Contents
Introduction
The surface area of a three-dimensional shape is the total area of all its faces. To find the surface area of a 3D shape, we can use a method called net. A net is a two-dimensional representation of a 3D shape that shows all its faces when unfolded. By using the net of a 3D shape, we can easily calculate the area of each face and then add them up to get the surface area.
What is a Net?
A net is a pattern made when the surface of a 3D shape is laid out flat. It shows each face of the shape in two dimensions. For example, the net of a cube is shown below:
A net can help us visualize how a 3D shape is formed by folding along its edges. A net can also help us measure the dimensions of each face of the shape. For example, in the net of the cube above, we can see that each face is a square with side length 4 cm.
A 3D shape can have more than one possible net. For example, the net of a rectangular prism can be arranged in different ways, as shown below:
How to Find the Surface Area Using a Net?
To find the surface area of a 3D shape using its net, we need to follow these steps:
- Identify the shape and its net.
- Find the area of each face using the appropriate formula for its shape.
- Add up the areas of all the faces to get the surface area.
Let’s see an example of how to apply this method.
Example: Find the surface area of a triangular prism using its net.
Solution:
- The shape is a triangular prism and its net is shown above.
- The net has five faces: two triangular bases and three rectangular lateral faces.
- To find the area of each face, we need to use the formula for the area of a triangle and the area of a rectangle.
- The area of a triangle is given by
�=����12�ℎA=frac12bh
, where
�b
is the base and
ℎh
is the height. - The area of a rectangle is given by
�=��A=lw
, where
�l
is the length and
�w
is the width. - The area of each face is calculated as follows:
| Face | Shape | Dimensions | Area |
| Base 1 | Triangle | Base = 6 cm, Height = 5 cm | �=����12(6)(5)=15������2A=frac12(6)(5)=15textcm2 |
| Base 2 | Triangle | Base = 6 cm, Height = 5 cm | �=����12(6)(5)=15������2A=frac12(6)(5)=15textcm2 |
| Lateral Face 1 | Rectangle | Length = 8 cm, Width = 6 cm | �=(8)(6)=48������2A=(8)(6)=48textcm2 |
| Lateral Face 2 | Rectangle | Length = 8 cm, Width = 5 cm | �=(8)(5)=40������2A=(8)(5)=40textcm2 |
| Lateral Face 3 | Rectangle | Length = 8 cm, Width = 5 cm | �=(8)(5)=40������2A=(8)(5)=40textcm2 |
- The surface area is the sum of the areas of all the faces:
���������������=15+15+48+40+40=�����158������2textSurfaceArea=15+15+48+40+40=boxed158textcm2
Conclusion
In this article, we learned how to find the surface area of a three-dimensional shape using its net. A net is a two-dimensional representation of a three-dimensional shape that shows all its faces when unfolded. By using the net of a shape, we can easily calculate the area of each face and then add them up to get the surface area. This method can be applied to any polyhedron, such as cubes, prisms, pyramids, etc. However, for shapes with curved surfaces, such as cylinders, cones, spheres, etc., we need to use different formulas to find their surface areas.
