Perimeter and Area of Plane Figures

Perimeter and Area of a Triangle

Perimeter of a triangle = a + b + c, where a, b and c are the sides of the 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 cm².

Therefore, the area of the triangle is 15 cm².

Perimeter and Area of Quadrilaterals

Four-sided 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.

Perimeter of a rectangle = 2(l + b)

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

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.

Perimeter of a parallelogram = 2(l + b)

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 = (d₁ × d₂)/2, where d₁ and d₂ 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

A trapezium has one pair of opposite sides parallel while the other is not parallel. To calculate the height, you need to draw a perpendicular from one parallel side to other.

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 non-parallel 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 = (d₁ × d₂)/2, where d₁ and d₂ 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 cm²

b. Perimeter of a square = 4 × s = 4 × 15 = 60 cm

Area of a square = s × s = 15 × 15 = 225 cm²

c. Perimeter of a square = 4 × s = 4 × 32 = 128 cm

Area of a square = s × s = 32 × 32 = 1024 cm²

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 cm²

b. Perimeter of a rectangle = 2(l + b) = 2(15 + 10) = 50 cm

Area of a rectangle = l × b = 15 × 10 = 150 cm²

c. Perimeter of a rectangle = 2(l + b) = 2(32 + 30) = 124 cm

Area of a rectangle = l × b = 32 × 30 = 960 cm²

Question 3: The area of a rhombus is 24 cm² and the height is 4 cm. Find the base.

Solution: Given, area = 24 cm², 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 cm².

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 cm²

Area Formulas of Plane Shapes

Triangle	½(b)(h)
Square	s²
Rectangle	b × h
Rhombus	½(d₁)(d₂), b × h
Parallelogram	b × h
Trapezium	[(a + b)h]/2
Isosceles Trapezium	[(a + b)h]/2
Kite	(d₁ × d₂)/2