Volume of Cone
The volume of a cone is the amount of space occupied by a cone in a three-dimensional plane. A cone has a circular base, which means the base is made of a radius and diameter. Then from the center of the base, you can go to the topmost part of the cone (of course, in the case of ice cream, this portion is at the bottom) that is measured as the height. In this article, we will learn how to calculate the volume of a cone and its formula using solved examples.
| 1. | What is the Volume of a Cone? |
| 2. | Volume of a Cone Formula |
| 3. | Derivation of Volume of Cone |
| 4. | How to Find Volume of Cone? |
| 5. | FAQs on Volume of Cone |
What is Volume of Cone?
The volume of a cone is defined as the amount of space or capacity a cone occupies. The volume of cone is measured in cubic units like cm3, m3, in3, etc. A cone can be formed by rotating a triangle around any of its vertices. A cone is a solid 3-D shape figure with a circular base. It has a curved surface area. The distance from the base to the vertex is the perpendicular height. A cone can be classified as a right circular cone or an oblique cone. In the right circular cone, a vertex is vertically above the center of the base whereas, in an oblique cone, the vertex of the cone is not vertically above the center of the base.
Volume of Cone Formula
The volume of a cone formula is given as one-third the product of the area of the circular base and the height of the cone. According to the geometric and mathematical concepts, a cone can be termed as a pyramid with a circular cross-section. By measuring the height and radius of a cone, you can easily find out the volume of a cone. If the radius of the base of the cone is "r" and the height of the cone is "h", the volume of cone is given as V = (1/3)πr2h.
Volume of Cone With Height and Radius
The formula to calculate the volume of a cone, given the height and its base radius is:
V = (1/3)πr2h cubic units
Volume of Cone With Height and Diameter
The formula to calculate the volume of a cone, given the measure of its height and base diameter is:
V = (1/12)πd2h cubic units
Volume of Cone With Slant Height
By applying Pythagoras theorem on the cone, we can find the relation between volume and slant height of the cone.
We know, h2 + r2 = L2
⇒ h = √(L2 - r2)
where,
- h is the height of the cone,
- r is the radius of the base, and,
- L is the slant height of the cone.
The volume of the cone in terms of slant height can be given as V = (1/3)πr2h = (1/3)πr2√(L2 - r2).
Derivation of Volume of Cone Formula
Here is an activity that shows how the formula for the volume of a cone is obtained from the volume of a cylinder. Let us take a cylinder of height "h", base radius "r", and take 3 cones of height "h". Fill the cones with water and empty out one cone at a time.
Each cone fills the cylinder to one-third quantity. Hence, such three cones will fill the cylinder. Thus, the volume of a cone is one-third of the volume of the cylinder.
Volume of cone = (1/3) × Volume of cylinder = (1/3) × πr2h = (1/3)πr2h
How to Find Volume of Cone?
Given the required parameters, the volume of a cone can be calculated by applying the volume of cone formula. The below-given steps can be followed when either the base radius or the base diameter, height, and slant height of cone are known.
- Step 1: Note down the known parameters, 'r' as the radius of the base of cone, 'd' as diameter, 'L' as slant height, and 'h' as the height.
- Step 2: Apply the formula to find the volume of cone,
Volume of cone using base radius: V = (1/3)πr2h or (1/3)πr2√(L2 - r2)
Volume of cone using base diameter: V = (1/12)πd2h = (1/12)πd2√(L2 - r2) - Step 3: Express the obtained result in cubic units.
Example: Find the volume of a cone whose radius is 3 inches and height is 7 inches. (Use π = 22/7).
Solution: As we know, the volume of the cone is (1/3)πr2h.
Given that: r = 3 inches, h = 7 inches and π = 22/7
Thus, Volume of cone, V = (1/3)πr2h
⇒ V = (1/3) × (22/7) × (3)2 × (7) = 22 × 3 = 66 in3
∴ The volume of cone is 66 in3.
Volume of Cone Tips
- The volume of a hemisphere with radius "r" is equal to the volume of a cone having radius "r" and height equal to '2r'. Thus, (1/3)πr2(2r) = (2/3)πr3.
- The volume of a cone can be calculated using the diameter, by dividing the diameter by 2 to find the radius, then applying the value into the volume of a cone formula (1/3)πr2h.
FAQs on Volume of Cone
What is the Volume of Cone?
The amount of space occupied by a cone is referred to as the volume of a cone. The volume of the cone depends on the base radius of the cone and the height of the cone. It can also be expressed in terms of its slant height wherever necessary.
What is the Volume of a Cone Formula?
The formula for the volume of a cone is one-third of the volume of a cylinder. The volume of a cylinder is given as the product of base area to height. Hence, the formula for the volume of a cone is given as V = (1/3)πr2h, where, "h" is the height of the cone, and "r" is the radius of the base.
☛Check:
What is Surface Area and Volume of a Cone?
As a cone has a curved surface, thus it has two surface area formulas, curved surface area as well as total surface area. These surface area formulas for the cone is listed below:
If the radius of the base of the cone is "r" and the slant height of the cone is "l", the surface area of a cone is given as:
- Total Surface Area of Cone, T = πr(r + l)
- Curved Surface Area of Cone, S = πrl
Whereas, the volume of a cone is one-third of the volume of a cylinder which is expressed as V = (1/3)πr2h cubic units. Here 'h' and 'r' refer to the height and radius of a cone.
☛Check:
How to Calculate Volume of a Cone Using Calculator?
To calculate the volume of a cone using a calculator the very important keynote is to remember the volume of a cone formula, i.e., V = (1/3)πr2h cubic units. By putting the values of h, r, and pi (constant 3.14 o 22/7) we can calculate the cone's volume using the volume of the cone calculator.
☛Check and practice the questions related to the volume of a Cone:
Can You Find the Volume of Cone with Slant Height?
Yes, we can find the formula of a cone with slant height. The formula for the volume of a cone is (1/3)πr2h, where, "h" is the height of the cone, and "r" is the radius of the base. In order to find the volume of the cone in terms of slant height, "L", we apply the Pythagoras theorem and obtain the value of height in terms of slant height as √(L2 - r2). This value is further substituted in the volume of cone formula as h = √(L2 - r2). Thus, the volume of the cone in terms of slant height is (1/3)πr2√(L2 - r2).
How Do You Find the Volume of Cone with Diameter and Slant Height?
The formula for the volume of a cone is (1/3)πr2h, where, "h" is the height of the cone, and "r" is the radius of the base. Thus, the volume of the cone in terms of slant height, "L" is (1/3)πr2√(L2 - r2). We can determine the volume of the cone with the diameter and slant height by substituting r = (D/2), where D is the diameter of the cone. Hence, the formula for the volume of the cone is (1/3)π(D/2)2√(L2 - (D/2)2).
☛ Try these for quick calculations:
What is Volume of a Cone in Terms of Pi?
The volume of a cone in terms of pi can be defined as the total amount of capacity required by the cone that is represented in terms of pi. The unit of volume of a cone in terms of pi is always expressed in terms of cubic units where the unit can be cm3, m3, in3, ft3, etc.
What Is the Volume of the Cone Formula for Partial Cone?
The volume of a cone formula for a partial cone is given as, volume of a partial cone, V = 1/3 × πh(R2 + Rr + r2). In the formula, small 'r' and capital 'R' are the base radii, such that R > r, and 'h' is the height.
What Is the Volume of a Cone Formula for Frustum of a Cone?
The volume of a cone formula for the frustum of a cone is defined as the number of unit cubes that can be fit into it. The volume (V) of the frustum of a cone is calculated using any one of the following formulas listed below.
- V = πh/3 [ (R3 - r3) / r ] (OR)
- V = πH/3 (R2 + Rr + r2)
How is the Volume of a Cone Affected By Doubling the Height?
The volume of the cone depends on the base radius, "r" of the cone, and the height, "h" of the cone. Thus, the volume of the cone gets doubled if the height of the cone is doubled as "h" is substituted by "2h" as V = (1/3)πr2(2h) = 2 ((1/3)πr2h).
What Happens to the Volume of a Cone When the Radius and Height are Doubled?
The volume of the cone will become eight times the original volume if the radius and height of the cone are doubled as, radius, "r" is substituted by 2r and height, "h" is substituted by 2h, V = (1/3)π(2r)2(2h) = 8((1/3)πr2(h)).
What Happens to the Volume of a Cone If the Height is Tripled and the Diameter of the Base is Doubled?
The volume of the cone will be twelve times the original volume if the height of the cone is tripled as "h" is substituted by 3h and diameter, D is substituted by 2D, V = (1/3)π(2D/2)2(3h) = πD2h = 12((1/3)π(D/2)2(h)).