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Surface Area Calculator

Surface Area Calculator

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Shape:

Base Shape:

Radius r1:

 

Radius r2:

 

Radius r3:

 

Let pi π

surface image

Height h:

 

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The Surface Area Calculator assists to find the Surface Area of three-dimensional solids like Capsules, Spheres, Cylinders, Cone, Prism, Pyramids, Hemisphere, Ellipsoid, etc.

Let’s have a look at the various geometrical shapes and their surface area!

What is Surface Area Definition?

The surface area of a solid geometric shape is the measure of the total area covered or occupied by an object. 

How to Find Surface Area?

Different formulas are used for finding the surface area of various solids. The surface area formula has two distinct parts: the base area and the lateral area. The units used to measure the surface area are inches, feet, mm, and cm. 

Base Area:

Base area refers to the area of the base of a solid. It can be different for various solids, as it can be rectangular, square, circular, triangular, etc. It can change the whole calculation of the surface area of a shape. 

Lateral Surface Area:

The lateral surface is the surface area of all the sides of the solids excluding the base and the top. There is a distinction necessary to establish, where you calculate the lateral area separately like a cylinder, cone, pyramid, or triangular prism. Finding the lateral area of any geometrical shape can be a little tricky, but if you are using our lateral surface area calculator, then you have just overwhelmed the issue and are about to get an instant calculation of each parameter in seconds.

Following is the surface area formula for various geometrical shapes:

The Surface Area of Sphere:

For finding the surface area of the sphere, you need to find the sphere’s radius or its diameter. The surface of a sphere can be problematic, but when you are using the sphere surface area calculator, it is easy to find the surface area of the sphere.
The Surface Area of Sphere:

The formula for the surface area of a sphere is:

  • S.A = 4 * π * r² 
  • Or
  • S.A =  π * d²

Where: 

r is the radius.

d is the diameter 

The Surface Area of Cone:

The surface area of a cone can be measured by summing the base surface area and the lateral surface area. The base surface area of the base is actually the SA of the circular base, the lateral surface area refers to the side to the apex of the cone. The surface area of a cone can be measured by the base area and lateral sides.
It can be rather tricky to find the surface area of a cone if we have no concept of the base area and lateral side of the cone.
How to find the surface area of a cone?
The Surface Area of Cone:
base SA = πr2

lateral SA = πr√r2 + h2

total SA = πr(r + √r2 + h2)

Where

r is the radius and h is the height

The Surface Area of Cube:

The Surface area of a cube can be measured by finding the area of a side and multiplying it by six. In a cube there are six equal sides, a cube having all the equal sides is a box. In this case, the surface area of a box is actually the SA of the cube. It is perfect to use the surface area of a cube calculator to measure the SA of a cube.
The Surface Area of Cube:

The formula for the Surface area of a cube:

SA = 6a2

Where:

 a is the edge length

The Surface Area of Cylinder:

The surface area of the cylinder can be measured by summing the base area and the lateral surface area of the cylinder. For the students, it can be problematic, how to find the surface area of a cylinder, and treat the base area and lateral area separately.
The Surface Area of Cylinder:

The formula for the Surface area of a cylinder:

base SA = 2πr2

lateral SA = 2πrh

total SA = 2πr(r + h) 

where 

r is the radius and h is the height

The Surface Area of Rectangular Prism:

To measure the surface area of a Rectangular Prism, consider the area of three rectangular sides. You may wonder how to find the surface area of a rectangle, all you need is to find the rectangular area of three rectangular sides, then add them, and at the end multiply the sum by two. You can find the area of the Rectangular Prism using the surface area of a rectangle calculator.
The Surface Area of Rectangular Prism:

SA of  = 2 * (A1 + A2 + A3)

where:

  • A1 = l * w
  • A2 = w * h
  • A3 = l * h

The formula for the Surface area of a cylinder:

A = 2 * (l * w + w * h + l * h)

The Surface Area of Capsule:

A capsule surface can be calculated by merging the surface area of the sphere and the lateral surface area of the cylinder. There is no base area involved in the surface area of the capsule. If you are finding any difficulty in finding the surface area of the capsule, insert values of “r” and “h” in the surface area calculator as well.
The Surface Area of Capsule:

The formula for the Surface area of a capsule:

SA = 4πr2 + 2πrh

Where:

r is the radius 

h is the height

The Surface Area of Hemisphere:

The surface area of the hemisphere is the curved surface of the hemisphere plus the base area of the circular portion. The detailed note on the hemisphere is described in the hemisphere calculator.
The Surface Area of Hemisphere:

The surface area of the Hemisphere = Curved surface area of hemisphere + circular base area

The curved surface area of the hemisphere = 2πr^2

Circular base area = πr^2

Then 

Surface area of the Hemisphere = 2πr^2 + πr^2 

Thus,

The surface area of a hemisphere = 3πr^2 square units

Where:

“r” is the radius of the hemisphere.

The Surface Area of Conical Frustum:

Let’s check the whole shape of the frustum, there are two circles at the base and at the apex. It means the surface area of the conical frustum is the sum of two circular portions and the lateral face. It is simple how to calculate surface area Frustum when there are two circular portions and the lateral part is involved in Conical Frustum. If you find difficulty in calculating the SA of the Frustum, then use the lateral surface area calculator.

The Surface Area of Conical Frustum:
The formula for the Surface area Conical Frustum:

The area of Conical Frustum  = Circle Top + Circular Bottom 

Circle top = π  r2

Circular bottom = π  R2

Then 

Area of Circular end SA = π(R2 + r2)

lateral SA = π(R+r)√(R-r)2 + h2

total SA = π(R2 + r2) + π(R+r)√(R-r)2 + h2

Where:

 R and r are the radii of the ends

 h is the height

The Surface Area of the Spherical Cap:

There are two portions of the Surface area of the Spherical Cap, the spherical cap and the base area. The Spherical Cap includes the height “h” and the area of the circle, and then multiply it by the height. It can be difficult to find the SA of various rare geometrical shapes like the Spherical Cap. To add the values of the spherical cap and the base area, use the total surface area calculator.

The Surface Area of Spherical Cap:

The formula for the Surface area Spherical cap:

Spherical cap SA = 2πRh

Base SA = πr2

SA of Spherical Cap= 2πRh + πr2

The Surface Area of Triangular Prism:

The surface area of a Triangular prism can be determined by the perimeter of the face of the Triangular Prism. Now multiply the perimeter by the length and add the base area to it.  If you are finding any difficulty to learn about the perimeter, you can use another perimeter calculator. The surface area finder can be used to find the perimeter and the base area of the triangular prism.

The Surface Area of Triangular Prism:

The formula for the Surface area Triangular Prism:

The SA of Triangular prism = (perimeter * length)+ (2 * base area)

The Surface Area of the Pyramid:

The various pyramids can be of different shapes due to their base, the pyramids can be subdivided into various types on the basis of their base shape:

The various type of pyramids are:

  • Equilateral Triangular base Pyramid
  • Square base Pyramid
  • Rectangular base Pyramid
  • Pentagon base Pyramid
  • Hexagonal base Pyramid

How to find the surface area of a pyramid?
The Surface Area the Pyramid:

A = A(base) + A(lateral) = A(base) + 4 * A(lateral face)

In the case of the square base 

Lateral Surface Area = a√(a2 + 4h2)

Base Surface Area = a2

Total Surface Area= L + B = a2 + a√(a2 + 4h2))

S.A Pyramid = A(base) + A(lateral) = A(base) + 4 * A(lateral face)
S.A Pyramid  =  a(a + √(a2 + 4h2))

It can be a little difficult when you are dealing with the various geometrical-shaped pyramids. The surface area of the pyramid can be measured by the surface area of a pyramid calculator

The Surface Area of Ellipsoid:

The Ellipsoid Surface area is not simple, you need to learn about the ellipse shape and the vertices. Choose and find the surface area calculator to measure the Surface area Ellipsoid easily, as the formula is a bit complicated. 

The Surface Area of Ellipsoid:

The formula for the Surface area Ellipsoid:

SA ≈ 4π 1.6√(a1.6b1.6 + a1.6c1.6 + b1.6c1.6)/3

where a, b, and c are the axes of the ellipse

Working of Surface Area Calculator:

We do need to follow the guidelines as under to use the surface area formula calculator.

Input:

  • Choose the Geometrical shape from the drop-down menu
  • Insert the relevant data for each shape 
  • Hit the calculate button to find the surface area

Output:

The surface area calculator generates the following answers:

  • The surface area of the geometric shape

FAQs:

Is Surface Area the Same as Area?

The main difference between the surface area and area is the surface area related to 3D shapes such as a sphere, cylinder, and so on. The area corresponds to the 2D shapes. The area of two 2-D shapes can be measured using a free area calculator.

Why we Consider Boundries in SA?

The surface is the outermost portion or uppermost layer of an object, when we are finding the surface area, we are considering the boundary of a geometrical object, we can use the surface area solver to find the area’s outer boundaries like sphere, rectangle or square, etc. 

What is the Relationship Between Surface Area and Net?

The surface area is the sum of net values of 3-D shapes, for example a cuboid 6 faces or rectangular net hands. We are adding all the nets or faces to calculate the value of the cuboid like SA=2lw+2lh+2hw, the surface area calculator actually adding all the net values of 3 D shape.

Why is Surface Area Important in Engineering?

Architectural Engineers utilize the surface area to volume ratio to design various buildings and objects. What is surface area to volume ratio helps to construct energy-efficient buildings.

Who Discovered Surface Area?

The Greek Mathematician  Archimedes discovered how to find the surface area of a sphere and the lateral surface area of a cylinder.

Is Surface Area Scalar or Vector?

Surface area is scalar quantity as you only need the magnitude and no directions.

Conclusion:

There are various applications of the surface area in everyday life like the total surface area of a building while painting or flooring. Estimate the surface area of the construction site by free surface area calculator and reduce the overall cost of construction.

References:

From the source of Wikipedia: Surface area,Definition

From the source of the tutorial.math.lamar.edu: Solid Area, Formula of SA