What is a Triangular Pyramid?
A triangular pyramid is a three-dimensional shape with a triangular base and three triangular lateral faces. The lateral faces of the pyramid share a common vertex known as the apex. In other words, all three vertices of the triangular base of the pyramid are connected to the apex.
Types of Triangular Pyramids
Triangular pyramids can be classified into regular and irregular.
The Regular Triangular Pyramid
The base of a regular triangular pyramid is an equilateral triangle, and its apex is aligned above the center of the base. All of its internal angles measure 60 degrees.
The Irregular Triangular Pyramid
The triangular faces of an irregular triangular pyramid are also triangular, but they are not equilateral. The internal angles of the faces add up to 180\(^{\circ}\).
[Note: Unless a triangular pyramid is specifically described as irregular, it is assumed that all triangular pyramids are regular.]
The Properties of a Triangular Pyramid
The properties of triangular pyramids allow us to quickly and easily identify them from a set of solid shapes.
- A triangular pyramid has 6 edges, 3 triangular lateral faces, 4 vertices, and a triangular base.
- At each vertex, three edges meet.
- There are no parallel faces in a triangular pyramid.
- All of the faces of a regular triangular pyramid are equilateral triangles.
- A regular triangular pyramid has six symmetry planes.
- The height of each lateral triangle is known as the slant height of the pyramid.
The Volume of a Triangular Pyramid
The volume of a triangular pyramid is given by:
Volume = (\(\frac{1}{3} \times\) Base Area \(\times\) Height) cubic units
The height is measured from the base to the apex.
The Surface Area of a Triangular Pyramid
The surface area of a triangular pyramid is the sum of the area of the base and the areas of the lateral faces.
The formula for calculating the total surface area of a triangular pyramid is:
The surface area of the triangular pyramid = Area of the base + Areas of the lateral faces.
Solved Triangular Pyramid Examples
Example 1:
Determine the surface area of the triangular pyramid given in the diagram.
Solution:
Area of the base: \(\frac{1}{2} \times 9 \times 6=27\)
Area of the lateral face: \(\frac{1}{2} \times 9 \times 11=49.5\)
Find the sum of the areas of the faces.
The surface area of the triangular pyramid = Area of the base + Areas of the lateral faces.
s = 27 + 49.5 + 49.5 + 49.5 [There are three identical lateral faces]
s = 175.5
So, the surface area is 175.5 square inches.
Example 2: Find the volume of a triangular pyramid with a base area of 28 cm\(^2\) and a height of 4.5 cm.
Solution:
As we know, the formula for the volume of a triangular pyramid is:
Volume = \(\frac{1}{3} \times\) Base Area \(\times\) Height
Now, substitute the values,
= \(\frac{1}{3} \times 28 \times 4.5\)
= \(\frac{1}{3} \times\) 126 [Simplify]
= 42 cm\(^3\) [Divide]
Hence, the required volume of a triangular pyramid is 42 cm\(^3\).
Example 3:
A triangular bipyramid is formed when two congruent triangular pyramids are stuck together along their base. How many faces, edges, and vertices does this bipyramid have?
Solution:
There are 6 triangular faces, 9 edges, and 5 vertices in this triangular bipyramid.
Example 4:
John completes the Pyraminx in under a minute. The Pyraminx is a triangular pyramid with a base area of 27 in\(^2\). If its height is 8 inches, determine its volume.
Solution:
As we know, the formula for the volume of a triangular pyramid is:
Volume = \(\frac{1}{3} \times\) Base Area \(\times\) Height
Now, substitute the values,
= \(\frac{1}{3} \times 27 \times 8\)
= \(\frac{1}{3} \times 216\) [Simplify]
= 72 in\(^3\) [Divide]
Hence, the volume of the Pyraminx is 72 in\(^3\).
A tetrahedron is a polyhedron with four faces, six edges, and four vertices, each of which is a triangle. It is also known as a triangular pyramid because its base is a triangle.
An oblique pyramid is a type of pyramid whose apex is not centered over its base.
A plane of symmetry divides a shape in half, resulting in each side of the plane being a mirror image of the other side.
A rectangular pyramid is a type of pyramid with a rectangle-shaped base and triangle-shaped lateral faces. A rectangular pyramid has five faces, five vertices, and eight edges.
A pyramid is a three-dimensional polyhedron with a single polygonal base that is attached to its lateral faces, which are always triangular. A prism is also a 3D polyhedron but with two identical bases which are perpendicular to the lateral faces, and the cross-section is the same across all faces.