Surface area formulas in geometry refer to the lateral surface and total surface areas of different geometrical objects. To recall, the surface area of an object is the total area of the outside surfaces of the three-dimensional object i.e, the total sum of the area of the faces of the object. It is measured in terms of square units. In other words, the surface area is the sum of all the areas of all the shapes that cover the surface of the object. On the other hand, the lateral surface area refers to the area of the sides of a shape, excluding its base and top area.

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## Surface Area Formulas:

Shape | Lateral Surface Area (LSA) | Total Surface Area (TSA) |
---|---|---|

Cuboid | 2h(l + b) | 2(lb + bh + lh) |

Cube | 4a^{2} |
6a^{2} |

Right Prism | Base perimeter × Height | LSA + 2 (area of one end) |

Right Circular Cylinder | 2πrh | 2πr(r + h) |

Right Pyramid | (1/2) Perimeter of base × Slant Height | LSA + Area of Base |

Right Circular Cone | πrl | πr(l + r) |

Solid Sphere | 4πr^{2} |
4πr^{2} |

Hemisphere | ½ × 4 × πr^{2} |
3πr^{2} |

### Download This Formula Sheet for Surface Areas:

The following table gives the surface area formulas for solid shapes or three-dimensional shapes. Scroll down the page if you need more explanations about the formulas, how to use them as well as worksheets.

## Solved Examples

**Example 1:**

What is the surface area of a cuboid with length, width and height equal to 4.4 cm, 2.3 cm and 5 cm, respectively?

**Solution: **

Given, the dimensions of cuboid are:

length, l = 4.4 cm

width, w = 2.3 cm

height, h = 5 cm

Surface area of cuboid = 2(wl+hl+hw)

= 2·(2.3 x 4.4 + 5 x 4.4 + 5 x 2.3)

= 87.24 square cm.

**Example 2:**

What is the volume of a cylinder whose base radii are 2.1 cm and height is 30 cm?

**Solution: **

Given,

Radius of bases, r = 2.1 cm

Height of cylinder = 30 cm

Volume of cylinder = πr^{2}h = π·(2.1)^{2}·30 ≈ 416.

## Frequently Asked Questions on Surface Area and Volume

### What are the formulas for surface area and volume of cuboid?

Volume = l × b × h

where l = length, b=breadth and h = height.

### What is the total surface area of the cylinder?

### How to calculate the volume of a cone-shaped object?

^{2}h

### What is the total surface area of the hemisphere?

Total surface area of hemisphere = 2 π r

^{2}+ π r

^{2}= 3 π r

^{2}

## What Is Surface Area Formula?

The surface area formula is used to find the sum of all the surface areas of any three-dimensional object. The surface area formula is classified under two categories:

- Curved surface area formula or Lateral surface area formula
- Total surface area formula

Look at the surface area formulas chart below depicting the surface area formulas of respective 3-D shapes.

Let us learn about the general surface area formulas of various shapes in detail.

## Surface Area Formulas of Different Shapes

Any three-dimensional object has lateral surfaces and the base surface(s). The Total surface area refers to the sum of both the lateral/curved surface area and the base surface area. In this section, we will learn about the surface area formulas of various 3-D shapes.

### Surface Area Formula of Cube

The surface area of the cube is the total area covered by all six faces of the cube. The general formula of the surface area of a cube is given as:

- The total surface area formula of the cube will be the sum of the area of vertical surfaces of the cube and the area of the base. The total surface area formula of cube = 6a
^{2}where “a” is the side length. - The lateral surface area formula of a cube is the sum of areas of all lateral side faces of the cube. LSA = 4a
^{2}where “a” is the side length.

### Surface Area Formula of Cuboid

The total surface area formula of the cuboid is obtained by adding the area of all 6 faces. The total surface area and lateral surface area can be expressed in terms of its dimensions: length (l), breadth(b), and height of cuboid(h) as:

- Total surface area of cuboid, S = 2 (lb + bh + lh) units
^{2} - Lateral surface area of cuboid, L = 2h (l + b) units
^{2}

### Surface Area Formula of Cone

A cone is a 3-D shape that has a circular base with a radius “r” and diameter “d”. It has a curved surface, thus we can have its curved surface area formula as well as total surface area formula. If the radius of the base of the cone is “r” and the slant height of the cone is “l”, the surface area of a cone is given as:

- Total surface area of a cone, T = πr(r + l)
- The curved surface area of a cone, S = πrl

### Surface Area Formula of Cylinder

A cylinder has a curved surface with two circular bases placed at both ends. If the radius of the base of the cylinder is “r” and the height of the cylinder is “h”, the surface area of a cylinder is given as:

- Total surface area of cylinder, T = 2πr(h + r)
- The curved surface area of a cylinder, S = 2πrh

### Surface Area Formula of Sphere

A sphere is a three-dimensional solid object with a round structure. The surface area of a sphere is the total area of the faces surrounding it. The surface area formula of the sphere is given as:

⇒ The surface area of Sphere, S = 4πr^{2} square units.

### Surface Area Formula of Hemisphere

Hemisphere is half of a sphere. The surface area of a hemisphere is the total area its surface covers. The surface area formula of the hemisphere can be classified into two categories:

- The curved surface area of a hemisphere(CSA) = ½ (curved surface area of a sphere) = ½ (4 π r
^{2}) = 2 π r^{2 }, where “r” is the radius of the hemisphere. - The total surface area of a hemisphere(TSA) = curved surface area + Base Area = 2 π r
^{2}+ π r^{2}= 3 π r^{2}, where “r” is the radius of the hemisphere.

### Surface Area Formula of Prism

The lateral surface area of a prism is the sum of the areas of all its lateral faces whereas the total surface area of a prism is the sum of its lateral area and the area of its bases. The surface area formulas of the prism can be given

- The lateral surface area of prism = base perimeter × height
- The total surface area of a prism = Lateral surface area of prism + area of the two bases = (2 × Base Area) + Lateral surface area or (2 × Base Area) + (Base perimeter × height).

There are seven types of prisms based on the shape of the bases of prisms. The bases of different types of prisms are different so are the formulas to determine the surface area of the prism. Check out prism to understand the concept behind the surface area formulas of various prisms.

### Surface Area Formula of Pyramid

If a pyramid has a regular polygon base with altitude passing through the center of the base, then the lateral surface area and total surface area formulas for the pyramid can be given as:

Consider a regular pyramid whose base perimeter is ‘P’, the base area is ‘B’, and the slant height (the height of each triangle) is ‘s’. Then,

- The lateral surface area of pyramid (LSA) = (1/2) Ps
- The total surface area of pyramid (TSA) = LSA + base area = (1/2) Ps + B

## Examples Using Surface Area Formula

**Example 1: **A** **cylindrical tank has a radius 4 yd and height 8 yd, using the surface area formula of the cylinder find its surface area. If the cost of the painting cylindrical tank is $6 per yd^{2}, what will be the total cost of the painting?

**Solution:**

We know that total Surface Area formula of cylinder = Curved Surface Area of cylinder + area of the top and bottom faces

=2πrh + 2πr^{2}

=2πr(r + h)

=2 × 22/7 × 4 × (4 + 8)

=301.68 yd^{2}

Cost of painting at $6 per yd^{2} = 301.68 × 6 = $1810.08**The cost of the painting is $1810.08. **

**Example 2:** Given that the radius of a cone is 6 inches and the slant height of a cone is 9 inches. Using the total surface area formula of the cone calculate the surface area of the cone.

**Solution:**

Given: Radius = 6 inches and slant height = 9 inches

The total surface area formula of cone = T = πr(r + l)

=3.14 × 6 × (6 + 9)

=282.6 inches^{2}**∴The surface area of cone will be 282.6 inches ^{2}**

**Example 3: **Using the surface area formula of the cube find the surface area of the cube whose side is 4 inches.

**Solution:** Given side length of cube = 4 inches

The surface area formula of cube = 6a^{2}

a = 4 inches

On substituting values in the surface area of a cube formula.

= 6 (4)^{2}

= 6 (16)

= 96 inches^{2}**∴The surface area of a cube will be 96 inches ^{2}**

## FAQs on Surface Area Formulas

### What Is the Surface Area Formula for Cuboid?

The surface area formula of the cuboid is 2(lb + bh + hl). Here “l”, “b”, and “h” denote the 3 dimensions: length, breadth, and height of the cuboid.

### What Is the Relation Between Curved Surface Area Formula for Sphere And Hemisphere?

The curved surface area formula of a hemisphere is half of the curved surface area of a sphere. It is given as:

CSA of hemisphere = ½ (curved surface area of a sphere) = ½ (4 π r^{2}) = 2 π r^{2 }, where “r” is the radius of the hemisphere/ sphere.

### What Is the Surface Area Formula of a Cone?

The total surface area formula of a cone is given as, T = πr(r + l).

The curved surface area formula of a cone is given as, S = πrl.

Here “r” is the radius of the base of the cone and “l” is the slant height of the cone.

### What Is the Surface Area Formula of a Cylinder?

The surface area formula of a cylinder is the total region covered by the surface of the cylindrical shape. It is mathematically expressed as 2πr(h+r), where, ‘r’ is the radius of the circular base of the cylinder and ‘h’ is the height of the cylinder. The surface area of a cylinder is given in square units, like m^{2}, in^{2}, cm^{2}, yd^{2}, etc.

## Area, Surface Area and Volume Formulas

Example 6.5.4

Find the surface area of a rectangular pyramid with a slant height of 10 yards, a base width (b) of 8 yards and a base length (h) of 12 yards.

## Partner Activity 1

- Find the area of a triangle with a base of 40 inches and a height of 60 inches.
- Find the area of a square with a side of 15 feet.
- Find the surface area of Earth, which has a diameter of 7917.5 miles. Use 3.14 for PI.
- Find the volume of a can a soup, which has a radius of 2 inches and a height of 3 inches. Use 3.14 for PI.

## Practice Problems

**(Problems 1 – 4) Find the area of each circle with the given parameters. Use 3.14 for PI. Round your answer to the nearest tenth.**

- Radius = 9 cm
- Diameter = 6 miles
- Radius = 8.6 cm
- Diameter = 14 meters

**(Problems 5 – 8) Find the area of each polygon. Round answers to the nearest tenth.**

12

**(Problems 13 – 17) Find the surface area of each figure. Leave your answers in terms of PI, if the answer contains PI. Round all other answers to the nearest hundredth.**

13

14

15

17

### Surface Area Of A Cube

A cube is a three-dimensional figure with six equal square sides. The figure below shows a cube with sides s.

If *s* is the length of one of its sides, then the area of each side of a cube is *s*^{2}. Since a cube has six square-shape sides, its total surface area is 6 times *s*^{2}.

Surface area of a cube = 6*s*^{2}

**How to find the surface area of a cube using the formula?**

Total surface area = 6*s*^{2} where *s* is the length of a side.

### Rectangular Solid Or Cuboids

A rectangular solid is also called a rectangular prism or a cuboid. In a rectangular solid, the length, width and height may be of different lengths.

The surface area of the above cuboid would be the sum of the area of all the surfaces which are rectangles.

Total area of top and bottom surfaces is *lw* + *lw = *2*lw*

Total area of front and back surfaces is *lh* + *lh = *2*lh*

Total area of the two side surfaces is *wh* + *wh = *2*wh*

Surface area of rectangular solid = 2*lw* + 2*lh* + 2*wh* = 2(*lw* + *lh* + *wh*)

**Formulas of Surface Area of Different Geometrical Figures**

The formulas for Lateral Surface Area (LSA) and Total Surface Area(TSA) of different 3-D Geometrical Figures are given below

**Surface area of Cube**

- A cube is a symmetrical figure consisting of six square side faces.
- Here each face has ‘a’ units of length as sides.

LSA of cube = 4×a^{2}TSA of cube = 6×a^{2}

**Surface area of Cuboid**

Cuboid

- A cuboid is a geometrical figure consisting of six rectangular side faces.
- Here faces have ‘l’ units of length, ‘b’ units of breadth, and ‘h’ units of height.

LSA of cube = 2×(hl + bh)TSA of cube = 2×(hl + bh + bh)

**3. Surface area of Sphere**

Sphere

- The sphere is a 3D circular figure.
- Here, the radius has ‘r’ units of length.

LSA of sphere = 4πr^{2}TSA of sphere = 4πr^{2}Here, TSA of sphere = LSA of sphere

**Surface area of the Hemisphere**

Hemisphere

- When a sphere gets sliced into two equal parts, each part is called a hemisphere.
- Here, the radius has ‘r’ units of length.

LSA of hemisphere = 2πr^{2}TSA of hemisphere = 3πr^{2}

**Surface Area of Cylinder**

Cylinder

- A cylinder is a 3D figure with two circular bases.
- Here circular base has ‘r’ units of radius and ‘h’ units of height.

LSA of cylinder = 2πrhTSA of cylinder = 2πr^{2}+ 2πrh

**Surface area of Pyramid**

Pyramid

- A pyramid is a 3D figure having triangular sides on a single given base.
- Here, the base can be a triangular, square, pentagon, or any 2-Dimensional shape.

LSA of Pyramid = 1/2 * (Perimeter of base) * HeightTSA of Pyramid = [ 1/2 * (Perimeter of base) * Height ] + Area of base

### Sample Questions

**Question 1: Find the Lateral surface of a Sphere with a radius of 4 cm.**

**Answer:**

The formula of Lateral Surface Area of Sphere = 4πr

^{2}.Given r is 4cm.

= 4×3.14 × r × r

= 4 × 3.14 × 4 × 4

= 200.96 cm

^{2}

**Question 2: Find the Total surface of a Hemi- Sphere with a radius of 6 cm.**

**Answer:**

Formula of Lateral Surface Area of Hemi- Sphere = 2πr

^{2}.Given that r is 6cm

= 2 × 3.14× r ×r

= 2 * 3.14 * 6 *6

= 226.08 cm

^{2}

**Question 3: Find the Total surface of a Cube with a side of 10 m.**

Answer:The formula of Total Surface Area of Cube = 6a

^{2}Given a is 10cm

= 6× a ×a

= 6× 10 × 10

= 600 m

^{2}

**Question 4: Find the Lateral surface of a Cuboid with a length of 10 cm, breadth of 8 cm, and height of 6 cm.**

**Answer:**

Formula of Lateral Surface Area of Cuboid = 2×(hl + bh)

Given l is 10cm, b is 8cm, h is 6cm

= 2× (h×l + b×h)

= 2× (6×10 + 8×6)

= 2× (60 + 48)

= 2×(108)

= 216 cm

^{2}

**Question 5: Find the Total surface of a Cylinder with a radius of 4 cm and height of 6 cm.**

**Answer:**

Formula of Total Surface Area of Cylinder = 2πr

^{2}+ 2πrhGiven r is 4cm and h is 6cm

= 2×3.14×r×r + 2×3.14×r×h

= 2×3.14×4×4 + 2×3.14×4×6

= 100.48 + 150.72

= 251.2 cm

^{2}

**Question 6: Find the Lateral surface of a Pyramid with the base as an equilateral triangle of side 5 cm and height of 8 cm.**

**Answer:**

Formula of Lateral Surface Area of Pyramid = 1/2 × (Perimeter of base) × Height

Given h is 8cm and the base is an equilateral triangle with a side of 5cm

= 1/2 × (Perimeter of base) ×Height

= 1/2 × (5 + 5 + 5) ×Height

= 1/2 × (5 + 5 + 5) × 8

= 1/2 × (15) × 8

= 60 cm

^{2}

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