Perimeter and Area of a Triangle
To find the area of a triangle, we need 2 things: the base and the height.
Area of a triangle
= ½ × b × h,
where b is the base and h is the height
of the triangle.
Solved Example
Question: In triangle ABC, the height is 5 cm and the
base is 6 cm. Find the area of the triangle.
Solution: Given, b = 6 and h = 5
Area of triangle ABC=½ × b × h
=½ × 6 × 5 = ½ × 30 = 15 cm^{2}.
Therefore, the area of the triangle is 15 cm^{2}.
Perimeter and Area of Quadrilaterals
Foursided polygons are called Quadrilaterals. They
have four sides, four angles and four vertices. The sum of the internal angles
of the quadrilateral is 360 degrees.
Depending on the length of the sides and measure of the
angles, there are 6 types
of quadrilaterals:
1. Rectangle
2. Square
3. Parallelogram
4. Rhombus
5. Trapezium
6. Kite
Let us study each of them in detail:
Perimeter and Area of a Rectangle
A rectangle has the opposite sides
equal and parallel to each other. Each angle of a rectangle is 90 degrees. If
ABCD is a rectangle, then Side AB = Side CD and Side BC= Side AD. One
pair of parallel sides is called the length and the other pair is called the
breadth. These are denoted by l and b, respectively.
Area of a rectangle = l × b
Perimeter and Area of a Square
A square has equal length of the sides and the equal
measure of angles. Every angle measures 90 degrees. The side is denoted
by s.
Perimeter of a square = 4 × s
Perimeter of a square = 4 × s
Area of a square = s × s
Perimeter and Area of a Parallelogram
A parallelogram has the opposite sides equal and
parallel to each other. Opposite angles of a parallelogram are equal.

Area of a parallelogram = l × h, where l is the base and h is the perpendicular height.
Perimeter and Area of a Rhombus
A rhombus is just like a square with equal sides. The
only difference is that the internal angles of a rhombus do not measure 90 degrees.
The area of rhombus can be calculated using two different formulas.
The first one is using the base and height:
Area of a rhombus
= b × h
The second one is using the length of the
diagonals:
Area of a rhombus
= (d_{1} × d_{2})/2, where d_{1} and d_{2} are
the length of the diagonals.
Perimeter of a
rhombus = 4 × s, where s is the side length of the rhombus.
Perimeter and Area of a Trapezium
Area of a trapezium
= [(a + b)h]/2, where a and b are the length of the parallel sides and h is the
height.
Perimeter of a
trapezium = sum of all the sides
Perimeter and Area of an Isosceles Trapezium
If the nonparallel sides of a trapezium are equal, then
the trapezium is called isosceles trapezium.
Area of an isosceles
trapezium = [(a + b)h]/2, where a and b are the length of the parallel sides
and h is the height.
Perimeter of an isosceles
trapezium = sum of all the sides
Perimeter and Area of a Kite
A quadrilateral with two pairs of equal adjacent sides
but unequal opposite sides is called a kite.
Perimeter of a kite = 2(a + b), where a and b are two
unequal adjacent sides.
Area of a kite = (d_{1} × d_{2})/2,
where d_{1} and d_{2} are the length of the diagonals.
Solved Examples on Perimeter and Area
Question 1: Find the perimeter and area of the
squares for the given sides.
a.
8
cm b. 15 cm c. 32 cm
Solution: a. Perimeter of a square = 4 × s = 4 ×
8 = 32 cm
Area of a square = s × s = 8 × 8 = 64 cm^{2}
b. Perimeter of a square = 4 × s = 4 × 15 = 60 cm
Area of a square = s × s = 15 × 15 = 225 cm^{2}
c. Perimeter of a square = 4 × s = 4 × 32 = 128 cm
Area of a square = s × s = 32 × 32 = 1024 cm^{2}
Question 2: Find the perimeter and area of the rectangles
for the given dimensions.
a.
8
cm by 5 cm b. 15 cm by 10 cm c. 32 cm by 30 cm
Solution: a. Perimeter of a rectangle = 2(l + b)
= 2(8 + 5) = 26 cm
Area of a rectangle = l × b = 8 × 5 = 40 cm^{2}
b. Perimeter of a rectangle = 2(l + b) = 2(15 + 10) = 50
cm
Area of a rectangle = l × b = 15 × 10 = 150 cm^{2}
c. Perimeter of a rectangle = 2(l + b) = 2(32 + 30) = 124
cm
Area of a rectangle = l × b = 32 × 30 = 960 cm^{2}
Question 3: The area of a rhombus is 24 cm^{2}
and the height is 4 cm. Find the base.
Solution: Given, area = 24 cm^{2}, and
h = 4 cm
Area of a rhombus = b × h
24 = b(4)
B = 24/4 = 6 cm
Question 4:
The height of a parallelogram is 10 cm and its base is 8 cm. Find its area.
Solution: Given, b = 8 cm and h = 10 cm
Area of a parallelogram = (b)(h) = (8)(10) = 80 cm^{2}.
Question 5: In a trapezium, the parallel sides
measure 12 cm and 7 cm, and the height is 4. Find the area of the trapezium.
Solution: Given, a = 12, b = 7 and h = 4
Area of a trapezium = [(a + b)h]/2 = [(12 + 7)4]/2
= [19 × 4]/2 = 76/2 = 38 cm^{2}
Area Formulas of Plane Shapes
Triangle

½(b)(h)

Square

s²

Rectangle

b × h

Rhombus

½(d_{1})(d_{2}), b × h

Parallelogram

b × h

Trapezium

[(a + b)h]/2

Isosceles Trapezium

[(a + b)h]/2

Kite

(d_{1} × d_{2})/2
