**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,

Where,A = Area of Triangle ABC

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

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

### 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 = 14cmb = 13cm

c = 15cm

Firstly, we will determine semi-perimeter(s)

s = (a + b + c)/2s = (14 + 13 + 15)/2

s = 21cm

**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/2s = 32/2

s =16cm

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

## 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:

**Total Surface Area****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.

**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, breadthb cmand heighth cm.Area of face

ABCD= Area of faceEFGH= (l * b)cm^{2}Area of face

AEHD= Area of faceBFGC= (b * h)cm^{2}Area of face

ABFE= Area of faceDHGC= (l * h)cm^{2}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)cm^{2}= 2(l * b + b * h + h * l)cm

^{2}= 2(lb + bh + hl)cm

^{2}As we know,

length, breadth and height of a cube are always equal.= 2(l * l + l * l + l * l)cm

^{2}= 2 ∗ 3

l^{2 }cm^{2}

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 cube= 6(Edge)^{2}here,

Edge = 8

Total Surface Area= 6 ∗ (8)^{2}Total Surface Area of the cube is 384

cm^{2}

**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 DHGCLateral Surface Area = 2(b * h) + 2(l * h)

^{ }cm^{2}As we know,

all sides of the cube are equalLateral Surface Area of cube = 2(l * l + l * l)

^{ }cm^{2}

Lateral Surface Area Of cube =4l^{2}= 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

^{ }cm^{2}Cost of painting = 5

^{ }cm^{2}

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.

**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 faceEFGH= (l * b)Area of face

AEHD= Area of faceBFGC= (b * h)Area of face

ABFE= Area of faceDHGC= (l * h)As we know,

Total Surface Area of Cuboid = Sum of the all the areas of all it’s six facesTotal Surface Area = 2(l * b) + 2(b * h) + 2(l * h)

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

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

**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 DHGCLateral Surface Area of Cuboid = 2(b * h) + 2(l * h)

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

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

### 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)

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

Total Surface Area = 544

Lateral Surface Area = 2(Length + Breadth) * HeightLateral Surface Area = 2(16 + 8) * 6

Lateral Surface Area = 288

**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.

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=

Internal Surface Area=

Area of top=

Area of base=Total Surface Area =

Total Surface Area =

Total Surface Area=

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

Curved Surface Area =

Curved Surface Area=

### 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 = 3cmTotal Surface Area =

Total Surface Area =

Total Surface Area=As we know the value of =

3.14(approx)

Total Surface Area= 471Curved Surface Area =

Curved Surface Area = 2*3.14*14*5

Curved Surface Area = 439.82

**RIGHT CIRCULAR CONE**

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

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 edge=

Area of the plane =

**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 Sector=

Area of the Base=Total Surface Area =

Total Surface Area=

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

Curved Surface Area=

### 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 = 7cmCurved Surface Area =

Curved Surface Area = 3.14 * 7 * 9

Curved Surface Area = 197.9

Total Surface Area = Curved Surface Area + Area of the BaseTotal Surface Area = 197.9 +

Total Surface Area = 197.9 + 153.9 = 351.83

**SPHERE**

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

Note: The sphere has only total surface area.Let, there be a sphere of radius r.

Surface Area of Sphere=

**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=

Total Surface Area of Hemisphere = Curved Surface Area + Area of BaseTotal Surface Area of hemisphere =

Total Surface Area of Hemisphere=

### 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) = 6cmArea of Sphere =

Area of sphere = 4 * 3.14 * 6 * 6

Area of sphere = 452.38

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

Total Surface Area of Hemisphere =

Total Surface Area of Hemisphere = 3 * 3.14 * 6 * 6 = 339.2

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