Frustum of a Cone

A frustum of a cone is the portion of a cone that remains after a smaller cone has been removed from the top, typically by slicing the cone parallel to its base. The result is a truncated cone with two circular bases: a larger base and a smaller top base. The frustum has a height, which is the perpendicular distance between the two bases, and a slant height, which is the distance along the slanted surface of the frustum between the edges of the two bases.

1.0What is the Frustum of a Cone?

The frustum of a right circular cone is essentially the portion of the cone that remains after slicing the cone parallel to its base. The cut results in two circular faces – one at the top and one at the bottom – and a slanted, curved surface connecting the two. Unlike a full cone, which has a single point at the top, a frustum has two circular bases, one larger than the other.

The two circular ends are known as: 

• The larger base with radius R,
• The smaller base with radius r,
• The height h, which is the perpendicular distance between the two bases.

2.0Net of Frustum of Cone

A frustum of a cone is a three-dimensional shape that results from slicing a cone parallel to its base. This shape has the following characteristics:

• Two circular bases (one larger and one smaller),
• A curved lateral surface that connects the two bases.

The net of a frustum of a cone is a two-dimensional representation of the surface area, consisting of two circles (representing the bases) and a sector (representing the lateral surface). When folded, the circles form the top and bottom, while the sector wraps around to create the curved surface, completing the frustum.

3.0Frustum of a Cone Volume Formula

The volume of a frustum of a cone is the space enclosed by the cone's truncated shape. It can be calculated using the following formula:

$V=\frac{1}{3}\pi H\left(R^2+Rr+r^2\right)$

Where:

• V = Volume of the frustum,
• R = Radius of the larger base,
• r = Radius of the smaller base,
• H = Height of the frustum.

This formula combines the areas of the two circular bases and the height, giving us the exact volume of the frustum. By using this, you can calculate how much space the frustum occupies.

4.0 Derivation of Volume of a Frustum of Cone 

To derive the formula for the volume of a frustum of a cone, let's start with the following information:

• The height of the frustum is H units.
• The slant height of the frustum is L units.
• The radii of the two circular bases are R (larger base) and r (smaller base).

Step 1: Concept of Similar Triangles

Consider the two triangles, ΔOO′D and ΔOPB, in the diagram.

Since the cones are similar, the corresponding angles are equal:

• $DO{O}^{\prime }=BOP$(common angle),
• $O^{\prime }DO=PBO$(corresponding angles).

This results in the similarity between the triangles ΔOO′D and ΔOPB by the AA (Angle-Angle) criterion.

Step 2: Applying the Condition for Similar Triangles

From the similarity of the triangles, the ratio of corresponding sides must be equal. This allows us to set up a ratio between the sides of the two triangles:

$\frac{O{O}^{\prime }}{OP}=\frac{O^{\prime }D}{PB}$. 

Step 3: Using the Ratios to Relate the Heights

Let the height of the smaller cone (the part that was "cut off") be denoted as h′. Using similar triangles, we can find a relationship between the height h′ and the height H of the frustum.

Substituting the values in the equation, we obtain the relationship:

$\frac{h^{\prime }}{H}=\frac{r}{R}$. 

Step 4: Deriving the Volume Formula

The formula for the volume of a frustum of a cone can be derived using the formula for the volume of a cone and the concept of the difference in volumes of two cones.

The volume of a cone is given by:

$V=\frac{1}{3}\pi r^{2}h$. 

For the frustum, the volume is the difference between the volumes of the larger cone (with radius R and height H + h') and the smaller cone (with radius r and height h′).

Thus, the volume of the frustum is:

$V=\frac{1}{3}\pi \left(R^2\left(H+h^{\prime }\right)-r^2{h}^{\prime }\right)$. 

Step 5: Simplifying the Expression

Substitute the value of h′ from the similarity condition:

$h^{\prime }=\frac{r}{R}H$. 

Now, substitute this expression for h′ into the volume formula:

$V=\frac{1}{3}\pi \left(R^2\left(H+\frac{r}{R}H\right)-r^2\frac{r}{R}H\right)$. 

Simplifying further:

$V=\frac{1}{3}\pi H\left(R^2+rR+r^2\right)$. 

Thus, the volume V of the frustum of a cone is given by the formula:

$V=\frac{1}{3}\pi H\left(R^2+Rr+r^2\right)$. 

This is the final formula for the volume of the frustum of a cone.

5.0Frustum of a Cone Surface Area Formula

The surface area of the frustum of a cone includes the areas of the two circular bases as well as the area of the slanted surface (the lateral surface). The formula for the surface area of a frustum of a cone is:

$A=\pi \left(R+r\right)\ell +\pi R^2+\pi r^2$

Where:

• A = Total surface area of the frustum,
• R = Radius of the larger base,
• r = Radius of the smaller base,
• ℓ = Slant height of the frustum (the distance along the slanted surface from the smaller base to the larger base),

The first part of the formula, π (R + r) ℓ, calculates the lateral surface area, while the other two terms  $\pi R^2$ and  $\pi r^2$  calculate the areas of the two circular bases.

6.0How to Find the Slant Height ℓ

The slant height ℓ is an essential part of calculating the surface area. You can find the slant height using the Pythagorean theorem if it's not directly given. The slant height is the hypotenuse of a right triangle where the base is the difference in the radii R – r, and the height is the vertical distance H. Therefore, the formula for the slant height is:

$\ell =\sqrt{H^2+{\left(R-r\right)}^2}$

7.0Solved Example on Frustum of Cone 

Example 1: A traffic cone is sliced horizontally, creating a frustum of a cone. The larger base has a radius of 5 cm, the smaller base has a radius of 3 cm, and the height of the frustum is 10 cm.

1. Calculate the volume of the frustum.
2. Calculate the surface area of the frustum, including both the curved surface and the two circular bases.

Solution: 

Given:

• Larger base radius, R = 5 cm,
• Smaller base radius, r = 3 cm,
• Height, H = 10 cm.

1. Using the volume formula:

$V=\frac{1}{3}\pi H\left(R^2+Rr+r^2\right)$

Substitute the given values:

$V=\frac{1}{3}\pi \left(10\right)\left(5^2+5\left(3\right)+3^2\right)$

$V=\frac{1}{3}\pi \left(10\right)\left(25+15+9\right)$

$V=\frac{1}{3}\pi \left(10\right)\left(49\right)=\frac{490}{3}\pi =513.13{\mathrm{cm}}^3$ 

2. Calculate the Slant Height $\ell$

Using the formula for the slant height:

$\ell =\sqrt{H^2+{\left(R-r\right)}^2}$

$\ell =\sqrt{10^2+{\left(5-3\right)}^2}$

$\ell =\sqrt{100+4}$

$\ell =\sqrt{104}=10.2\mathrm{cm}$

Calculate the Surface Area

Now, using the surface area formula:

$A=\pi \left(R+r\right)\ell +\pi R^2+\pi r^2$

Substitute the given values:

$A=\pi \left(5+3\right)\left(10.2\right)+\pi \left(5^2\right)+\pi \left(3^2\right)$

$A=\pi \left(8\right)\left(10.2\right)+\pi \left(25\right)+\pi \left(9\right)$

$A=\pi \left(81.6\right)+\pi \left(25\right)+\pi \left(9\right)=\pi \left(115.6\right)=363.61{\mathrm{cm}}^2$

Thus, the surface area is approximately $363.61{\mathrm{cm}}^2$ .

Example 2: A frustum of a cone is a bucket with the radii of the larger and smaller ends as 15 cm and 7 cm, respectively. The height of the frustum is 20 cm. Find the volume of the frustum. (Use $\pi =\frac{22}{7}$ ).

Solution:

Given:

• Radius of the larger base $R=15\mathrm{cm}$,
• Radius of the smaller base $r=7\mathrm{cm}$,
• Height of the frustum $H=20\mathrm{cm}$,
• $\pi =\frac{22}{7}$.

The formula for the volume of the frustum of a cone is:

$V=\frac{1}{3}\pi H\left(R^2+Rr+r^2\right)$

Substitute the given values:

$V=\frac{1}{3}\times \frac{22}{7}\times 20\times \left(15^2+15\times 7+7^2\right)$

Now, calculate each term inside the parentheses:

$15^2=225,15\times 7=105,7^2=49$

$225+105+49=379$

Substitute this back into the volume formula:

$V=\frac{1}{3}\times \frac{22}{7}\times 20\times 379$

Simplify the expression:

$V=\frac{1}{3}\times 22\times 20\times 379.7$

$V=\frac{1}{3}\times 22\times 20\times 54.14$

$V=\frac{1}{3}\times 2384.16$

$V=794.72{\mathrm{cm}}^3$

Answer:

The volume of the frustum is approximately 794.72 cm³.

Example 3: The radii of the two ends of a conical water tank are 12 m and 6 m, respectively, and the height of the tank is 18 m. Calculate the volume of water the tank can hold. (Use $\pi =\frac{22}{7}$ ).

Solution:

Given:

• Radius of the larger base R = 12 m,
• Radius of the smaller base r = 6 m,
• Height of the frustum h=18 m.

The formula for the volume of the frustum of a cone is:

$V=\frac{1}{3}\pi H\left(R^2+Rr+r^2\right)$

Substitute the given values:

$V=\frac{1}{3}\times \frac{22}{7}\times 18\times \left(12^2+12\times 6+6^2\right)$

Now, calculate each term inside the parentheses:

$12^2=144,12\times 6=72,6^2=36$

$144+72+36=252$

Substitute this back into the volume formula:

$V=\frac{1}{3}\times \frac{22}{7}\times 18\times 252$

Simplify the expression:

$V=\frac{1}{3}\times 22\times 18\times 252.7$

$V=\frac{1}{3}\times 22\times 18\times 36=\frac{1}{3}\times 14256$

$V=4752m^3$

Answer:

The volume of the tank is 4752 m³.

Example 4: A conical bucket is in the shape of a frustum of a cone. The radius of the larger circular base is 10 cm, and the radius of the smaller circular base is 4 cm. The slant height of the frustum is 12 cm. Calculate the total surface area of the bucket. (Use $\pi =\frac{22}{7}$​).

Solution:

Given:

• The radius of the larger base R = 10 cm,
• The radius of the smaller base r = 4 cm,
• The slant height ℓ = 12 cm,
• The height of the frustum H = 15 cm.

The formula for the total surface area of a frustum of a cone is:

$A=\pi \left(R+r\right)\ell +\pi R^2+\pi r^2$

$A=\pi \left(10+4\right)\times 12+\pi \times 10^2+\pi \times 4^2$

$A=\pi \times 14\times 12+\pi \times 100^{}+\pi \times 16$

Now calculate each term:

$A=\pi \times 168+\pi \times 100^{}+\pi \times 16$

$A=\pi \times \left(168+100^{}+16\right)$

$A=\pi \times 284$

Substitute $\pi =\frac{22}{7}$:

$A=\frac{22}{7}\times 284$

Multiply:

$A=\frac{22\times 284}{7}$

$A=\frac{6248}{7}$

$A=892{\mathrm{cm}}^3$

Answer:

The total surface area of the frustum is 892 cm².

Example 5: A frustum of a cone is formed by cutting a cone in two, leaving a smaller cone on top. The radius of the larger base of the frustum is 8 cm, and the radius of the smaller base is 5 cm. The slant height of the frustum is 10 cm. Find the lateral surface area of the frustum. (Use $\pi =\frac{22}{7}$​).

Solution:

Given:

• The radius of the larger base R = 8 cm,
• The radius of the smaller base r = 5 cm,
• The slant height ℓ = 10 cm.

The formula for the lateral surface area of a frustum of a cone is:

$A_{lateral}=\pi \left(R+r\right)\ell$

Substitute the given values into the formula:

$A_{lateral}=\pi \left(8+5\right)\times 10$

Simplify:

$A_{lateral}=\pi \times 13\times 10=\pi \times 130$

Now substitute $\pi =\frac{22}{7}$:

$A_{lateral}=\frac{22}{7}\times 130$

$A_{lateral}=\frac{2860}{7}$

$A_{lateral}=408.57{\mathrm{cm}}^3$

Answer:

The lateral surface area of the frustum is approximately 408.57 cm².

8.0Similar Questions

1. How do you find the volume of the frustum of a cone?

Ans: The volume V is:

$V=\frac{1}{3}\pi H\left(R^2+Rr+r^2\right)$

Where R and r are the radii of the two bases, and H is the height.

2.  How do you find the surface area of a frustum of a cone?

Ans: The total surface area A is:

$A=\pi \left(R+r\right)\ell +\pi R^2+\pi r^2$

Where ℓ is the slant height.

3. What is the lateral surface area of the frustum of a cone?

Ans: The lateral surface area is:

$A_{lateral}=\pi \left(R+r\right)\ell$

Where R and r are the radii and ℓ is the slant height.

4. How do you find the slant height of the frustum of a cone?

Ans: Use the Pythagorean theorem:

$\ell =\sqrt{H^2+{\left(R-r\right)}^2}$

Where H is the height, and R and r are the radii of the bases.