The Definition of Area
The area of a shape is the extent of the region covered by the shape on a plane. The surface area of a three-dimensional shape refers to the total amount of region occupied by all of its faces.
We usually find the surface area of three-dimensional shapes by considering the nets of the shapes. A net is an arrangement of two-dimensional shapes which when folded forms a three-dimensional shape or a solid. In other words, the net of a shape is what a 3D shape would look like if it was opened out and laid flat.
List of Formulas
These are the formulas that we will look at and they are related to the surface areas of multiple solid shapes that are also prisms.
- The surface area of a triangular prism \(= ~bh~+~(a~+~b~+~c)~H\)
- The surface area of a square prism \(=~ 2a^2~+~4ah\)
- The surface area of a rectangular prism (cuboid) \(=~ 2(lb~+~lh~+~bh)\)
- The surface area of a cube \(=~ 6a^2\)
Formula of surface area of prisms and cube
The formula used to find the surface area of a prism depends on the shape of its base. To find the surface area of the three-dimensional shape, we can find the areas of the individual two-dimensional shapes in its net and then add them up.
A triangular prism has two triangular faces that are parallel to each other, and three rectangular faces that connect the triangular faces.
We use the formula \(bh~+~(a~+~b~+~c)~H\) to find the surface area of a triangular prism. Here, b and h are the base and height of the triangular faces, a, b and c are the lengths of the sides of the triangular faces (also the sides of the rectangular faces), and H is the height of the triangular prism.
The first part of the equation, bh, gives us the area of the two triangular faces, and the second part of the equation, \((a~+~b~+~c)~H\) gives us the area of the three rectangular faces.
A square prism is a three-dimensional shape in which the bases are squares and the other four faces are made up of rectangles. The formula used to find the surface area of a square prism is \(2a^2~+~4ah\). Here, a is the length of the side of the square face, and h is the height of the prism.
The first part of the formula,\(2a^2\), gives us the area of the two square faces, and the second part of the formula \((4ah)\) gives us the area of the four rectangular faces.
A rectangular prism, or a cuboid, is a three-dimensional shape made up of six identical rectangular faces. The formula used to find the surface area of a cuboid is \(2(lb+lb+bh)\). Here, l, b, and h are the length, breadth, and height of the rectangular faces, respectively.
A cube is a special type of prism in which the six faces are congruent squares. So, the formula used to calculate the surface area of a cube is \(6a^2\), which is six times the area of one square.
Solved Examples
Example 1: Which three-dimensional shape will be formed by the following net?
Solution: The given net has 6 two-dimensional shapes. That means the three-dimensional shape formed by this net will have 6 faces.
There are two rectangular bases having the same dimensions. So, they will form the bases of a prism.
Since all shapes in the net are rectangular, we can conclude that the shapes in this net when folded will form a rectangular prism.
Example 2: Which prism does this net represent?
Solution: A triangular prism is a polyhedron made up of two identical triangular faces. It also has three rectangular faces that connect the two triangular faces.
So, the given net is of a triangular prism.
Example 3: A dice is 0.5 inches long, 0.5 inches wide and 0.5 inches tall. Find the surface area of the dice.
Solution:
A dice is in the shape of a cube, which is a prism that has 6 congruent square faces.
Length of the dice = 0.5 inches
Width of the dice = 0.5 inches
Height of the dice = 0.5 inches.
So, side of the cube, a = 0.5 inch
Surface area \( =~6a^2 =~6\times~0.5^2\)
\( =~6\times~0.25\)
\(=~1.5\) square inches
Therefore, the surface area of the dice is 1.5 square inches.
Example 4: A chocolate company packs its chocolates in the shape of a triangular prism. The dimensions of the prism are given below. Find the surface area of the chocolate packaging.
Solution: The given net forms a triangular prism as it has three rectangular faces that connect to identical triangular faces.
The surface area of a triangular prism \(=~bh~(a~+~b~+~c)~H\)
Here, b = 14 cm, h = 24 cm, a = c = 25 cm and H = 50 cm
Therefore, surface area = 14 \(\times\) 24 + (25 + 14 + 25) \(\times\) 50
= 336 + 3200
= 3536 square centimeters
So, the surface area of the chocolate packaging is 3536 square centimeters.
Example 5: Jane wants to gift her friend some clothes. She wants to pack it in a gift box that is in the shape of a rectangular prism. Find the surface area of the gift wrapping paper required.
Solution:
The gift box is in the shape of a rectangular prism or a cuboid. To find the surface area of the gift wrapping paper required to wrap the gift, we need to find the surface area of the cuboid.
Here, l = 9 inches
b = 3 inches
h = 4 inches
Total surface area of the cuboid = \( 2~(lb~+~lb~+~bh)\)
= 2 \(\times\) (9 \(\times\) 3 + 9 \(\times\) 4 + 3 \(\times\) 4)
= 2 \(\times\) (27 + 36 + 12)
= 2 \(\times\) 75
= 150 square inches
Therefore, Jane needs 150 square inches of gift wrapping paper to wrap the box.
Cubes are a special type of prism in which the six sides are made of identical squares. So, any face of a cube can be considered as the base of the prism.
Cubes are made up of square faces and cuboids are made up of rectangular faces. Since the definition of a square fits the definition of a rectangle, we can consider a cube as a special case of cuboids in which all dimensions are equal.
All sides of a cube are made up of squares and have the same dimensions. On the other hand, the bases of a square prism are made up of squares and the rest four sides are made up of rectangles.
A net is an arrangement of two-dimensional shapes that can be folded to form a three-dimensional shape or a solid. Since we can see each two-dimensional shape that makes up the three-dimensional shape, it is easy to find their individual areas. We can add the individual areas of the faces to get the surface area of the prism.
A prism is a three-dimensional figure that has two identical bases. Whereas a pyramid is a three-dimensional figure that has only one base.
Now that you have learned different types of prisms and the formulas used to find their surface areas, go ahead and check out these topics:
- Surface Areas of Three-Dimensional Figures: Pyramids Formula
- Volumes of Three-Dimensional Figures: Rectangular Prism and Cubes Formulas
- Surface and Lateral area of Three-Dimensional Figures: Prisms Formula
- Surface and Lateral Area of Three-Dimensional Figures: Cylinders Formula
- Surface and Lateral Area of Three-Dimensional Figures: Pyramids Formula
- Volumes of Three-Dimensional Figures: Prisms Formula
- Volumes of Three-Dimensional Figures: Pyramids Formula
- Volume of Three-Dimensional Figures: Cylinders Formula
- Volume of Three-Dimensional Figures: Cones Formula
- Volume of Three-Dimensional Figures: Spheres Formula
- Area of Rectangles Formula
- Areas of Polygons: Parallelograms Formula
- Areas of Polygons: Triangles Formula
- Areas of Polygons: Trapezoids and Kites Formula