**Introduction to Square Pyramid**

The great pyramid of Giza and many other small pyramids (replica of pyramids of Giza) are square pyramids. Besides, in this topic, we will discuss the square pyramid their definition, properties, areas, and volumes.

**Square Pyramids**

Square pyramid shares all the properties of being a polyhedron with a polygonal base and triangle sides. Also, all the triangular side reach to the top called the Apex.

Moreover, it is 3-D (three dimensional) figure with flat polygons as sides; also a polygon is a enclosed, flat, and plain shape with at least three sides and angles.

In simple words, a square pyramid is a pyramid that has four sides (faces). Also, it has a square as a base. Due to the square base and four sides it is called square pyramids.

Besides, it has five vertices, eight edges. Also, its top point is the apex and the base is a square. Earlier we mention that it is a polyhedron but to be more specific is actually a pentahedron because it is a polyhedron that has five sides. Besides, Penta (Latin) means five.

Furthermore, if all the triangular sides of the square pyramid are equilateral then it means that all the sides and angles of the triangles are also equal. Also, all the edges of the square pyramid will be of equal length too. Moreover, in reality, we refer to this as Johnson solid.

Besides, any other geometric forms or figures can be Johnson solid if they are convex polyhedrons with regular polygon faces but not uniform faces.

**Properties of the Square Pyramid**

The various properties of square pyramids are:

If we talk about the volume of the square pyramid then it is one third (1/3) of the product of the height and area of the square.

Besides, the lateral surface area of the square pyramid is twice the product of the base edge and the slant height. In addition, if we talk about the surface area then it is half of the product of the perimeter of the base and slant height of the pyramid.

Most noteworthy, it is a 3-D (three dimensional) solid that has five triangular faces, five vertices, and eight edges.

**Volume of the Square Pyramid**

As we discussed earlier, the volume of the square pyramid is one third (1/3) of the product of the base area and the height of the pyramid. Also, height can be understood as the length of altitude is the perpendicular distance from the vertex to the plane of the square base.

Now, we know that the volume of the pyramid is = 1 / 3 (Height × Base)

So, let’s take b as the base edge and h as the height of the pyramid. Then,

Base area = (Side)^{ 2} = b^{2} (area of the square)

Volume = 1 / 3 (b^{2} × h) = 1 / 3 (b^{2}h)

As a result, the volume of the square pyramid = 1 / 3 (b^{2}h)

**Surface Area of the Square Pyramid**

Square pyramids surface area is the combination of the square are of two times the product of slant height and the base side.

Given that, the surface area of pyramid = Base Area + 1 / 2 (slant height × perimeter of base) (when the faces of sides are similar)

Now, let s be the side of the square, a be the slant height, and h be the height of the square pyramid. Then

Base area = (side)^{ 2} = s^{2}

Perimeter of base = 4 × side = 4s

So, the surface area of square pyramid = s^{2} + 1 / 2 (a × 4s) = s^{2} + 2sa

**Solved Question for You**

**Question.** What will be the volume of the square pyramid if its base and height is 5 cm and 7 cm respectively?

- 85.324 cm
^{2} - 60.521 cm
_{2} - 58.333 cm
^{2} - 72.589 cm
^{2}

**Answer.** The correct answer is option C.

**Solution:** volume = 1 / 3 (b^{2}h)

Volume = 1 / 3 (5^{2}7) = 1 / 3 (25 × 7) = 1 / 3 (175)

Volume = 1 / 3 × 175 = 175 / 3 = 58.333 cm^{2}

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