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You may have heard the term equilateral triangular pyramid but may not fully grasp this knowledge. To help readers have a better understanding of the definition of what an equilateral triangular pyramid is, as well as the knowledge about what a triangular pyramid is, we have compiled some useful content for your reference.
What is an equilateral triangular pyramid? Images and sample problems about triangular pyramid
I. Exploring the Equilateral Triangular Pyramid
1. Concept of Equilateral Triangular Pyramid
- An equilateral triangular pyramid is a pyramid with an equilateral triangle as its base, and its lateral faces (side edges) are equal. The projection of the apex onto the base coincides with the center of the equilateral triangle.
2. Properties
- The base of this pyramid is an equilateral triangle.
- All side edges are equal.
- The lateral faces of this pyramid form isosceles triangles, not necessarily equilateral.
- The height aligns with the center of the base (the center of the base is the centroid of the triangle).
3. Distinguishing between an Equilateral Triangular Pyramid and a Regular Tetrahedron
- A regular tetrahedron is essentially an equilateral triangular pyramid; however, in a regular tetrahedron, the side edge equals the base edge, meaning all faces are equilateral triangles.
II. Visualizing the Equilateral Triangular Pyramid
III. Easy 3-Step Guide to Drawing an Equilateral Triangular Pyramid
Step 1: Draw the base of the pyramid as equilateral triangle ABC (though not necessarily with completely equal sides, it can be a regular triangle), with AC drawn as a dashed line.
Step 2: Draw the medians CF and AI intersecting at O, where O is the foot of the altitude coinciding with the centroid.
Step 3: From O, draw a vertical line to get the apex S, and connect S to vertices A, B, C.
=> Complete the construction of the equilateral triangular pyramid SABC with SH as the altitude, and SA = SB = SC.
Formulas related to the equilateral triangular pyramid
- Formula for calculating the surface area of an equilateral triangular pyramid (base):
S = (a^2 x √3) : 4
- Formula for calculating the height of the equilateral triangle:
h = (a x √3) : 2
- Formula for calculating the volume of an equilateral triangular pyramid:
V = 1/3. h. Sbase
- Explanation of symbols:
+ S is the area of the equilateral triangle
+ Sbase is the area of the base
+ a is the length of one side of the triangle
+ h is the height
Exercise on the equilateral triangular pyramid: Given the equilateral triangular pyramid SABC with a base edge b, where the lateral face forms a 60-degree angle with the base. Calculate the volume of the pyramid SABC.
* Instructions:
- Students construct the equilateral triangular pyramid SABC as illustrated above.
- Let point I be the center of the base => SI is perpendicular to the plane of the base ABC.
=> Volume of SABC = 1/3. SI. Striangle ABC
- Calculate: SABC = b^2√3 : 4
- Calculate SI:
+ Angle formed by the lateral face (SBC) and the base (ABC) = angle SDI = 60 degrees
We have: ID = 1/3. AD = 1/3. (b√3 : 2) = b : 2√3 (property of altitude, median AD in an equilateral triangle)
+ Consider the right triangle SID: tanSDI = opposite/adjacent = SI : ID
=> The surface area (SI) is calculated using the formula SI = (b : 2√3) . √3
=> SI equals b/2
=> Volume of pyramid VSABC = 1/3 . b/2. b2√3 : 4 = b3√3/24 (unit).
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Through our article, we believe readers now have a clearer understanding of what an equilateral triangular pyramid is, how to draw it easily, and the formulas to solve problems related to equilateral triangular pyramids. You can also actively solve some exercises based on our suggestions. Additionally, students should reinforce their knowledge of calculating the area of triangles, which is a fundamental concept to grasp.
Furthermore, refer to the article on formulas for calculating the volume of a pyramid in the pyramid series of exercises. We hope it will be very helpful for students in the learning program.
