Determining the Surface Area of Solid Shapes
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 space occupied by all of its faces. We can use the concept of nets to find the surface area of three-dimensional solids. A net is an arrangement of two-dimensional shapes that can be folded to obtain a three-dimensional shape. Basically, the net of a shape is the same as what a 3D shape would look like if it was opened and laid out.
List of Formulas
We can find the area of a pyramid by adding the area of its base and the lateral surface area. As the lateral surfaces are congruent triangles, we can use a formula to find the lateral surface area in one go.
- The surface area of a pyramid = Area of base + Sum of areas of lateral surfaces
- The surface area of a regular pyramid = \(\frac{1}{2}Pl+B\)
Here, \(P\) is the perimeter of the base, \(l\) is the height of the triangular faces, which is also the slant height of the pyramid, and \(B\) is the area of the base.
The first equation is similar to the second equation. In the first equation, we find the base area and the areas of the individual lateral surfaces. Then we add these areas together to get the surface area.
In the second equation, we are calculating the total area of the lateral surfaces and the area of the base. We can use this idea as the triangular faces in a regular pyramid are congruent. Instead of using the \(\frac{1}{2}\times b \times h\) equation for each triangle, we find the sum of the bases of the triangles — which is the same as the perimeter of the base — and use the equation \(\frac{1}{2}\times P \times l\) to find the total area of the lateral surfaces. Here, P is the perimeter of the base, and l is the height of the triangle, which is also the slant height of the pyramid.
Alternatively, we can use the nets of the pyramids to find the total area of the pyramid by calculating the sum of the areas of the individual faces.
Solved Examples
Example 1: Find the total area of the pyramid.
Solution:
The length of the edges of the base of the pyramid is 5 inches. Since the pyramid has a square base, it is a regular pyramid. The slant height of the pyramid is 8 inches.
The surface area of a regular pyramid = \(\frac{1}{2}Pl+B\)
The perimeter of the base, P = 5 + 5 + 5 + 5 = 20 inches
The slant height of the triangular face, \(l=8\) inches
The lateral surface area = \(\frac{1}{2}Pl\)
= \(\frac{1}{2}\times 20 \times 8\)
= 80 square inches
Base area, B = \(a^2\)
= \(5^2\)
= 25 square inches
So, the total surface area of the pyramid = \(\frac{1}{2}Pl+B\)
= 80 + 25
= 105 square inches
Example 2: Find the surface area of a square pyramid whose base length is 16 inches and the slant height is 17 inches
Solution:
Since it is a square pyramid, we can use the formula used to find the surface area of a regular pyramid.
The surface area of a regular pyramid = \(\frac{1}{2}Pl+B\)
Base perimeter, P = 16 + 16 + 16 + 16
= 64 inches
The area of the base, B = \(a^2\)
= \(16^2\)
= 256 square inches
So, the surface area of a regular pyramid = \(\frac{1}{2}Pl+B\)
= \(\frac{1}{2}\times 64\times 17+256\)
= \(32\times 17+256\)
= 544 + 256
= 800 square inches
Hence, the surface area of the pyramid is 800 square inches.
Example 3: A structure in the shape of a triangular pyramid is placed on a flat surface which needs to be covered with a piece of cloth. The base of the triangular base is 3 feet long and the slant height of the pyramid is 5 feet. Find the area of the cloth used to cover the pyramid.
Solution:
To find the area of the cloth used to cover the pyramid, we need to find the area of the lateral surfaces of the pyramid, as the base of the pyramid is not covered with cloth.
Base length of the triangular base of the pyramid = 3 feet
The slant height of the pyramid = 5 feet
Area of one triangular face = \(\frac{1}{2}\times b \times h\)
= \(\frac{1}{2}\times 3 \times 5\)
= 32.5
= 7.5 square feet
Since this is a triangular pyramid, there are three lateral surfaces.
So, lateral surface area = 7.53
= 22.5 square feet
So, the area of the cloth used to cover the structure is 22.5 square feet.
Example 4: Find the surface area of the regular pyramid below.
Solution:
The length of the base edge of the pyramid is 6 inches, and the distance between the apex of the pyramid and the vertex is 5 inches.
Since this is a regular pyramid we will use the formula to find the surface area of a regular pyramid.
The surface area of a regular pyramid = \(\frac{1}{2}Pl+B\)
The perimeter of the base, P = 6 + 6 + 6 + 6 = 24 inches
Area of the base, B = \(6^2=36\) square inches
In order to determine the surface area of the pyramid, we need to find the slant height of the pyramid using the Pythagorean Theorem.
To use the Pythagorean Theorem, take a look at one of the faces of the pyramid. The slant height is perpendicular to the edge of the base.
Hence, \(a^2+b^2=c^2\)
Here, \(b=\frac{6}{2}=3\) inches
\(c=5\) inches
\(a^2+3^2=5^2\)
\(a^2=25-9\)
\(a^2=16\)
\(a=\sqrt{16}\)
\(a=4\) inches
So, the slant height (\(l\)) of the triangular faces is 4 inches.
Therefore, the surface area of a regular pyramid = \(\frac{1}{2}Pl+B\)
= \(\frac{1}{2}\times 24 \times 4+36\)
= 48 + 64
= 112 square inches
So, the surface area of the pyramid is 112 square inches.
The slant height of a pyramid is the perpendicular height of the triangular faces. In other words, it is measured along the center of a lateral face. It is the distance from the apex to the midpoint of the edge of the base.
A pyramid is a three-dimensional figure that has only one base. Whereas a prism is a three-dimensional figure that has two identical bases.
The area of all faces of a solid is known as its surface area. On the other hand, the area of all surfaces excluding the bases is known as the lateral surface area of the solid.
Regular pyramids are the ones that have a regular polygon as their base. Irregular pyramids have irregular polygons as the base.
The difference between a square pyramid and a triangular pyramid is that the base of a square pyramid is a square and hence has 4 lateral faces. In comparison, the base of a triangular pyramid is a triangle and, hence, has 3 lateral faces.