Area: It is defined as the surface enclosed by its sides.

Square: It is defined as a four sided shape that is made up of four straight sides that are the same lengths and that has four right angles.

Triangle: The plane figure formed by connecting three points not in a straight line by straight line segments like a three sided polygon.

(i) Sum of angles of a triangle is 180 degrees.

(ii) The sum of any two sides of a triangle is greater than the third side.

(iii) The line joining the mid – point of a side of a triangle to the opposite vertex is called the meridian.

(iv) The point where three medians of a triangle meet, is called centroid. The centroid divides each of the medians in the ratio 2 : 1.

(v) In an isosceles triangle, the altitude from the vertex bisects the base.

(vi) The median of a triangle divides it into two triangles of the same area.

(vii) The area of a triangle formed by joining the mid – points of the sides of a given triangle is one – fourth of the area of the given triangle.

Rectangle: The plane figure formed with four straight sides and four right angles, especially one with unequal adjacent sides, in contrast to a square.

Circle: A line that is curved so that its ends meet and every point on the line is the same distance from the centre.

Quadrilateral: Quadrilateral means four sided flat shape.

(i) The diagonals of a parallelogram bisect each other.

(ii) Each diagonal of a parallelogram divides it into two triangles of the same area.

(iii) The diagonals of a rectangle are equal and bisect each other.

(iv) The diagonals of a square are equal and bisect each other at right angles.

(v) The diagonals of a rhombus are unequal and bisect each other at right angles.

(vi) A parallelogram and a rectangle on the same base and between the same parallels are equal in area.

(vii) Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.

Example:
Given a square where the length of each side (edge) is 5cm. Find the area of this square.

Solution:

\(A = L^{2}\)

= \(5^{2}\)

= 5 x 5

= \(25 \ cm^{2}\)

Example:
Given a rectangle with the length of 4ft and the width of 3ft. Find its area.

Solution:

\(A = lw\)

= (4)(3)

= \(12ft^{2}\)

Example:
Given a circle with the radius 3cm. Find its area. Take \(\pi\) = 3.14.

Solution:

\(A = \pi r^{2}\)

= (3.14)\((3)^{2}\)

= (3.14)(9)

= 28.26 \(cm^{2}\)

Example:
Given a parallelogram with the base 5 in and the height 3 in. Find the area of this parallelogram.

Solution:

\(A = bh\)

= (5)(3)

= \(15 in^{2}\)

Example:
Given a triangle with the base 4cm and the height 2cm. Find its area.

Solution:

\(A = \frac{1}{2}bh\)

= \(\frac{1}{2}(4)(2)\)

= \(\frac{1}{2}(8)\)

= \(4cm^{2}\)

Example:
Given a trapezoid with the height of 4 in, and two parallel sides of 2 in and 3 in respectively. Calculate its area.

Solution:

\(A = (\frac{a + b}{2})h\)

= \((\frac{2 + 3}{2})4\)

= \((\frac{5}{2})4\)

= \(10in^{2}\)

1 Pythagoras theorem: In a right-angled triangle,
\((hypotenuse)^2\) = \((base)^2\) + \((height)^2\).

2: Area of a triangle = (length x breadth)
Therefore, length = \(\frac{area}{breadth}\) and breadth = \(\frac{area}{length}\).

3: Perimeter of a rectangle = 2(length x breadth).

4: Area of a square = \((side)^2 \) = \(\frac{1}{2}(diagonal)^2\).

5: Area of four walls of a room = 2(length x breadth) x height.

6: Area of a triangle = \(\frac{1}{2}\) x base x height.

7: Area of a triangle = \(\sqrt{s(s – a)(s – b)(s – c)}\), where \(a, b, c\) are sides
of the triangle and \(s\) = \(\frac{1}{2}(a + b + c)\).

8: Area of an equilateral triangle = \(\frac{\sqrt{3}}{4} * (side)^2\).

9: Radius of incircle of an equilateral triangle of side \(a\) is \(\frac{a}{2\sqrt{3}}\).

10: Radius of circumference of an equilateral triangle of side \(a\) is \(\frac{a}{\sqrt{3}}\).

11: Radius of incircle of a triangle of area \(\bigtriangleup\) and semi-perimeter \(s\) = \(\frac{\bigtriangleup}{s}\).

12: Area of a parallelogram = base x height.

13: Area of a trapezium = \(\frac{1}{2}\) x (sum of parallel sides) x distance between them.

14: Area of a circle = \(\pi R^2\), where R is the radius.

15: Circumference of a circle = 2\(\pi\)R.

16: Length of an arc = \(\frac{2\pi R\theta}{360}\), where \(\theta\) is the central angle.

17: Area of a sector = \(\frac{1}{2}\)(arc x R) = \(\frac{\pi R^2\theta}{360}\).

18: Area of a semi circle = \(\frac{\pi R^2}{2}\).

19: Area of isosceles triangle = \(\frac{a}{4}\sqrt{4b^2 – a^2}\)square units, where \(a, b\) are linear units.

20: Circumference of a semi-circle = \(\pi R\).

21: Side of a rhombus = \(\frac{1}{2}\sqrt{(d_{1})^2 + (d_{2})^2}\) linear units, where \(d_{1}\) and \(d_{2}\) are the lengths of diagonals.

1. The area of four walls of a room is 600\(m^2\) and its length is twice its breadth. If the height of the room is 11 m, then find the area of the floor in \(m^2\)?

Solution:

Let, breadth of a room = \(x\) m

Given,

Length of the room = 2\(x\) m

Height of the room = 11 m

Total area of 4 waits of room = 2(2\(x\) + \(x\)) x 11 \(m^2\) = 66\(m^2\)

By hypothesis, 66\(x\) = 660

⇒ \(x\) = 10 m

Hence, area of floor = 2\(x\) + \(x\) = 2\(x^2\) = 2\((10)^2\) = 2 x 100 = 200\(m^2\)

2. The perimeter of two squares are 60 cm and 44 cm. Find the perimeter of a third square whose area is equal to the difference of the areas of two squares?

Solution:

Given,

Side of first square = \(\frac{60}{4}\) = 15 cm

Side of second square = \(\frac{44}{4}\) = 11 cm

Area of third square = \((15)^2 – (11)^2\) \((cm)^2\)

= (225 – 121)\((cm)^2\)

= 104 \((cm)^2\)

Side of third square = \(\sqrt{104}\) cm = 10.19 cm

Therefore, required perimeter = (10.19 x 4) cm = 40.76 cm.

3. Find the area of the square, one of whose diagonals is 6 m long?

Solution:

Given,

Diagonal = 6 m

Now, consider

Area of square = \(\frac{1}{2}(diagonal)^2\)

⇒ Area of square = \(\frac{1}{2}(6)^2\)

⇒ Area of square = \(\frac{1}{2}36\)

⇒ Area of square = 18 \(m^2\)

Therefore, Area of square = 18 \(m^2\)

4. If the length of a certain rectangle is decreased by 5 cm and the width is increased by 4 cm, a square with the same area as the original rectangle would result . Find the perimeter of the original rectangle?

Solution:

Let,

\(x\) and \(y\) be the length and breadth of the rectangle respectively.

Then, \(x\) – 5 = \(y\) + 4

\(x\) – \(y\) = 9 ——– (i)

Now, Area of rectangle = length x breadth and

Area of square = (\(x\) – 5)(\(y\) + 4)

Therefore, (\(x\) – 5)(\(y\) + 4) = \(xy\)

⇒ 4\(x\) – 5\(y\) = 20 ——– (ii)

By soling (i) and (ii),

\(x\) = 25 and \(y\) = 16.

Therefore, Perimeter of the rectangle = 2(\(x\) + \(y\)) = 2(16 + 25) = 82 cm.

5. Find the area of the triangle whose base sides measure 10 cm, 13 cm and 15 cm?

Solution:

Given that,

Let base sides \(a\) = 10 cm, \(b\) = 13 cm and \(c\) = 15 cm

Now, consider \(s\) = \(\frac{1}{2}(a + b + c)\)

⇒ \(s\) = \(\frac{1}{2}(10 + 13 + 15)\)

⇒ \(s\) = 19

Therefore,

(s – a) = 19 – 10 = 9

(s – b) = 19 – 13 = 6

(s – c) = 19 – 15 = 4

Therefore, area = \(\sqrt{s(s – a)(s – b)(s – c)}\) = \(\sqrt{19 * 9 * 6 * 3}\) = \(\sqrt{3078}\) = 55.47 \((cm)^2\)