When studying absolutely any spatial figure, it is important to know how to calculate its volume. This article gives the formula for the volume of a regular quadrangular pyramid, and also shows how to use this formula using the example of solving problems.
What kind of pyramid will be discussed?
Every high school student knows that a pyramid is a polyhedron consisting of triangles and a polygon. The latter is the base of the figure. Triangles have one common side with the base and intersect at a single point, which is the top of the pyramid.
Each pyramid is characterized by the length of the sides of the base, the length of the side ribs and the height. The latter is a perpendicular segment, lowered to the base from the top of the figure.
A regular quadrangular pyramid is a figure with a square base, the height of which intersects this square in its center. Perhaps the most famous example of pyramids of this type are the ancient Egyptian stone structures. Below is a photograph of the pyramid of Cheops.
The studied figure has five faces, four of which are the same isosceles triangles. It is also characterized by five peaks, four of which belong to the base, and eight edges (4 edges of the base and 4 edges of the side faces).
The formula for the volume of a quadrangular pyramid is correct
The volume of the figure in question is a part of the space that is limited by five sides. To calculate this volume, we use the following dependence of the area parallel to the base of the slice pyramid S z on the vertical coordinate z:
S z = S o * (h - z / h) 2
Here S o is the area of the square base. If we substitute z = h in the written expression, then we get a zero value for S z . This value of z corresponds to a slice that will contain only the top of the pyramid. If z = 0, then we get the value of the base area S o .
It is easy to find the volume of the pyramid, knowing the function S z (z), for this it is enough to cut the figure into an infinite number of layers parallel to the base, and then carry out the integration operation. Following this technique, we get:
V = ∫ 0 h (S z ) * dz = -S 0 * (hz) 3 / (3 * h 2 ) | 0 h = 1/3 * S 0 * h.
Since S 0 is the area of the square base, then, denoting the side of the square by the letter a, we obtain the formula for the volume of the regular quadrangular pyramid:
V = 1/3 * a 2 * h.
Now let us show by examples of solving problems how this expression should be applied.
The task of determining the volume of a pyramid through its apothem and side edge
The apothem of the pyramid is the height of its lateral triangle, which is lowered to the side of the base. Since all triangles are equal in the regular pyramid, their apofems will also be the same. Denote its length by h b . The side edge is denoted by b.
Knowing that the apothem of the pyramid is 12 cm, and its side edge is 15 cm, find the volume of the regular pyramid of the quadrangular.
The formula for the volume of the figure written in the previous paragraph contains two parameters: side length a and height h. At the moment, we do not know any of them, so we will deal with their calculations.
The length of the side of the square a can be easily calculated using the Pythagorean theorem for a right triangle whose hypotenuse is an edge b, and the apothem h b and half of the base side a / 2 will be the legs. We get:
b 2 = h b 2 + a 2/4 =>
a = 2 * √ (b 2 - h b 2 ).
Substituting the known values from the condition, we obtain the value a = 18 cm.
To calculate the height h of the pyramid, you can do two ways: consider a right triangle with a hypotenuse-side rib or with a hypotenuse-apothem. Both methods are equal and involve the execution of the same number of mathematical operations. Let us dwell on a triangle, where the hypotenuse is the apothem h b . The legs in it are h and a / 2. Then we get:
h = √ (h b 2 -a 2/4) = √ (12 2 - 18 2/4) = 7.937 cm.
Now you can use the formula for volume V:
V = 1/3 * a 2 * h = 1/3 * 18 2 * 7.937 = 857.196 cm 3 .
Thus, the volume of a regular quadrangular pyramid is approximately 0.86 liters.
The volume of the Cheops pyramid
Now we will solve an interesting and practically important problem: we will find what the volume of the largest pyramid in Giza is equal to. From the literature it is known that the original height of the structure was 146.5 meters, and the length of its base is 230.363 meters. These numbers allow us to apply the formula for calculating V. We get:
V = 1/3 * a 2 * h = 1/3 * 230,363 2 * 146.5 ≈ 2591444 m 3 .
The resulting value is almost 2.6 million m 3 . This volume corresponds to the volume of the cube, the side of which is 137.4 meters.