Before getting to the main formula of calculating the volume of a square pyramid, let’s discuss what the pyramid is.

Pyramid is a polyhedron, one of the faces of which (called the base) is an arbitrary polygon, and the other faces (called lateral faces) are triangles with a common vertex. By the number of angles of the base the pyramids can be triangle (tetrahedron), quadrangular, and so on. Pyramid is a particular case of the cone.

The start of pyramid geometry began in ancient Egypt and Babylon, however, its active development happened in ancient Greece. The volume of the pyramid (including the volume of a square pyramid) was known to the ancient Egyptians. The first Greek mathematician who established what the volume of the pyramid (volume of a square pyramid) equals to was Democritus, and Eudoxus of Cnidus proved that. Ancient Greek mathematician Euclid systematized knowledge of the pyramid in the 12th volume of his «Principia» and gave the first definition of the pyramid: it is the corporeal figure bounded by the planes that from one plane converge at a single point.

Elements of a Pyramid

- Apothem is the height of the side face of the right pyramid, conducted from the top of it.
- Lateral sides are triangles converging at the apex.
- Lateral edges are the common sides of the side faces.
- Top of the pyramid is the point connecting the lateral edges and not lying in the plane of the base.
- Height is a segment of the perpendicular drawn through the top of the pyramid to its plane of the base (the ends of this segment are the top of the pyramid and the base of perpendicular).
- The diagonal section of the pyramid is a section of a pyramid passing through the top and the diagonal of the base.
- Base is a polygon, which does not have the top of the pyramid.

Square pyramid is a volume figure with a base in the form of a square and triangular side faces. The top of the square pyramid is projected in the center of the base. If a is the side of the square base, h is the height of the pyramid (the perpendicular dropped from the top of the pyramid to its base center), then the volume of a square pyramid can be calculated by the formula: a2 x (1/3) h. This formula for finding the volume of a square pyramid is valid for a square pyramid of all sizes (from souvenir pyramids to the Egyptian pyramids).

At first, find the side of a base. Since the base of the pyramid is a square, all sides of the base are equal. It is therefore necessary to find the length of any side of the base. For example, let’s assume we have a pyramid with the base side that equals to 5 cm.

If the sides of the base are not equal to each other, then you are dealing with a rectangular rather than square pyramid. However, the formula for calculating the volume of the rectangular pyramid is similar to the formula for computing the volume of a square pyramid. If l and w are two adjacent (unequal) sides of the rectangle in the bottom of the pyramid, the volume of the pyramid can be calculated as follows: (l x w) x (1/3) h.

Now, calculate the area of the square base by multiplying the side of the square by the other side of the square (or, in other words, squaring the side of the square). In our example: 5 x 5 = 52 = 25 cm2.

Do not forget that the area is measured in square unities (square centimeters, square meters, square kilometers, and so on).

The next step is to multiply the area of the base by the height of the pyramid. The height is the perpendicular dropped from the top of the pyramid to its base. When multiplying these values, you will get the volume of a cube with the same base and height as that of the pyramid. In our example, the height is 9 cm, so 25 cm 2 x 9 cm = 225 cm3. Remember that the volume is measured in cubic units, in this case, in cubic centimeters.

Now, in order to find the volume of a square pyramid, divide the result by 3. In our example: 225 cm3 / 3 = 75 cm3. The volume is measured in cubic units.

If you are given either the area or the height of the pyramid and its apothem, you can find the volume of a pyramid, using the Pythagorean Theorem. The apothem is the height of the sloping triangular face of the pyramid, drawn from the vertex of the triangle to its base. To find the apothem, use the base of the pyramid and its height. Apothem divides the base in half and intersects it at a right angle.

Now study the right-angled triangle formed with apothem, height, and the segment connecting the center of the base and the middle of its side. In such a triangle, apothem is the hypotenuse, which can be found using the Pythagorean Theorem. The segment connecting the center of the base and the middle of its side is equal to half of the base (this segment is one of the legs; the second leg is the height of the pyramid).

Recall that the Pythagorean Theorem is written as follows: a2 + b2 = c2, where a and b are legs, and c is the hypotenuse of a right triangle.

For example, there is a pyramid with a base of 4 cm and the apothem of 6 cm. In order to find the height of the pyramid, put these values in the Pythagorean Theorem: a2 + b2 = c2; a2 + (4/2)2 = 62; a2 = 32; a = √32 = 5,66 cm. So, you have found the second leg of a right triangle, which is the height of the pyramid (likewise, if there was a value of the apothem and the height of the pyramid, you could have found the half of the base of the pyramid).

Use the values to find the volume of a square pyramid by the formula: a2 x (1/3) h. In our example, you have calculated that the height of the pyramid is equal to 5.66 cm. Substitute the values into the formula to calculate the volume of a square pyramid: a2 x (1/3) h; 42 x (1/3) (5,66); 16 x 1,89 = 30,24 cm3.

If don’t have a value of the apothem, use the edge of the pyramid to calculate the volume of a square pyramid. The edge is a segment connecting the top of the pyramid with the top of the square at the base of the pyramid. In this case, you will get a right-angled triangle, legs of which are the height of the pyramid and the half of the diagonal of the square at the base of the pyramid, and the hypotenuse is the edge of the pyramid. Since the diagonal of a square is equal to √2 x side of the square, you can find a side of the square (base) by dividing the diagonal by √2. Then you’ll find the volume of a square pyramid by the formula described above.

For example, there is a square pyramid with the height of 5 cm and the edge of 11 cm. Calculate the half of the diagonal as follows: 52 + b2 = 112; b2 = 96; b = 9,80 cm.

You have found the half of the diagonal, a diagonal is therefore: 9.80 cm x 2 = 19.60 cm.

The side of the square (base) is equal to √2 × diagonal, so 19,60 / √2 = 13,90 cm. Now find the volume of a square pyramid according to the formula: a2 x (1/3) h; 13,902 x (1/3) (5); 193,23 x 5/3 = 322,05 cm3.

In a square pyramid, its height, the base, and the apothem are linked with the Pythagorean Theorem: (side ÷ 2)2 + (height)2 = (apothem)2.

In any right pyramid, the apothem, the base, and the edge are connected with the Pythagorean Theorem: (side ÷ 2)2 + (apothem)2 = (rib)2.

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