Surface Area Formulas - Derivation, Examples (2022)

The surface area formula is a mathematical solution to find the total area ofany three-dimensional object occupied by all of its surfaces. Let us understand in detail about surface area formulas of different 3-D shapes.

What Is Surface Area Formula?

The surface area formula is usedto find the sum of all the surface areas of any three-dimensional object. The surfacearea formula is classifiedundertwocategories:

  • 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.

Surface Area Formulas - Derivation, Examples (1)

(Video) Area of Surfaces of Revolution | Derivation & Example

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 isthe 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 = 6a2where "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 = 4a2where "a" is the side length.

Surface Area Formula of Cuboid

The total surfacearea formula of the cuboid is obtained by adding the area of all 6faces.The total surface area and lateral surface area can be expressed in termsof 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 witha radius "r" and diameter "d". It has a curved surface, thus we can haveits curved surface area formula as well as total surface area formula.If theradiusof the base of the cone is "r" and the slant height of the cone is "l",the surface areaof aconeis 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 cylinderhas a curved surface with two circular bases placedat both ends. If theradiusof the base of the cylinder is "r" and the height of the cylinderis "h",the surface areaof a cylinderis 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 roundstructure.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 =4πr2square 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:

(Video) The Surface Area formula for Parametric Surfaces // Vector Calculus

  • The curved surface area of a hemisphere(CSA) =½ (curved surface area of a sphere) = ½ (4 π r2) = 2π r2, where "r" is the radius of the hemisphere.
  • The total surface area of a hemisphere(TSA) =curved surface area + Base Area =2π r2+π r2= 3 π r2, where "r" is the radius of the hemisphere.

Surface Area Formula ofPrism

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 prismcan be given

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

There are seventypes ofprismsbased on the shape of the bases of prisms. Thebases of different types of prisms are different so arethe formulas to determinethe surface areaof the prism. Check out prism tounderstand theconcept behind the surface area formulas of various prisms.

Surface Area Formula ofPyramid

If a pyramid hasaregular polygonbase withaltitude passingthrough the center of the base, thenthe lateral surface area and total surface area formulas for the pyramid canbegiven as:

Consider a regular pyramid whose baseperimeteris 'P', the base area is 'B', and theslant 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

Surface Area Formulas - Derivation, Examples (2)

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(Video) 82. Surface Area - Part 1 - Derivation of the Formula, Examples

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Examples UsingSurface Area Formula

Example 1: Acylindrical tank has aradius 4ydand height 8yd, using the surface area formula of the cylinderfind its surface area.If the cost of the painting cylindrical tankis $6peryd2,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)
=301.68yd2
Cost of painting at $6per yd2= 301.68× 6= $1810.08
The cost of the painting is$1810.08.

Example 2:Given that the radius of a cone is 6inches and theslantheight of a cone is9inches. Using the total surface area formula of the cone calculate the surface area of the cone.

Solution:

Given: Radius = 6inches and slant height = 9inches
The total surface area formula of cone =T = πr(r + l)
=3.14 × 6 × (6+ 9)
=282.6inches2
∴The surface area of cone will be 282.6inches2

(Video) Surface Area Formula Derivation

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

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 RelationBetween Curved Surface Area Formula for Sphere And Hemisphere?

The curved surface area formula of a hemisphere is half of thecurved surface area of a sphere. It is given as:
CSA of hemisphere=½ (curved surface area of a sphere) = ½ (4 π r2) = 2π r2, where "r" is the radius of the hemisphere/ sphere.

What Is the Surface Area Formula of a Cone?

Thetotal surface area formula of a cone is given as, T =πr(r + l).
Thecurved 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 cylinderis given in square units, like m2, in2, cm2, yd2, etc.

FAQs

How is the formula for surface area derived? ›

The surface surface area equals pi times the quantity f of b plus f of a time's. The quantity B

How do you calculate surface area example? ›

We can use the formula for the area of a square. Which is area equals side squared since the length

What is derivation of curved surface formula? ›

=π[(r1+r2)l+r12+r22]

How do you derive the surface area of a cube? ›

Surface area of cube is the sum of areas of all the faces of cube, that covers it. The formula for surface area is equal to six times of square of length of the sides of cube. It is represented by 6a2, where a is the side length of cube.It is basically the total surface area.

How is surface area of sphere derived? ›

The formula to find the surface area of sphere is 4 times of pi (π) and radius-squared (r2).

What is the formula for surface area 6th grade? ›

Surface Area of Rectangular Prisms (Grade 6) - YouTube

What is the formula for surface area of a square? ›

The area of a square is equal to (side) × (side) square units. The area of a square when the diagonal, d, is given is d2÷2 square units. For example, The area of a square with each side 8 feet long is 8 × 8 or 64 square feet (ft2).

How do you calculate the surface of an object? ›

Count the Surfaces - Sample Questions and Answers - YouTube

What are types of surface area? ›

There are three types of surface area: lateral surface area, curved surface area, and total surface area.

How do you derive the curved surface area of a cylinder? ›

The CSA of a cylinder having its base radius 'r' and height 'h' is given by: Curved surface area (CSA) of cylinder = 2πrh sq.

What is the formula for surface area of a prism? ›

The formula for the surface area of a prism is SA=2B+ph, where B, again, stands for the area of the base, p represents the perimeter of the base, and h stands for the height of the prism.

What is the formula for SA of a rectangular prism? ›

Therefore, the lateral surface area (LSA) of a rectangular prism = 2 ( l + w ) h square units. Therefore, the surface area of a rectangular prism formula is given as, Surface Area of a rectangular prism = 2 (lh +wh + lw ) Square units.

How do u find the surface area of a rectangle? ›

To find the surface area of a rectangle, you require its length and width:
  1. Convert the length and width into the same unit. We must measure them both in the same unit. ...
  2. Multiply this length and width to obtain the area in corresponding units.
  3. Double-check your answer using the surface area of a rectangle calculator.
14 Mar 2022

How do you find the total surface area of a cylinder? ›

A cylinder's volume is π r² h, and its surface area is 2π r h + 2π r².

How do you derive the volume of a cube? ›

Therefore, the volume of cube is equal to product of its length, width and height. It is measured in cubic units.

How do you derive the formula for a sphere? ›

The general formula for the volume of sphere in terms of its radius is given as V = (4/3) π r3. Let's say 'd' is its diameter, according to the definition of diameter, we have d = 2r. From this, we get the value of radius = (d/2).

What is the surface area of a 7cm cube? ›

The surface area of a 7 cm will be 294 cm2.

What is the formula for surface area 7th grade? ›

7th Grade Math 9.4b, Finding the Surface Area of a Prism, Rectangular

What is the easiest way to find surface area? ›

Easy Way To Find The Surface Area Of Rectangular Prism - YouTube

How do you solve surface area and volume problems easily? ›

Surface Areas And Volumes
  1. To find the surface area of a solid, add the areas of all the faces. You can remember the formula as sum of areas of the all the faces. ...
  2. Volume of a prism is area of its base times height. ...
  3. Volume of a pyramid is one third of area of its base times height.

What is the formula for area of all shapes? ›

Area of 2D Shapes Formula
ShapeAreaTerms
Circleπ × r2r = radius of the circle
Triangle½ × b × hb = base h = height
Squarea2a = length of side
Rectanglel × wl = length w = width
3 more rows

What is the surface area of a ball? ›

And the formula for the surface area of a sphere of radius R is 4*Pi*R2.

What is the surface area of a triangle? ›

The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e. A = 1/2 × b × h.

What is the surface area of the figure? ›

The surface area of a figure (in square units) is the number of unit squares it takes to cover the entire surface without gaps or overlaps. If a three-dimensional figure has flat sides, the sides are called faces. The surface area is the total of the areas of the faces.

Why is calculating surface areas important? ›

Measure, analyse, control

Surface area analysis refers to the measurement of a particle's available surface. It is important because it is the means by which a solid interacts with its surroundings, whether they are gases, liquid, or other solids.

How do you find volume from area? ›

Therefore, the formula to calculate surface area to volume ratio is: SA/VOL = surface area (x2) / volume (x3) SA/VOL = x-1 , where x is the unit of measurement. Below is a table showing how to calculate the surface area to volume ratio of some common three-dimensional objects.

Who discovered surface area? ›

What specific works did Archimedes create? Archimedes wrote nine treatises that survive. In On the Sphere and Cylinder, he showed that the surface area of a sphere with radius r is 4πr2 and that the volume of a sphere inscribed within a cylinder is two-thirds that of the cylinder.

What is difference between surface area and total surface area? ›

Curved Surface Area (CSA) – It includes the area of all the curved surfaces. Lateral Surface Area (LSA) – It includes the area of all the surface excluding the top and bottom areas. Total Surface Area (TSA) – It includes the area of all the surfaces of the object including the bases.

Is surface area same as area? ›

Area vs Surface Area

The area is the measurement of the space occupied by any two-dimensional geometric shape. The surface area is the sum of areas of all the faces of the three-dimensional figure. Plane 2D figures represent the area. Example: circles, rectangles, and triangles.

How is the surface area of revolution derived? ›

Area of Surfaces of Revolution | Derivation & Example - YouTube

What is the formula for surface area simple? ›

Variables:
Surface Area FormulaSurface Area Meaning
SA=2B+PhFind the area of each face. Add up all areas.
SA=B+12sPFind the area of each face. Add up all areas.
SA=2B+2πrhFind the area of the base, times 2, then add the areas to the areas of the rectangle, which is the circumference times the height.
2 more rows
17 Sept 2020

What are all the surface area formulas? ›

Surface Area Formulas:
ShapeLateral Surface Area (LSA)Total Surface Area (TSA)
Right Circular Cylinder2πrh2πr(r + h)
Right Pyramid(1/2) Perimeter of base × Slant HeightLSA + Area of Base
Right Circular Coneπrlπr(l + r)
Solid Sphere4πr24πr2
4 more rows

What is the formula for surface area of a square? ›

The area of a square is equal to (side) × (side) square units. The area of a square when the diagonal, d, is given is d2÷2 square units. For example, The area of a square with each side 8 feet long is 8 × 8 or 64 square feet (ft2).

How do you find surface area using integration? ›

Surface Area of Revolution By Integration Explained ... - YouTube

How do you find the surface area of a cylinder using integration? ›

surface area of cylinders (KristaKingMath) - YouTube

How do you find the surface area of a double integral? ›

Computing Surface areas using double integrals - YouTube

What are the formulas of surface area and volume? ›

ShapeSurface AreaVolume
CubeS A = 6 s i d e 2V o l u m e = s i d e 3
CuboidS A = 2 l w + w h + l h Where, length, width, heightV o l u m e = l e n g t h × w i d t h × h e i g h t
CylinderS A = 2 π r h + 2 π r 2 Where radius of the circular base HeightV o l u m e = π r 2 h
2 more rows

How do you calculate surface area of a rectangle? ›

To find the surface area of a rectangle, you require its length and width:
  1. Convert the length and width into the same unit. We must measure them both in the same unit. ...
  2. Multiply this length and width to obtain the area in corresponding units.
  3. Double-check your answer using the surface area of a rectangle calculator.
14 Mar 2022

How do you solve surface area and volume problems easily? ›

Surface Areas And Volumes
  1. To find the surface area of a solid, add the areas of all the faces. You can remember the formula as sum of areas of the all the faces. ...
  2. Volume of a prism is area of its base times height. ...
  3. Volume of a pyramid is one third of area of its base times height.

Which formula is L * B * H? ›

Volume of a cuboid = l × b × h [cubic units]

l = length. b = breadth. h = height.

What is the surface area of a ball? ›

And the formula for the surface area of a sphere of radius R is 4*Pi*R2.

What is the surface area of a triangle? ›

The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e. A = 1/2 × b × h.

What is the surface area of a solid? ›

The total surface area of a solid is the sum of the areas of all of the faces or surfaces that enclose the solid. The faces include the tops and bottoms (bases) and the remaining surfaces.

Videos

1. Surface Area of Revolution By Integration Explained, Calculus Problems, Integral Formula, Examples
(The Organic Chemistry Tutor)
2. Surface Area of Solid of Revolution (about x-axis, formula explained)
(blackpenredpen)
3. Derivation of Formulas For Surface Area and Volumes of Cube ,Cuboid ,Cone and Cylinder
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4. Arc Length & Surface Area examples
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5. Surface Area of Right Circular Cylinder | Formula derivation with practical example | Class 9 Maths
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6. Area of a circle, formula explained
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