Find the formula
for the total length of the edges of a cube. (NCERT) l
Solution: The length of an edge of the cube = l
l
Number of edges in a cube = 12 l
Therefore, the total length of the edges of the given cube = 12l.
Example 9: Explain the associative property of addition and distributive property of multiplication
over addition for three numbers a, b and c. Also verify them using a = 3, b = 5 and
c = 7.

Solution: We know that the associative property of addition is (a + b) + c = a + (b + c). Now,
let’s verify it using a = 3, b = 5 and c = 7.
LHS = (a + b) + c = (3 + 5) + 7 = 8 + 7 = 15
RHS = a + (b + c) = 3 + (5 + 7) = 3 + 12 = 15

Now, a × (b + c) = a × b + a × c is the distributive property of multiplication over
addition. Let’s verify it using a = 3, b = 5 and c = 7.
LHS = a × (b + c) = 3 × (5 + 7) = 3 × (12) = 36
RHS = a × b + a × c = 3 × 5 + 3 × 7 = 15 + 21 = 36

Algebraic Expressions and Its Formation
An algebraic expression contains variables and a constant. It cannot be evaluated as such unless
we know the value of the variables.

Example 10: Make three expressions with numbers 3, 4 and 7 (not variables) using operations of
addition, subtraction or multiplication. All numbers should be used, but not more
than once.
Solution: (a) (3 + 4) – 7 (b) 3 × (4 + 7) (c) (4 + 7) + 3
Example 11: Which out of the following are expressions with numbers only?
(a) x + 4 (b) 7y – (8 × 3) (c) 5(21 – 7) + 7 × 2 + x

(d) 8 (e) 3 + 4n (f) (3 × 51) + (2 × –3)

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