In our daily life, we see many objects like books, pencil boxes, cones, football, cylinder. These all are three-dimensional objects (solid shapes). All these objects occupy some shape and have three dimensions Length, Breadth, Height, or Depth. We often find some shapes with two or more identical (congruent) faces. For example, the cube has squared faces on each side.

**Surface Area**

The surface area of an object is the sum of all the areas of the shapes that cover the surface of the object. For example, the surface area of a cube is the sum of the area of six squares that cover it. If the area of one side is a^{2 }then the area of six sides is 6a^{2} which is its surface area.

**Cube**

Cube is a 3-dimensional box-like figure represented in the 3-dimensional plane. Cube has 6 squared-shape equal faces. Each face meets another face at 90 degrees each. Three sides of the cube meet at the same vertex.

We know that the cube consists of 6 square faces. Let us consider, if each side of a cube is a, then the,

The total surface area of the cube is 6a^{2}

**Length of Edge of Cube**

From the surface area of the cube, we can find the length of the edge of the cube by rearranging the formula,

Area = 6 * side

^{2}=> side

^{2}= Area / 6=> side = √Area / 6

**Examples**

**Example 1. If the sidewall of a cubic structure is 5m, then find the Total Surface Area?**

**Solution:**

Given, length of sidewall = 5m

Total surface Area = 6a

^{2}= 6 (5

^{2})= 6 (25)

= 150 sq.m

**Example 2. Find the length of the edge of the cube, if its Area is 294 sq.m?**

**Solution:**

Given, Area = 294 sq.m

Length of edge of cube = √ Area / 6

= √ (294 / 6)

= √ 49

= 7 m

**Example 3. What is the surface area of a cube of side 6cm?**

**Solution:**

Given,

Side of cube = a = 6cm

Surface Area of cube = 6a

^{2}= 6 * 6

^{2}cm^{2}= 216 cm

^{2}

**Example 4. Find the side of a cube whose surface area is 384 cm ^{2}?**

**Solution:**

Given,

Surface area of a cube = 384 cm

^{2}Let ‘a’ be the side of a cube .

we know that , surface area of cube = 6a

^{2}=> 6a

^{2}= 384=> a

^{2}= 384 / 6=> a

^{2}= 64=> a = √64

=> a = 8

Therefore, side of a cube = 8cm.

Note:The lateral surface area of a cube is the Total surface area of a cube, which is equal to the sum of areas of all its sides.

**Cuboid**

A cuboid is made up of six rectangular planes, with different values of length, width, and height. It may be a brick or a box which has a rectangular view from aside.** **A cuboid is a three-dimensional solid with six rectangular faces. It has 8 vertices and 12 edges. The opposite faces of cuboids are always equal.

A cuboid has three measures Total surface area, Lateral surface area, and volume. The surface areas are measured in square units.

**Total Surface Area of a Cuboid**

The Total Surface Area of a cuboid is the sum of its 6 rectangular faces, given by

2 (lw + wh + lh)

l = length, w = width, h = height

**Lateral Surface Area of a Cuboid**

The Lateral Surface Area of a cuboid is the sum of its 4 rectangular faces leaving its base and top.

LSA = 2 (lh + wh)

= 2h (l + w) sq.units

**Examples**

**Example 1. Find the total surface area of a cuboid having dimensions as length = 4cm , width = 3cm , height = 2cm?**

**Solution:**

Given h = 2cm, l = 4cm, w = 3cm

TSA = 2 (lw + wh + hl)

= 2 (4 * 3 + 3 * 2 + 2 * 4)

= 2 (12 + 6 + 8)

= 2 (26)

= 52 sq.cm

**Example 2. Find the Lateral Surface area of the cuboid, if its length, breadth, and height are 8cm, 9cm, and 10cm respectively?**

**Solution: **

Given length = 8cm

breadth = 9cm, height = 10cm

Lateral Surface Area = 2h (l + b)

= 2 * 10 (8 + 9)

= 2 * 10 (17)

= 2 * 170

= 340 cm

^{2}

**Example 3. Find the Total and Lateral Surface Areas of the cuboid with the following dimensions?**

**Length = 4cm **

**width = 5cm**

**height = 6cm**

**Solution:**

Total Surface Area = 2 (lw + wh + lh)

= 2 (4 * 5 + 5 * 6 + 4 * 6)

= 2 (20 + 30 + 24)

= 2 (74)

= 148 sq.cm

Lateral Surface Area = 2h (l + w)

= 2 * 6 (4 + 5)

= 12 (9)

= 108 sq.cm

**Cylinder**

A cylinder is a three-dimensional object with two circular bases parallel to each other. Cylinder is a curved surface with two circular bases.

- Curved Surface Area of Cylinder = 2π * r * h sq.units,
- Area of two circular bases = 2 (πr
^{2})

The Total Surface Area of a cylinder is the area occupied by its surface in three-dimensional space. It is defined as the sum of the area of curved surfaces and the Area of circular bases.

Total Surface Area of Cylinder = 2π * r * h + 2 (πr

^{2})= 2πr (h + r)

Where:r = radius of cylinder

h = height of cylinder

π (pie) = 3.14 or 22/7

**Examples**

**Example 1. Find the curved surface area of the cylinder with a diameter of 6cm and a height of 4cm?**

**Solution:**

Given,

diameter = 6cm , radius = 3cm, height = 4cm

Curved Surface Area = 2π * r * h Square Units.

= 2 * 3.14 * 3 * 4

= 75.36 cm

^{2}

**Example 2. Calculate the cost required to paint a container that is in the shape of a cylinder having a base radius of 10cm and height of 15 cm. If the painting cost of the container is INR 3 / cm ^{2}?**

**Solution: **

Given, Radius of container = 10cm

Height = 15cm

Total Surface Area of container = 2πr (h + r)

= 2 * 3.14 * 10 (15 + 10)

= 2 * 3.14 * 10 (25)

= 1570 cm

^{2}Painting cost per 1cm

^{2}= Rs. 3Total cost of painting the container = 3 * 1570

= Rs.4710

**Example 3. Find the total surface area and curved surface area of a cylinder with a diameter of 12cm and height of 10cm.**

**Solution:**

Given,

diameter = 12cm, radius = 6cm

height = 10cm

Total Surface Area = 2πr (h + r)

= 2 * 3.14 * 6 (10 + 6)

= 602.88 sq.cm

Curved Surface Area = 2π * r * h

= 2π * 6 * 10

= 376.8 sq.cm

**Example 4. If the lateral Surface Area of a cylinder is 484 cm ^{2} and its height is 12 cm, then find its radius of the base?**

**Solution:**

Given,

Lateral Surface Area of cylinder = 484

2π * r * h = 484

2 * 3.14 * r * 12 = 484

75.36 * r = 484

r = 484 / 75.36

r= 6.42

Therefore the radius of the cylinder is 6.42 cm.

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