While a cylinder is formed by a pair of congruent circles joined by a curved surface, a cone is formed from a single circle with a curved surface that meets itself at the apex (or vertex).
Use the button to pour liquid from the cone into the cylinder.
How many times does the liquid in the cone fill the cylinder?
How do you think the formula for the volume of a cone will relate to the formula for the volume of a cylinder?
The volume of a cone is exactly one-third the volume of a cylinder formed from the same base with the same perpendicular height. That is, the volume of a cone is given by the general formula:
We can rewrite this formula more specifically for cones as:
Find the volume of this cone and round your answer to two decimal places.
A cone is sawed in half to create the following solid:
What is the volume of the solid?
A cone has a volume of 160 \text{ mm}^{3}. If the perpendicular height and radius of the cone are equal in length, find the radius r in mm. Round your answer to two decimal places.
An ice cream shop currently has one size ice cream cone. They are adding a new, larger size option.
The new cone will have 2 times the volume of the original cone.
The height of the new cone will remain the same as the height of the original cone.
How does the new radius compare to the radius of the original cone?
The volume of a cone can be found by taking one-third the volume of a cylinder with the same base area and height. The formula for the volume of a cone is given by:
This can be rewritten specifically for cones as:
A pyramid is a figure formed from a polygonal base and a set of triangular faces. The triangular faces connect to one side of the base and all join together at the apex.
Drag the sliders to fold the pyramids in to the cube and change the size of the figure.
How many pyramids fit into the prism?
If the volume of a prism is found using the formula V=Bh, how do you think we can find the volume of a pyramid?
Similarly to the volume of a cylinder and a cone, the volume of a pyramid is also calculated by taking one-third the volume of a prism that has the same height and base area.
A more specific formula is dependent on the shape of the base.
The Pyramid of Giza is a square pyramid, that is 280 Egyptian royal cubits high and has a base length of 440 Egyptian royal cubits. What is the volume of the Pyramid of Giza?
A small square pyramid of height 4 \text{ cm} was removed from the top of a large square pyramid of height 8 \text{ cm} forming the solid shown. Find the exact volume of the solid.
The height and base side of a triangular pyramid are tripled. The triangular base is equilateral. Determine the effect this has on the volume of the pyramid.
The volume of a pyramid can be found by taking one-third the volume of a prism with the same base area and height. The formula for the volume of a pyramid is given by:
Similar to prisms, we can consider the nets of pyramids and cones to determine their surface area and lateral surface area.
Flatten the cone using the sliders and then straighten the arc and dissect the sector created by the lateral face of the cone. Use the sliders to explore what happens.
The formula for the lateral surface area of a cone is given by LA=\pi r h_s, where r is the radius of the base of the cone and h_s is the slant height of the cone. How does this formula relate to the area of the rearranged lateral face on the cone?
The surface area of a pyramid or cone is the sum of the area of the base and the area of the lateral face. The formula can be written generally as:
For a cone we must find the area of the circular base and add that to the area of the sector that creates the lateral face.
\text{Area of the sector}=\pi l^{2} \cdot \frac{r}{l}=\pi rl
We can then combine this area with the area of the circular base to get a formula for the surface area of a cone given by:
\begin{aligned} \text{Surface area of right cone} &= \text{Area of base} + \text{Area of sector} \\ SA &= \pi r^2 + \pi r l \end{aligned} where r is the cone's base radius and l is the slant height.
Consider the diagram of the right square pyramid and its net shown:
Let b= the base of each triangle and l= the height of each triangle.
Explain a process for finding the surface area of the square pyramid.
Now, consider the regular right octagonal pyramid shown below, where a= the distance from the midpoint of a side of the base to the center of the base:
Draw the net of the right octagonal pyramid and describe how to find the surface area of the pyramid.
Write a formula for finding the surface area of a regular, right n-gon pyramid. Explain your reasoning.
What is the surface area of the following cone?
The Pyramid of Giza is a square pyramid, that is 280 Egyptian royal cubits high and has a base length of 440 Egyptian royal cubits. What is the surface area of the exposed walls of Pyramid of Giza?
A decorative lampshade is designed in the shape of a right circular cone with an original height of 8\text{ in} and a radius of 4\text{ in}. To fit a larger lamp base, the height of the lampshade is increased by 6\text{ in}. Determine the new total surface area of the cone.
Provide your answer rounded to the nearest whole number.
The surface area of a cone or pyramid can be calculated using the formula: