Coordinate Geometry

Areas of Triangle and Quadrilateral

Do you think finding the area of a triangle and a quadrilateral out there is a rocket science? Then you are sorely mistaken. It’s actually a piece of cake. Here’s how we can have that piece easily. Let’s get right into it!

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Area of a Triangle and a Quadrilateral

Before jumping straight into finding the area of a triangle and a quadrilateral, let us first brush up on the basics.

Triangles

A triangle is a polygon with 3 sides, 3 vertice, and 3 angles. Breaking down the word tri/angle it literally means something that has 3 angles.  It is one of the basic shapes of geometry. The sum of the internal angles of the triangles is 180 degrees.

Types of Triangle:

  1. Equilateral triangle: Has equal sides and angles. The measure of each angle is 60 degrees.
  2. Isosceles triangle: Has 2 equal sides and 2 equal angles. The angle formed by the equal sides is different.
  3. Scalene triangle: Does not have any congruent side. Length of every side is different.
  4. Acute triangle: All the angles of these triangles are less than 90 degrees.
  5. Obtuse triangle: Only one of the angles is more than 90 degrees.
  6. Right-angled Triangle: One angle has the measure of 90 degrees.

Area of a triangle

To find the area of a triangle you need 2 things: the base and the height. The height of the triangle is the perpendicular drawn on the base of the triangle. Formula for finding the area is,

Area of Triangle= ½ × base× height

Learn more about Section Formula here in detail.

Solved Example

Q:  In triangle ABC the height is denoted by h and its value is 5. The base is 4. Find the area of the triangle.

Sol: Given, b=4, and h=5

To find: A(triangle ABC)

Solution: Area of triangle ABC=½ ×b×h

=½ ×4×5 =½ ×20=10.

Therefore the area of the triangle is 10.

Quadrilaterals

Four-sided polygons are called Quadrilaterals. They have four sides, four vertices, and four angles. The sum of the internal angles of the quadrilateral is 360 degree.

Depending on the length of the sides and measure of the angle there are seven types of quadrilaterals:

  1. Square
  2. Rectangle
  3. Rhombus
  4. Parallelogram
  5. Trapezium
  6. Isosceles Trapezium
  7. Kite

Let us study each of them in detail:

Square

This quadrilateral has equal length of the sides and the equal measure of angles. Every angle is 90 degrees. The side is denoted by s. The opposite sides are parallel to each other.

Area of Square= s×s

Rectangle

This type of quadrilaterals has the opposite sides equal to each other in length and are parallel. The angle of each angle of a rectangle is 90 degrees. If ABCD is a quadrilateral then Side AB = Side CD and Side BC= Side AD.  One pair of parallel sides is called the length while the other pair is the breadth. These are denoted by l and b.

Area of Rectangle = l×b

Rhombus

A rhombus is just like a square with equal sides. The only difference is that the internal angles do not form a 90-degree angle. So a rhombus is also called as a slanted square. The area of rhombus can be calculated using 2 different formulae.

The first one is using the base and height

Area of rhombus= b×h

The second one is using the length of the diagonals,

Area of rhombus= (d1×d2)/2

Parallelogram

A parallelogram is a slanted rectangle with the length of the opposite sides being equal just like a rectangle.

Area of parallelogram= b×h

Trapezium

A trapezium has one pair of sides parallel while the other isn’t. A trapezium is basically a triangle with the top sliced off.  The parallel sides of this quadrilateral are called the base and to calculate the height you need to draw a perpendicular from one parallel side to other.

The formula for area of a trapezium is,

Area of Trapezium= [(a+b)h]/2

Isosceles Trapezium

A trapezium with a line of symmetry dividing it. Both the parts of the trapezium look like mirror images of each other. The area of isosceles trapezium is,

Area of Isosceles Trapezium=[(a+b)h]/2

Kite

A quadrilateral with two pairs of equal adjacent sides but unequal opposite sides.

Area of a Kite= (d1×d2)/2

Solved Examples

Q: The area of a rhombus is 12 and the height is 6. Find the base.

Sol: Given, area=12, and h=6

To find: base b

Formula: b×h

Area of rhombus=b×h

12=b(6)

b=12/6 = 2

 

Q: The height of a parallelogram is 15 and its base is 12. Find the area.

Sol: Given, b=12 and h=15

To find: A(parallelogram)

Formula: b×h

Area of parallelogram= (b)(h) = (12)(15) = 180 square units

 

Q: In a trapezium, the base1 is 7, base2 is 9 and the height is 2. Find the area of the trapezium.

Sol: Given, a=7, h=2 and b=9

To find: A(trapezium)

Formula:[(a+b)h]/2

Area of trapezium= [(a+b)h]/2 = [(7+9)2]/2

= [16×2]/2 = 32/2 = 1

Formuale At A Glance

 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
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Derivation of section formula in all four quadrants

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