# Surface Area Formulas

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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 4a2 6a2
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 4πr2 4πr2
Hemisphere ½ × 4 × πr2 3πr2

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 = πr2h = 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?

Surface area of cuboid = 2(lb+bh+hl)
Volume = l × b × h
where l = length, b=breadth and h = height.

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

The total surface area of the cylinder = 2 π r(r+h), where r is the radius of the circular base and h is the height of the cylinder.

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

If r is the radius of the circular base of the cone-shaped object and h is the height, then the formula to find the volume of the cone is given by: V = 1/3π r2 h

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

The total surface area of the hemisphere is equal to the sum of half of the surface area of sphere and the area of its circular base.
Total surface area of hemisphere = 2 π r2+ π r2 = 3 π r2

## 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 = 6a2 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 = 4a2 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) units2
• Lateral surface area of cuboid, L = 2h (l + b) units2

### 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 = 4πr2 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 r2) = 2 π r, where “r” is the radius of the hemisphere.
• The total surface area of a hemisphere(TSA) = curved surface area + Base Area = 2 π r2 + π r2 = 3 π r2 , 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 yd2, 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 + 2πr2 =2πr(r + h) =2 × 22/7 × 4 × (4 + 8) =301.68 yd2 Cost of painting at$6 per yd2 = 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 inches2
∴The surface area of cone will be 282.6 inches2

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 = 6a2
a = 4 inches
On substituting values in the surface area of a cube formula.
= 6 (4)2
= 6 (16)
= 96 inches2
∴The surface area of a cube will be 96 inches2

## 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 r2) = 2 π r, 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 m2, in2, cm2, yd2, 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

1. Find the area of a triangle with a base of 40 inches and a height of 60 inches.
2. Find the area of a square with a side of 15 feet.
3. Find the surface area of Earth, which has a diameter of 7917.5 miles. Use 3.14 for PI.
4. 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.

2. Diameter = 6 miles
4. 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 s2. Since a cube has six square-shape sides, its total surface area is 6 times s2.
Surface area of a cube = 6s2

How to find the surface area of a cube using the formula?
Total surface area = 6s2 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 = 2lw

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

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

Surface area of rectangular solid = 2lw + 2lh + 2wh = 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×a2
TSA of cube = 6×a2

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 =  4πr2
TSA of sphere = 4πr2

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 = 2πr2
TSA of hemisphere = 3πr2

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 = 2πrh
TSA of cylinder = 2πr2 + 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) * Height
TSA 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.

The formula of Lateral Surface Area of Sphere = 4πr2

Given r is 4cm.

= 4×3.14 × r × r

= 4 × 3.14 × 4 × 4

= 200.96 cm2

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

Formula of Lateral Surface Area of Hemi- Sphere  = 2πr2

Given that r is 6cm

= 2 × 3.14× r ×r

= 2 * 3.14 * 6 *6

= 226.08 cm2

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

The formula of Total Surface Area of Cube = 6a2

Given a is 10cm

= 6× a ×a

= 6× 10 × 10

= 600 m2

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

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 cm2

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

Formula of Total Surface Area of Cylinder = 2πr2 + 2πrh

Given 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 cm2

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.

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 cm2

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