Question

Choose either pyramids or cones, and explain how the formula for their volume can be described using other solids

Answer 1

A pyramid is a three-dimensional geometric solid with a polygonal base and triangular faces that meet at a common point called the apex. The volume of a pyramid can be described using other solids, specifically prisms.

Imagine placing a pyramid inside a prism in such a way that the pyramid's apex coincides with one vertex of the prism's base, and the pyramid's base lies flat on one of the faces of the prism. Let's assume that the pyramid and the prism have the same height, and the base of the pyramid and the corresponding face of the prism have the same shape and size.

Now, let's add more pyramids inside the prism in the same way, stacking them on top of each other, until the topmost pyramid's apex reaches the height of the prism's top face. The pyramid layers will form a stepped pattern within the prism.

At this point, you'll notice that the stepped pattern formed by the pyramids inside the prism exactly fills the entire volume of the prism, without any gaps or overlaps.

Now, here's the key observation: The volume of the prism is equal to the sum of the volumes of the individual pyramids inside it. Since the base and height of all the pyramids are the same, we can write the formula for the volume of the prism as:

Volume of Prism = (Area of Base of Pyramid) × Height of Prism

Since the base of the pyramid is a polygon, the area of the base can be calculated using the appropriate formula for that polygon (e.g., area of a triangle, rectangle, etc.).

So, when you calculate the volume of a pyramid, you're essentially finding the volume of the prism that contains it. This relationship between the volume of a pyramid and the volume of a prism helps us understand and describe the volume of pyramids in terms of other familiar solids, making it easier to visualize and understand their dimensions.

Answer 2

Pyramid = one-third the volume of a prism with same base and height

Cone = one-third the volume of a cylinder with same base and height.

Explanation:

Pyramids and cones are both solids that have a base and a vertex, but they differ in the way that their bases are shaped. Pyramids have a polygonal base, while cones have a circular base.

Pyramid: The volume of a pyramid is equal to one-third the volume of a prism with the same base and height. The prism can be made by slicing the pyramid parallel to the base and rearranging the slices.

The formula for the volume of a pyramid is:

`\boxed{\tt V = (1)/(3)B h}`

where:

• V is the volume of the pyramid
• B is the area of the base
• h is the height of the pyramid

Cone: The volume of a cone is equal to one-third the volume of a cylinder with the same base and height. The cylinder can be made by slicing the cone along a radius and flattening the resulting shape.

The formula for the volume of a cone is :

`\boxed{\tt V =(1)/(3) \pi r^2 h}`

where:

• π is the mathematical constant pi (approximately equal to 3.14)
• r is the radius of the base
• h is the height of the cone