Volume of a Geometric Solid
How much water fits in an aquarium? How much ice cream is in an ice cream cone? How many lead balls can you make by melting down one big sphere? All of these questions are about volume: the space a solid takes up.
Units of Volume and Capacity
Volume is measured in cubic units: $1 \text{ cm}^3$ is a little cube with 1 cm edges. Each unit is 1,000 times the next one (because $10 \times 10 \times 10$).
An aquarium is in the shape of a rectangular prism with the following dimensions: $60\text{ cm}$ in length, $30\text{ cm}$ in width and $40\text{ cm}$ in height.…
An aquarium is in the shape of a rectangular prism with the following dimensions: $60\text{ cm}$ in length, $30\text{ cm}$ in width and $40\text{ cm}$ in height. Calculate how many liters of water fit in the aquarium if it is completely filled. (Remember that $1\text{ dm}^3 = 1\text{ liter}$).
Prisms and Cylinders
A prism or a cylinder is a "stack" of identical bases. Its volume is base area times height:
$$V = A_{base} \times h \qquad \text{Rectangular prism: } V = a \times b \times c \qquad \text{Cylinder: } V = \pi r^2 h$$There are several strategies for arriving at the same formula: counting $1 \text{ cm}^3$ cubes layer by layer (the number that fit in the base, times the number of layers), stacking "coins" as in the drawing, or breaking the base into triangles and rectangles. They all say the same thing: base times height.
A package containing a gaming keyboard is in the shape of a rectangular prism. Its dimensions are $45\text{ cm}$ in length, $15\text{ cm}$ in width and $5\text{ cm}$ in…
A package containing a gaming keyboard is in the shape of a rectangular prism. Its dimensions are $45\text{ cm}$ in length, $15\text{ cm}$ in width and $5\text{ cm}$ in height. Calculate the volume of air inside the box in cubic centimeters.
A cylindrical drink can has a diameter of $6\text{ cm}$ and a height of $12\text{ cm}$. Calculate its volume using the value of $\pi \approx 3.14$.
A cylindrical drink can has a diameter of $6\text{ cm}$ and a height of $12\text{ cm}$. Calculate its volume using the value of $\pi \approx 3.14$.
Calculate the volume of a prism whose base is a right-angled triangle with legs $6\text{ cm}$ and $8\text{ cm}$, knowing that the height of the prism is $15\text{ cm}$.
Calculate the volume of a prism whose base is a right-angled triangle with legs $6\text{ cm}$ and $8\text{ cm}$, knowing that the height of the prism is $15\text{ cm}$.
Pyramids and Cones: One Third
If you fill a pyramid with water and pour it into a prism with the same base and the same height, you need exactly three pyramids to fill it. The same thing happens with the cone and the cylinder.
$$V_{pyramid} = \frac{A_{base} \times h}{3} \qquad\qquad V_{cone} = \frac{\pi r^2 h}{3}$$A decorative glass pyramid has a square base of side $8\text{ cm}$ and a height of $10\text{ cm}$. Calculate the volume of glass needed to manufacture it.
A decorative glass pyramid has a square base of side $8\text{ cm}$ and a height of $10\text{ cm}$. Calculate the volume of glass needed to manufacture it.
An ice cream cone has the shape of a cone. The radius of the base is $3\text{ cm}$ and its height is $12\text{ cm}$. Calculate how many cubic centimeters of ice cream…
An ice cream cone has the shape of a cone. The radius of the base is $3\text{ cm}$ and its height is $12\text{ cm}$. Calculate how many cubic centimeters of ice cream fit inside it (without overflowing at the top).
An ice cream cone is to be made whose slant height measures $15 \text{ cm}$ and whose base radius is $6 \text{ cm}$. Calculate the volume of ice cream that fits inside…
An ice cream cone is to be made whose slant height measures $15 \text{ cm}$ and whose base radius is $6 \text{ cm}$. Calculate the volume of ice cream that fits inside the cone. If a perfect hemisphere of ice cream of the same radius is added on top of the base when served, what will be the total volume of the ice cream?
The Sphere
$$V_{sphere} = \frac{4}{3}\pi r^3$$The PE teacher stores handball balls in a box. If each ball has a radius of $9\text{ cm}$, calculate the volume of one of them rounding to two decimal places. (Use…
The PE teacher stores handball balls in a box. If each ball has a radius of $9\text{ cm}$, calculate the volume of one of them rounding to two decimal places. (Use $\pi \approx 3.14$).
Three metal balls of radius $3 \text{ cm}$ are melted down to create a single larger sphere. Assuming no material is lost in the process, what will be the radius of the…
Three metal balls of radius $3 \text{ cm}$ are melted down to create a single larger sphere. Assuming no material is lost in the process, what will be the radius of the resulting new sphere? Round the answer to two decimal places.
Cone, Sphere, and Cylinder: Archimedes' Relationship
Take a cylinder with radius $r$ and height $2r$ (the sphere of radius $r$ fits snugly inside). Now fit into it, one at a time, a cone with the same base and the same height, and the sphere. What fraction of the cylinder does each one fill?
If you do it with water, you can see it without formulas: the sphere fills two cones, and one cone plus the sphere fill the cylinder. Conversely, this relationship is a way to remember (and justify) the sphere formula: if the cylinder measures $2\pi r^3$, the sphere is two thirds of it, $\frac{2}{3} \times 2\pi r^3 = \frac{4}{3}\pi r^3$.
Cavalieri's Principle states that if two solids have the same height and the area of their cross-sections at any height is equal, then they have the same volume. Use…
Cavalieri's Principle states that if two solids have the same height and the area of their cross-sections at any height is equal, then they have the same volume. Use this idea to compare the volume of a sphere of radius $R$ with the volume contained between a cylinder of radius $R$ and height $2R$, and a double cone inscribed in it. Show mathematically that the volume of the sphere is indeed $V = \frac{4}{3}\pi R^3$ by first calculating the volume of the cylinder and subtracting that of the two cones.
An hourglass consists of two identical cones joined at their vertices inside a protective glass cylinder. The cones have a base radius of $6\text{ cm}$ and a height of…
An hourglass consists of two identical cones joined at their vertices inside a protective glass cylinder. The cones have a base radius of $6\text{ cm}$ and a height of $10\text{ cm}$. If the upper chamber is filled with sand up to half its height, calculate the volume of sand in $cm^3$. If the sand flows into the lower chamber at a constant rate of $5\text{ cm}^3$ per minute, how long will it take for the upper part to empty? (Remember to use similar cones to find the radius of the sand's surface).
Composite Solids
Many real objects combine several solids: a silo (cylinder + cone), a capsule (cylinder + two hemispheres), a piece with a hole in it. You break them down, calculate each part, and then add the volumes (or subtract the hole).
A cylindrical glass with an internal diameter of $6 \text{ cm}$ contains water up to a height of $10 \text{ cm}$. An irregularly shaped stone is placed inside it and the…
A cylindrical glass with an internal diameter of $6 \text{ cm}$ contains water up to a height of $10 \text{ cm}$. An irregularly shaped stone is placed inside it and the water level rises to $12.5 \text{ cm}$ without overflowing. What is the volume of the stone expressed in cubic centimeters?
Calculate the volume of a geometric solid formed by a cylinder of height $10 \text{ cm}$ and radius $4 \text{ cm}$, topped by a square pyramid whose base is inscribed in…
Calculate the volume of a geometric solid formed by a cylinder of height $10 \text{ cm}$ and radius $4 \text{ cm}$, topped by a square pyramid whose base is inscribed in the circle of the cylinder and whose height is $6 \text{ cm}$.
(Hint: the side of the square base of the pyramid is equal to the side of a square inscribed in a circle of radius $4 \text{ cm}$).
The municipality has installed concrete planters in the shape of a regular hexagonal prism. If each side of the base hexagon measures $30 \text{ cm}$ and the height of…
The municipality has installed concrete planters in the shape of a regular hexagonal prism. If each side of the base hexagon measures $30 \text{ cm}$ and the height of the planter is $60 \text{ cm}$, calculate the volume of soil (in liters) needed to fill it completely.
(Given: remember that you must first find the apothem of the base using the Pythagorean theorem).