Q&A for How to Calculate the Volume of a Square Pyramid

You cannot solve for height if all you know is the volume.

Because that's the formula for the volume: one-third of the product of the base area and the height.

To find the length of one side of the square base, you must know the height and the volume of the pyramid. To get the length, multiply the volume by three, divide that by the height, and then take that number and find its square root.

Draw a picture of the pyramid's cross section through the vertex. It should look like an isosceles triangle with two slant heights and one base length. If you draw an altitude from vertex to base, you have a right triangle with hypotenuse equal to slant height and a leg whose length is half the base length. Then you can use the Pythagorean Theorem to find the other leg, which is the pyramid's height you are looking for.

If you know the length of one of the bases, and it's a square pyramid, then you can draw a right triangle with the slant, edge, and half the base. The edge length will be the hypotenuse, and twice the length of the base.

In the case of a square pyramid, the surface area can be considered as consisting of just the four triangular sides, or as those four sides and the square base.

You cannot find the surface area of a pyramid if you know only the height.

The volume is one-third the base area multiplied by the perpendicular height. If the base is a rectangle measuring 34 x 24, the base area is 816 square units. One-third of that is 272. Multiplying by 15 gives a volume of 4,080 cubic units.

Assuming a square base, the height of a pyramid is three times the volume divided by the square of the length of a side of the base (that is, three times the volume divided by the base area).

If "only" the sides are given, you cannot find the area of the sides of the pyramid. You also need to know the height. If you are told that you have a "regular" square pyramid, then the edge height of the pyramid will be equal to the sides of the square. That is, each side of the pyramid will be an equilateral triangle, and that will be enough to find the area. If the side of the base is "s," then the edge height is also "s." You need to find the vertical height of the equilateral triangle. Using the relationships of 30-60-90 triangles, this height is s*(sqrt(3))/4. Since the area of a triangle is A=1/2bh, the area = (1/2)(s)(s*(sqrt(3))/4). This simplifies to A=(s^2)(sqrt(3))/8.

Without the volume you can't find the height.

You don't need the height. To find the diagonal, square the length of one side of the base, multiply by 2, then find the square root of that number.

I assume you mean that you have the height of the pyramid itself. You need to use the Pythagorean Theorem to calculate the slant height of the pyramid. Imagine a right triangle cutting through the center of the pyramid. Its base is half the side of the pyramid. Its height is the height of the pyramid. Its hypotenuse is the slant height. Assume the side is 6 and height is 10. Then (side/2)^2 + h^2 = slant ht.^2. So 3^2 + 10^2 = slant^2, or 109=slant^2. Then slant=10.4. Lateral Surface Area = 1/2*side*slant*4=1/2*6*10.4*4=124.8.

V=1/3*l*w*h, where l and w are the length and width of the rectangular base of the pyramid, and h is the height. Suppose the rectangle base measures 4cm x 3cm and the height is 7 cm. Then the volume is 1/3*4*3*7=28 cubic cm.

You would also have to know the volume. Assuming a square base, square the edge length to find the base area. Then divide that into three times the volume to find the height.

Your only options are to use one of the formulas in the above article, and they all involve a height measurement.

Multiply the volume by three, then divide that by the base area. So the height (h) = 3V / s², where V is the volume and s is the side of the (square) base.

In your example, where the height is equal to the base edge (s), V = 150 = s³ / 3. So 50 = s³, and s = the cube root of 50, or 3.684.