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Heron’s Formula
  • Last Updated : 19 Jan, 2021

Heron’s Formula is a very popular formula for finding the area of a triangle when the three sides are given. This formula was given by “Heron” in his book “Metrica”. We can apply this formula to all the types of triangles, be it right-angled, equilateral, or isosceles. The Heron’s Formula is,

A = \sqrt{s(s-a)(s-b)(s-c)}

Where,

A = Area of Triangle ABC

a, b, c = Lengths of the sides of the triangle



s = semi-perimeter = (a + b + c)/2

Heron’s Formula

Examples on Heron’s Formula

Example 1: Calculate the area of a triangle whose lengths of sides a,b and c are 14cm,13cm, and 15 cm respectively?

Solution:

Given:- a = 14cm

             b = 13cm

             c = 15cm

Firstly, we will determine semi-perimeter(s)

s = (a + b + c)/2



s = (14 + 13 + 15)/2

s = 21cm

A = \sqrt{s(s-a)(s-b)(s-c)}

A=\sqrt{21*(21-14)(21-13)(21-15)}

A = \sqrt{21*7*8*6}

A=\sqrt{3*7*7*4*4*3}

A = 7^2+4^2+3^2

A = 84cm^2

Example 2: Find the area of the triangle if the length of two sides is 11cm and 13cm and the perimeter is 32cm?

Solution:

Let a, b and c be the three sides of the triangle.

a = 11cm

b= 13 cm

c =?

Perimeter = 32cm

As we know, Perimeter equals to the sum of the length of three sides of a triangle.

Perimeter = (a + b + c)

32 = 11 + 13 + c

c = 32 – 24

c= 8cm

Now as we already know the value of perimeter,

s = perimeter/2

s = 32/2

s =16cm

a = 11cm, b = 13cm, c = 8cm, s = 16cm

A=\sqrt{s(s-a)(s-b)(s-c)}

A=\sqrt{16(16-11)(16-13)(16-8)}

A = \sqrt{16*5*3*8}

A = \sqrt{8*8*30}

A = 8\sqrt{30}cm^2

Surface Area

The area occupied by the surface of the solid object can be called a surface area. Generally, the Area is of two types:

  1. Total Surface Area
  2. Lateral Surface Area

We will discuss both of them in detail in the latter part of the article.

CUBE

A solid three-dimensional object whose length, breadth, and height all are equal is called a cube. A cube consists of  6 faces, 12 edges & 8 vertices.

Cube

Formula 1: Total Surface Area of cube = Sum of its six faces

Deriving the formula for the surface area of the cube:

Consider a cube of length l cm, breadth b cm and height h cm.

Area of face ABCD = Area of face EFGH = (l * b)cm2

Area of face AEHD = Area of face BFGC = (b * h)cm2

Area of face ABFE = Area of face DHGC = (l * h)cm2

As we know, Total Surface Area of Cube = Sum of the all the areas of all it’s six faces

Total Surface Area = 2(l * b) + 2(b * h) + 2(l * h)cm2

= 2(l * b + b * h + h * l)cm2

= 2(lb + bh + hl)cm2

As we know, length, breadth and height of a cube are always equal.

= 2(l * l + l * l + l * l)cm2

2 ∗ 3l2 cm2

Total Surface Area = 6(Edge)2

Example

Find the surface area of a cube whose edge is 8cm?

Solution:

As we know, Total surface area of cube6(Edge)2

here, Edge = 8

Total Surface Area 6 ∗ (8)2

Total Surface Area of the cube is 384 cm2

Formula 2: Lateral Surface Area of Cube = Sum of the areas of four faces(Leaving the bottom and top faces)

Lateral Surface Area Of Cube = Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC

Lateral Surface Area = 2(b * h) + 2(l * h) cm2

As we know, all sides of the cube are equal

Lateral Surface Area of cube = 2(l * l + l * l) cm2

Lateral Surface Area Of cube = 4l2 = 4(Edge)2 

Example

Find the cost of painting a cubical box of side 6cm which is open from the top at the rate of 5cm/sq?

Solution:

Side = 6cm

Lateral Surface Area of box 4(Edge)2

Lateral Surface Area = 4∗ (6)2

The lateral surface area of box = 144 cm2 

Cost of painting = 5 cm2

Total cost = 144 * 5 = 720 Rupees

CUBOID

A solid three-dimensional object having length, breadth, and height with six rectangular faces placed at the right angle. Like, cube a cuboid also consists of 6 faces, 12 edges & 8 vertices.

Cuboid

Formula 1: Total Surface Area of cuboid = Sum of its six faces

Deriving the formula for the surface area of the cuboid:

Consider a cuboid of length l cm, breadth b cm and height h cm.

Area of face ABCD = Area of face EFGH = (l * bcm^2

Area of face AEHD = Area of face BFGC = (b * hcm^2

Area of face ABFE = Area of face DHGC = (l * hcm^2

As we know, Total Surface Area of Cuboid = Sum of the all the areas of all it’s six faces

Total Surface Area = 2(l * b) + 2(b * h) + 2(l * h) cm^2

Total Surface Area of Cuboid= 2(l * b + b * h + h * l) cm^2

Total Surface Area of Cuboid = 2(length * breadth + breadth * height + height * length) cm^2

Formula 2: Lateral Surface Area of Cuboid = Sum of the areas of four faces(Leaving the bottom and top faces

Lateral Surface Area of Cuboid = Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC

Lateral Surface Area of Cuboid  = 2(b * h) + 2(l * h) cm^2

Lateral Surface Area of Cuboid  = 2(l + b) * h cm^2

Lateral Surface Area of Cuboid  = 2(Length + Breadth) * Height cm^2

Example

Calculate the Total surface area and Lateral Surface area of a chalk box of length, breadth, and height 16cm, 8cm, and 6cm respectively?

Solution:

As we know that chalk box is cuboidal in shape.

Given, length = 16cm, breadth = 8cm, height = 6cm  

Total Surface Area = 2(length * breadth + breadth * height + height * length)

Total Surface Area = 2(16 * 8 + 8 * 6 +16 * 6) cm^2

Total Surface Area = 2(128 + 48 + 96) cm^2

Total Surface Area = 544 cm^2

Lateral Surface Area = 2(Length + Breadth) * Height

Lateral Surface Area = 2(16 + 8) * 6 cm^2

Lateral Surface Area = 288 cm^2

RIGHT CIRCULAR CYLINDER

A solid shape generated by the revolution of a rectangle about one of its sides is called a right circular cylinder. Example: Straw, Rubber Pipes. A solid bounded by two coaxial cylinders of the same height and different radius is called a hollow cylinder.

Cylinder

Let R and r be the external and internal radius of a hollow cylinder and h be their height.

Formula 1: Total Surface Area = External Surface Area + Internal Surface Area + Area of top + Area of base

Deriving the formula of the surface area of a Right Circular Cylinder:

External Surface Area 2\pi Rh

Internal Surface Area 2\pi rh

Area of top\pi (R^2 - r^2)

Area of base\pi (R^2 - r^2)

Total Surface Area = 2\pi Rh + 2\pi rh+2*\pi(R^2-r^2)

Total Surface Area = 2\pi h(R+r) +2\pi(R-r)

Total Surface Area 2\pi(R+r)(h+R-r)sq.units

Formula 2: Curved(Lateral) Surface Area = External Surface Area + Internal Surface Area

Curved Surface Area =  2\pi Rh + 2\pi rh

Curved Surface Area2\pi h(R+r)sq.units

Example

Find the Total Surface Area and Curved Surface Area of a hollow right circular cylinder of height 14 cm and Internal radius = 2cm and External radius = 3cm?

Solution:

Given, Height = 14cm , r = 2cm , r = 3cm

Total Surface Area = 2\pi(R+r)(h+R-r)

Total Surface Area = 2\pi(3+2)(14+3-2)  cm^2

Total Surface Area2\pi*5*15     cm^2

As we know the value of   \pi  = 3.14(approx)

Total Surface Area = 471 cm^2

Curved Surface Area = 2\pi h(R+r)

Curved Surface Area = 2*3.14*14*5

Curved Surface Area = 439.82 cm^2

RIGHT CIRCULAR CONE

A Three-Dimensional solid object having a flat base and an apex. Example: Birthday Caps

Cone

Height = The length of the line joining a vertex to the centre of the base.

Slant height = The length of the line joining a vertex to any point of the circular edge.

Radius = Radius of the base.

Deriving Formula for the Surface area of Right Circular Cone :

Let there be a cone of radius r, height h and slant height l.

Therefore, Length of Circular edge2\pi r

Area of the plane = \pi r^2

Formula 1: Total Surface Area Of Cone = Area of the Sector + Area of the Base

Area of the Sector = 1/2 * (arc length) * (radius)

Area of the Sector1/2*2\pi r*l

Area of the Base \pi r^2

Total Surface Area =  \pi rl+\pi  r^2

Total Surface Area\pi r(l+r)

Formula 2: Curved(Lateral) Surface Area Of Cone = Area Of the Sector

Curved Surface Area\pi rl

Example

Find the Total Surface Area and Curved Surface Area of a Cone of slant height 9cm and diameter 14cm?

Solution:

Given,  slant height(l) = 9cm,  radius(r) = diameter /2 = 7cm

Curved Surface Area =  \pi rl

Curved Surface Area = 3.14 * 7 * 9 cm^2

Curved Surface Area = 197.9 cm^2

Total Surface Area = Curved Surface Area + Area of the Base

Total Surface Area = 197.9 +  \pi r^2

Total Surface Area = 197.9 + 153.9 = 351.83 cm^2

SPHERE

A Three-Dimensional solid object having all its points equidistant from a fixed point and is round in shape. Example: Ball.

Sphere & Hemisphere

Note: The sphere has only total surface area.

Let, there be a sphere of radius r.

Surface Area of Sphere =  4\pi r^2 sq.units

Hemisphere

A plane through the center of a sphere divides the sphere into two equals parts. Each of them is called a hemisphere.

Curved Surface Area Of Hemisphere =  2\pi r^2 sq.units

Total Surface Area of Hemisphere = Curved Surface Area + Area of Base

Total Surface Area of hemisphere = 2\pi r^2+ \pi r^2

Total Surface Area of Hemisphere3\pi r^2

Example

Find the area of a sphere of radius 6cm also find the Total Surface Area and Curved Surface Area if the sphere divided into two equal halves?

Solution:

Given,  radius(r) = 6cm

Area of Sphere = 4\pi r^2

Area of sphere = 4 * 3.14 * 6 * 6 cm^2

Area of sphere = 452.38 cm^2

We know if we divide a sphere into two equal parts, we eventually get two hemispheres of the same area.

Curved Area of the Hemisphere 2\pi r^2

Curved Surface Area = 2 * 3.14 * 6 * 6 = 226.19 cm^2

Total Surface Area of Hemisphere =  3\pi r^2

Total Surface Area of Hemisphere = 3 * 3.14 * 6 * 6 = 339.2  cm^2

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