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Surface Area Formulas

Last Updated : 19 Feb, 2024
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Surface Area Formulas are the formulas in mensuration that help us calculate the surface area of any 3D geometric shape. Surface area refers to space occupied by the three-dimensional shape. It is denoted by the sum of the individual surfaces of the sides of a three-dimensional figure. The surface area of 3-D figures is of two types, Lateral Surface Area/Curved Surface Area and Total Surface Area.

Let’s learn the Surface Area Formulas of various geometric figures.

Surface Area Definition

Surface Area of any figure is defined as the area of the faces of the figure. It is the total area of all the faces of the figure. Surface Area can be calculated for both 2-D figures and 3-D figures. For 3-D figures, we can have two types of Surface Areas, i.e. Lateral/Curved Surface Area, and Total Surface Area.

Aspect Lateral Surface Area (LSA) / Curved Surface Area (CSA) Total Surface Area
Definition The area of the curved or side surfaces of a figure. The area of all surfaces of the figure, including the top, base, and sides.
Also Known As Curved Surface Area

TSA

Formula (General Concept) LSA = Area of Side Faces Total Surface Area = LSA + Area of Top Surface + Area of Base Surface
Application Used for objects with curved sides like cylinders, cones, etc. Used for all 3D figures to determine the complete outer area.

Surface Area Formulas

Surface Area Formulas is given for total surface area and the lateral surface area. The total surface area includes the area of all surfaces of the figure/ object (base + sides) whereas the lateral surface area of geometrical figures includes the only surface of the sides. There are various surface area formulas and some of the surface area of the important figures are added in table below:

Surface-Area-Formulas

Surface Area Formulas

Surface Area Formula List

The following table contains the surface area formulas of different shapes

Shape

Figure

Lateral Surface Area (LSA)

Total Surface Area (TSA)

Cube

Surface Area of Cube

4a2

6a2

Cuboid

Surface Area of Cuboid

2h(l+b)

2(lb + lh + bh)

Cylinder

Surface Area of Cylinder

2πrh

2πr(r + h)

Cone

Surface Area of Cone

πrl

πr(l + r)

Sphere

Surface Area of Sphere

4πr2

4πr2

Hemisphere

Surface Area of Hemisphere

2πr2

3πr2

Pyramid

Surface Area of Pyramid

1/2 × (Base Perimeter) × (Slant Height)

LSA + Area of Base

Prism

Surface Area of Prism

(Base Perimeter) × (Height)

LSA + 2(Area of Base)

Surface Area of Different Shapes

Let’s discuss the formulas for Lateral Surface Area (LSA) and Total Surface Area (TSA) of different 3D Geometrical Figures below:

Surface Area Formula of Cube

A cube is six faced 3D shape in which all the faces are equal. A cube is a three-dimensional shape with several key characteristics:

  1. Faces: It has six square faces, all the same size and shape.
  2. Edges: It has twelve edges, each connecting two adjacent faces.
  3. Vertices: It has eight corners, where three edges meet.
  4. Properties: All its angles are right angles (90 degrees), and opposite faces are parallel.

Here are some additional details about cubes:

  • Regular hexahedron: It is also known as a regular hexahedron because all its faces are regular polygons (squares) and all its edges are the same length.
  • Platonic solid: It is one of the five Platonic solids, which are regular solids with specific properties.

The following image shows a typical cube:

Cube Surface Area

Formulas for Surface Area of Cube are given by:

Lateral Surface Area (LSA) of Cube =  4a2

Total Surface Area (TSA) of Cube = 6a2

where:

  • a is Side of a Cube

Surface Area Formula of Cuboid

Cuboid is a 3D figure in which opposite faces are equal. A cuboid, also known as a rectangular prism, is a 3D geometric shape very similar to a cube, but with some key differences:

  • Faces: Similar to a cube, a cuboid has six faces, but unlike a cube, these faces are rectangles instead of squares. So, they can have different lengths and widths.
  • Edges: It still has twelve edges, connecting the faces, but unlike a cube, not all edges have to be the same length.
  • Vertices: Like a cube, it has eight corners or vertices where three edges meet.
  • Properties: While not every edge is equal, opposite faces are still parallel and angles remain right angles (90 degrees).

The following image shows a typical cuboid:

Cuboid Surface Area

Formulas for Surface Area of Cuboid are given by:

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

Total Surface Area (TSA) of Cuboid = 2 × (hl + bh + bh)

where:

  • l is Length of Cuboid
  • b is Breadth of Cuboid
  • h is Height of Cuboid

Surface Area Formula of a Sphere

Sphere is 3D figure that is similar to real life ball. A sphere is a three-dimensional, perfectly round object with several key characteristics:

  1. Surface: It has a smooth, curved surface with no edges or corners. Every point on the surface is the same distance away from the center of the sphere. This distance is called the radius.
  2. Shape: Imagine cutting a circle out of a piece of paper and then rotating it around its center 360 degrees. The resulting solid shape is a sphere.

Other properties:

  • Symmetry: Spheres are highly symmetrical, meaning they look the same from any angle.
  • Minimising surface area: Spheres have the smallest possible surface area for a given volume. This is why bubbles and water droplets tend to be spherical in nature.

The following image shows a typical sphere:

Sphere Surface Area

Formula for the Surface Area of Sphere is:

Surface Area of Sphere =  4πr2

where:

  • r is Radius of Sphere

Surface Area Formula of a Hemisphere

Hemisphere is 3D figure that is half of the Sphere. It is created by slicing it through its center with a flat plane.

Key Details:

  1. Shape: It has one smoothly curved surface and one flat circular base. Unlike a sphere, it has an edge where the curved surface meets the flat base.
  2. Properties: Just like a sphere, it has no vertices or corners. The line segment connecting two opposite points on the base and passing through the center is its diameter. The line segment from the center to any point on the curved surface is the radius.
  3. Dividing a sphere: One sphere can be divided into exactly two hemispheres.

The following image shows a typical hemisphere:

Hemisphere Surface Area

Surface Area of Hemisphere formula is:

Curved Surface Area (CSA) of Hemisphere = 2πr2

Total Surface Area (TSA) of Hemisphere = 3πr2

where:

  • r is Radius of Sphere

Surface Area Formula of a Cylinder

A cylinder is a 3D figure with two circular bases and a curved surface.

Key Details:

  1. Faces: It has two circular bases, perfectly flat and congruent (identical in shape and size) to each other.
  2. Curved surface: Connecting the two bases is a smoothly curved surface, like rolling a rectangle and connecting the longer sides.
  3. Types of cylinders: While the classic type has circular bases, other variations exist, like elliptical cylinders where the bases are ellipses instead of circles.

The following image shows a typical cylinder:

Cylinder Surface Area

Surface Area of Cylinder formula is:

Curved Surface Area (CSA) of Cylinder = 2πrh

Total Surface Area (TSA) of Cylinder = 2πr2 + 2πrh = 2πr(r+h)

where:

  • r is Radius of base of Cylinder
  • H is Height of Cylinder

Surface Area Formula of a Cone

A cone is 3D geometric shape with a circular base and a pointed edge at the top called the apex. A cone has one face and a vertex.

Key Details:

  1. Base: It has one base, which is typically circular (but can also be elliptical in some cases). This base is flat and forms the bottom of the cone.
  2. Apex: It has a single point at the top, called the apex or vertex.
  3. Slant height: This is the shortest distance from the apex to any point on the circumference of the base.
  4. Height: This is the distance from the apex to the center of the base, perpendicular to the base.
  5. Types of cones: The most common type is the right circular cone where the base is a circle and the height forms a right angle with the base. Other types include oblique cones and elliptical cones.

The following image shows a typical cone:

Cone Surface Area

The Surface Area of Cone formulas is:

Curved Surface Area (CSA) of Cone = πrl

Total Surface Area (TSA) of Cone = πr(r + l)

where:

  • r is Radius of Base of Cone
  • l is Slant Height of Cone

Surface Area Formula of Pyramid

A pyramid is a 3D figure having triangular faces and a triangular base. It is a three-dimensional polyhedron with a polygonal base and triangular sides that meet at a common point called the apex.

Key Features:

  1. Base: The base can be any polygon shape, like triangular, square, pentagonal, hexagonal, or even more complex shapes. The most common type of pyramid, however, has a square base.
  2. Sides: Each side of a pyramid, except for the base, is a triangle. These triangular sides are called lateral faces.
  3. Apex: The top point where all the lateral faces meet is called the apex.
  4. Edges: The lines where two faces meet are called edges. A pyramid has the same number of edges as the perimeter of its base.
  5. Properties: Unlike prisms, pyramids have only one base. All their faces (except the base) come to a point at the apex. Some pyramids have right angles where the lateral faces meet the base, while others have slanted sides.
  6. Types of pyramids: There are different types of pyramids classified based on the shape of their base and the angles of their sides. Some common types include regular pyramids (all base sides equal), right pyramids (base is perpendicular to the apex), and oblique pyramids (base is not perpendicular to the apex).

The following image shows a typical pyramid:

Pyramid Surface Area

The Surface Area of Pyramid formula is:

Lateral Surface Area (LSA) of Pyramid =  1/2 × (Perimeter of Base) × Height

Total Surface Area (TSA) of Pyramid = [1/2 × (Perimeter of Base) × Height] + Area of Base

Solved Questions on Surface Area Formulas

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

Solution:

Given,

  • Radius of Sphere (r) = 4 cm

Formula of Lateral Surface Area of Sphere = 4πr2

LSA = 4 × 3.14 × r × r = 4 × 3.14 × 4 × 4

LSA = 200.96 cm2

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

Solution:

Given,

  • Radius of Hemisphere (r) = 6 cm

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

LSA = 2 × 3.14× r × r = 2 × 3.14 × 6 × 6

LSA = 226.08 cm2

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

Solution:

Given,

  • Side of Cube (a) = 10 cm

Formula of Total Surface Area of Cube = 6a2

TSA = 6 × a × a = 6 × 10 × 10

TSA = 600 m2

Related:

Practice Questions on Surface Area Formulas

Q1. Find the surface area of cube of side 22 m.

Q2. Find the surface area of cuboid with dimensions length, breadth, and height to be 10, 12, 1and 14 units.

Q3. Find the surface area of cylinder with base radius 14 m and height 10 m.

Q4. Find the surface area of cone with base radius 10 mm and height of cone is 12 mm.

Surface Area Formulas MCQs Practice Problems

To learn more about Surface Area Formulas Practice Surface Area and Volume Quiz

Practice Problems on Surface Area of Shapes

1. What is the formula for finding the surface area of a cube?

  1. 4a
  2. 6a2
  3. 8a
  4. 3a2

2. Which of the following is the formula for calculating the surface area of a cylinder?

  1. 2πr
  2. 2πr2
  3. πr2h
  4. πrh

3. What is the formula for the surface area of a rectangular prism?

  1. 2(l + w)
  2. lwh
  3. 2lw + 2lh + 2wh
  4. l2 + w2 + h2

4. Which formula represents the surface area of a sphere?

  1. 4πr2
  2. 2πr2
  3. πr2
  4. (4/3)πr3

5. What is the surface area of a cone with radius ‘r’ and slant height ‘l’?

  1. πr2
  2. πrl
  3. 2πr2 + πr2
  4. 2πr2 + πrl

6. The surface area of a pyramid with a square base is calculated by which formula?

  1. 4s
  2. s2
  3. 2s2
  4. 2s2 + 4s

7. What is the surface area of a triangular prism with base area ‘B’ and height ‘h’?

  1. Bh
  2. 2B + 3h
  3. Bh + 2B
  4. 2Bh + 2B

8. How do you find the surface area of a regular hexagonal prism?

  1. 6s2
  2. 3s2√3
  3. 6s2√3
  4. 3s2

9. The surface area of a regular tetrahedron is calculated by which formula?

  1. s2√3
  2. 3s2
  3. 2s2
  4. 4s2

10. Which formula represents the surface area of a rectangular pyramid?

  1. (lwh)/2
  2. lwh
  3. 2lw + 2lh + 2wh
  4. l2 + w2 + h2

Answers

1. 6a2

6. 2s2 + 4s

2. 2πr2

7. Bh + 2B

3. 2lw + 2lh + 2wh

8. 6s2√3

4. 4πr2

9. s2√3

5. 2πr2 + πrl

10. (lwh)/2

FAQs on Surface Area Formulas

What is Surface Area Formula?

Surface area formulas are the formulas that are used to find the Lateral(Curved) Surface Area and Total Surface Area of various figures.

What is Surface Area of Cube Formula?

For a cube of side “a”, Surface Area of Cube is calculated using the formula,

Surface Area of Cube = 6a2

What is Surface Area of Cuboid Formula?

For a cuboid of side “l”, “b”, and “h”, Surface Area of Cuboid is calculated using the formula,

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

What is Surface Area of Cone Formula?

For a cone of base radius “r” and slant height “l”, Surface Area Formulas of Cone is calculated using the formula, Total Surface Area of Cone = πr(r + l) and Lateral Surface Area = πrl

What is Surface Area of Cylinder Formula?

For a cylinder of base radius “r” and height (h), Surface Area of Cylinder is calculated using the formula, Total Surface Area of Cylinder = 2πr(h + r) and Lateral Surface Area = 2πrh

What is Volume of a 3 D Figure?

Volume of the 3-D figure is the total space occupied by the 3-D figure. It is also explained as the amount of material required to make that solid figure. Formulas for the volume of some common figures are,

  • Volume of Cylinder = πr2h
  • Volume of Cone = 1/3πr2h
  • Volume of Cube = a3
  • Volume of Cubiod = l.b.h

What is Surface Area of Sphere?

The equation that gives the surface area of sphere is,

Surface Area of Sphere = 6πr2

What is Surface Area of Hemisphere Formula?

The Surface Area Formula of Hemisphere is

Surface Area of Hemisphere = 3πr2

What is Surface Area of Prism Formula?

The Surface Area Formulas of Prism is,

Surface Area of Prism = (Perimeter of Base) × (Height)

What is Surface Area of Triangular Prism Formula?

The surface area formulas for Triangular Prism is given as, Total Surface Area = (Perimeter × Length) + (2 × Base Area) and Lateral Surface Area = Perimeter of base × Length



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