Eighth Grade Mathematics: Volume of Cones and Relationship to Cylinders
In geometry, whenever a three-dimensional solid tapers to a single sharp point called an apex, its volume contracts significantly compared to a prism or cylinder of identical base and height. A cone is the circular counterpart to a pyramid: it features a flat circular base that tapers smoothly upward to an apex vertex. The mathematical relationship between the volume of a cone and the volume of a cylinder sharing the exact same base and height is one of the most elegant integer fractions in all of geometry: exactly one-third. In this chapter, you will investigate this one-third relationship, master the cone volume formula, and calculate slant height versus perpendicular height.
The One-Third Principle: Cones versus Cylinders
Imagine a hollow cylinder and a hollow cone that have the exact same circular base radius r and the exact same perpendicular height h.
The Pouring Experiment:
Cone (Radius r, Height h) Cylinder (Radius r, Height h)
* Apex +---------+
/ \ / / \ +-------------+
h / \ h | |
/ \ | |
+---------+ +-------------+
\ / \ /
+-----+ +---------+
If you fill the cone completely with water or sand and pour it into the cylinder, it will fill exactly one-third of the cylinder's volume. You must fill and pour the cone three complete times to fill the cylinder to its brim.
Volume Relationship:
Volume of Cylinder = pi * r^2 * h
Volume of Cone = 1/3 * (Volume of Cylinder)
= 1/3 * pi * r^2 * h
This one-third ratio holds universally for all pyramids and cones in Euclidean geometry.
The Crucial Distinction: Height (h) versus Slant Height (l)
One of the most frequent errors in calculating the volume of a cone is confusing the perpendicular height h with the slant height l:
* Apex
/| / | Slant Height l/ |h \ Slant Height l (Along the outside edge)
/ | /____|_____ r Radius r
(Perpendicular 90-degree right angle at center)
- Perpendicular Height (h): The vertical distance measured at a 90-degree right angle from the center of the circular base straight up to the apex. This is the height required for the volume formula!
- Slant Height (l): The diagonal distance measured along the outside curved surface from the perimeter of the circular base to the apex.
- Radius (r): The distance from the center of the base to its outer edge.
Notice that the radius r, the perpendicular height h, and the slant height l form a right triangle inside the cone. By the Pythagorean Theorem:
r^2 + h^2 = l^2
If a problem provides the slant height l and the radius r, you MUST use the Pythagorean Theorem to solve for the vertical height h before calculating the volume: h = sqrt(l^2 - r^2).
Computing the Volume of a Cone
To calculate the volume of a cone, follow this step-by-step procedure:
Step 1: Confirm you have the radius r and perpendicular height h.
Step 2: Square the radius: r^2 = r * r.
Step 3: Multiply by height h: (r^2 * h).
Step 4: Multiply by 1/3 (or divide the product by 3).
Step 5: Multiply by pi (or attach pi for the exact answer).
Example: A cone has a base radius r = 6 cm and height h = 10 cm.
Step 1: r = 6, h = 10.
Step 2: r^2 = 6^2 = 36.
Step 3: r^2 * h = 36 * 10 = 360.
Step 4: 1/3 * 360 = 120.
Step 5: Exact Volume = 120*pi cubic centimeters.
Approximate Volume (pi = 3.14) = 120 * 3.14 = 376.8 cubic centimeters.
Notice how dividing 360 by 3 first (yielding 120) simplifies manual calculation before dealing with pi.
Chapter Exercises
Exercise 1: Calculate the exact volume in terms of pi and the approximate volume using 3.14 for an ice cream cone with a base diameter of 8 centimeters and a perpendicular height of 9 centimeters.
Exercise 2: A party hat shaped like a cone has a base radius of 5 inches and a slant height of 13 inches. Use the Pythagorean Theorem to find the perpendicular height h of the cone, and then calculate the exact volume of the hat in terms of pi.
Exercise 3: A cylindrical container and a conical container have identical base radii and identical heights. The cylinder holds 540 cubic centimeters of liquid. How much liquid does the cone hold? Explain your reasoning in paragraph form using the fractional relationship between cones and cylinders.
Solutions and Explanations
Solution 1: The cone has a diameter of 8 centimeters, so its base radius is r = 8 / 2 = 4 centimeters, and its perpendicular height is h = 9 centimeters. Apply the cone volume formula V = (1/3) * pi * r^2 * h. First, square the radius: r^2 = 4^2 = 16. Next, multiply by the height: 16 * 9 = 144. Now divide by 3 (or multiply by 1/3): 144 / 3 = 48. Multiplying by pi gives the exact volume: V = 48pi cubic centimeters. To find the decimal approximation using 3.14, multiply 48 by 3.14: 48 * 3.14 = 150.72 cubic centimeters. The exact volume is 48pi cm^3 and the approximate volume is 150.72 cm^3.
Solution 2: In the conical party hat, the radius r = 5 inches and the slant height l = 13 inches form a right triangle with the unknown perpendicular height h: r^2 + h^2 = l^2. Substituting the known values gives 5^2 + h^2 = 13^2, which simplifies to 25 + h^2 = 169. Subtracting 25 from both sides gives h^2 = 169 - 25 = 144. Extracting the principal square root gives the perpendicular height: h = sqrt(144) = 12 inches (a classic (5, 12, 13) Pythagorean triple). Now calculate the volume using V = (1/3) * pi * r^2 * h: V = (1/3) * pi * (5^2) * 12 = (1/3) * pi * 25 * 12. Multiply 25 by 12 to get 300, and multiply by 1/3 to get 100. Therefore, the exact volume of the party hat is 100*pi cubic inches.
Solution 3: The volume of any cone is mathematically defined as exactly one-third of the volume of a cylinder that shares the exact same base radius and perpendicular height: V_cone = (1/3) * V_cylinder. Given that the cylinder has a volume of 540 cubic centimeters, the volume of the matching cone is calculated by taking one-third of 540: V_cone = (1/3) * 540 = 540 / 3 = 180 cubic centimeters. This one-third relationship is invariant across all cones and cylinders with identical dimensions because a cone tapers uniformly from its circular base to an apex vertex, enclosing exactly one-third of the three-dimensional space enclosed by the straight-walled cylinder. Therefore, the cone holds exactly 180 cubic centimeters of liquid.