AREA OF A TRIANGLE/COMMON SHAPES

Area of a triangle/common shapes

Area of a triangle/common shapes

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What is area?


First of all, let’s understand the basic concept of area. Area is defined as the total surface/region bounded by a 2D shape/object. (Note: Only a 2D object has area, a 3D object has volume). Now, to measure area, certain unit denotations are used. They are called square units, specifically square centimetres (cm²) and square metres (m²). 


Now, let’s look at the area of some common shapes.




Area of rectangle


The area of a rectangle which has a length l and width w has an area A = l*w. (Length is the largest side of the rectangle and width/breadth is the shorter side)








Area of Square 


The area of a square with a side length s is A = s*s. (A square is a form of rectangle whose length and breadth/width dimensions are equal)










Area of Triangle 


The area of a triangle whose height and base length are given is


A = 12b.h (Note: h is the height of the perpendicular drawn to the base of the triangle and b is the base length)








Area of Circle


The area of a circle with radius r is A = .r² (Note: If the diameter of the circle is given and it is asked to find the area, you can simply do so by dividing the diameter by 2 to get the radius and then find the area)












Area of Trapezoid


To calculate the area of a trapezoid which has individual sides of length a, b, c and d and a distance between two parallel sides as h is 


A = 12(a+c).h 










Area of Parallelogram




A parallelogram is a 2-D figure with parallel sides having equal length. So, the area of a parallelogram with base of the length b and height measurement h is 


A = b.h






Area of a Triangle if three sides of a triangle are entered


Now, let’s find out how to find the area of a triangle when all of its three sides are given. 




Let the length of the three sides be a, b and c respectively. Now to find the area, first we have to find the perimeter, which is found by


p = (a+b+c)


Now the area of the triangle is A = pp-ap-bp-c


 


This method is specially effective when finding the height of the triangle is difficult. You can just find the perimeter and then subsequently, find the area.


 


Area of a Rhombus



The area of a rhombus in a two-dimensional plane is defined as the amount of space enclosed or surrounded by a rhombus in that plane. Unlike other types of parallelograms, a rhombus has all of its sides exactly equal to one another. 


The internal angle of a rhombus distinguishes it from a square, which allows it to be distinguished from both. It is not necessary for the internal angle of a rhombus to be a right angle in particular. 


It is possible to compute the area of a rhombus in a variety of methods, depending on the characteristics that we are aware of.


 


Some properties of a rhombus are:


In order to define a rhombus, the following characteristics must be met:




  • Because all of the sides of a rhombus are of equal length, it is classified as an equilateral quadrilateral.




  • At right angles to one another, diagonals form the shape of a rhombus (see figure).




  • Angle bisectors are represented by the diagonals.




  • how to find the area of a triangle

    The area of a rhombus can be calculated in a variety of ways, including utilising the base and height of the rhombus, diagonals, and trigonometry.




 


Area of Rhombus with given Base and Height


A parallelogram is what a rhombus is. We already know that the area of a parallelogram may be calculated by multiplying the base and height. The rhombus is subjected to the same rules as the triangle.


The area of a Rhombus is equal to A = b*h


 


Area of Rhombus with known Diagonals


When two diagonals are multiplied together, the area of a rhombus is half the product of their lengths. The following is the formula for calculating the area of a rhombus by utilising diagonals:


A = d1*d22      d1 and d2 being the two diagonals.


 


Area of Rhombus with known side and angle measurements


When we know the sides and angles of a triangle, we can use the concept of trigonometry to calculate the area of the triangle. The fact that we can utilise any angle is due to the fact that the angles are either equal or supplementary, and supplementary angles have the same sine. The area of a rhombus calculated with the help of side and angle is given as:


A = side².sin(A) sq. units, where 'A' is an internal angle of the Rhombus.


 


Pythagorean Theorem


 


It is known as Pythagorean theorem, or Pythagoras' theorem, because it establishes a basic relationship between the three sides of a right triangle. 


Given a right triangle, which is defined as a triangle in which one of the angles is 90°, the Pythagorean theorem states that the area of the square formed by the longest side of the right triangle (the hypotenuse) is equal to the sum of the areas of the squares formed by the other two sides of the right triangle. The right triangle is defined as follows.


In other words, provided that the hypotenuse is the longest side of the triangle and that the other two sides are a and b, the following is true:


a²+b²=c²


The Pythagorean equation is named after the ancient Greek thinker Pythagoras, and it describes how to solve this problem. When two sides of a right triangle are known, this relationship is useful because the Pythagorean theorem may be used to compute the length of the third side.


To calculate the length of the third side, there are many Pythagorean theorem calculators available on the internet. You can use them as well.


 


Area of Triangle Worksheet


Below we have given some problems based on the area of triangle. Hope you could solve it after this discussion.




  1. Calculate the area of a triangle whose sides are 24 cm, 32 cm, and 40 cm in length, width, and height, respectively.




  2. The three sides of a triangle are in the ratio 2: 3: 4, and the perimeter of the triangle is 225 m. Find out its area.




  3. Calculate the area of a triangle whose sides are 10 cm and 9 cm in length and whose perimeter is 36 cm.




  4. The sides of a triangle are in the ratio 14: 18: 26, and the triangle's perimeter is 580 cm in circumference.  Additionally, determine the altitude that corresponds to the shortest side.



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