Conical Frustum Calculator

The simple formulas to compute the conical frustum height, slant height, radius, surface area, and volume are given here. Have a look at them and follow them when finding the unknown measures. These are the conical frustum formulas in terms of both circles radius and height.

• The Volume of a Conical Frustum Formula:
  • Volume V = (1/3) * π * h * (r² + R² + (r * R))
• Lateral Surface Area of a Conical Frustum Formula:
  • Lateral Surface Area L = π * (r + R) * s = π * (r + R) * √(r - R)² + h²)
• Top Surface Area of a Conical Frustum Formula:
  • Top Surface Area T = πr²
• Base Surface Area of a Conical Frustum Formula:
  • Base Surface Area B = πR²
• Total Surface Area of a Conical Frustum Formula:
  • Total Surface Area A = π * (r² + R² + (r + R) * s) = π * [ r² + R² + (r + R) * √((r - R)² + h²) ]
• Slant Height of a Conical Frustum Formula:
  • Slant Height s = √((r - R)² + h²)

Conical Frustum Calculations:

These formulas are helpful to calculate the unknown measures easily.

• If you know lateral surface area of the frustum, and two circles radii, then
  • Slant height s = L / (π * (r + R))
  • Height h = √(s² - (r - R)²)
  • Volume V = (1/3) * π * √(s² - (r - R)²) * (r² + R² + (r * R))
  • Top surface Area T = πr²
  • Bottom surface Area B = πR²
  • Total Surface Area A = L + B + T = L + πR² + πr²
• If radius of top, bottom circles and volume of frustum are given, then
  • Height h = (3 * V) / (π * (r² + R² + (r * R)))
  • Slant Height s = √((r - R)² + (3 * V) / (π * (r² + R² + (r * R)))²)
  • Top Surface Area T = πr²
  • Base Surface Area B = = πR²
  • Lateral Surface Area L = π * (r + R) * s
  • Total Surface Area A = T + B + L
• If slant height, radius of both circles are given, then
  • Height h = √(s² - (r - R)²)
  • Volume V = (1/3) * π * h * (r² + R² + (r * R))
  • Base Surface Area B = πR²
  • Lateral Surface Area L = π * (r + R) * s
  • Total Surface Area A = π * (r² + R² + (r + R) * s)
• If total surface area, radius of circles are known, then
  • Slant height s = [A/π - r² - R²] / (r + R)
  • h = √(s² - (r - R)²)
  • Volume V = (1/3) * π *√(s² - (r - R)²) * (r² + R² + (r * R))
• If radius of the top and bottom circles and height of the srustm are given, then
  • Lateral Surface Area L = π * (r + R) * s = π * (r + R) * √(r - R)² + h²)
  • Slant Height s = √((r - R)² + h²)
  • Volume V = (1/3) * π * h * (r² + R² + (r * R))
  • Top Surface Area T = πr²
  • Base Surface Area B = πR²
  • Total Surface Area A = π * (r² + R² + (r + R) * s) = π * [ r² + R² + (r + R) * √((r - R)² + h²) ]

Where,

r is the radius of the top circle

R is the radius of the bottom circle

h is the frustum height

V is volume

L is the frustum lateral surface area

T is the top surface area

B is the bottom surface area

A is the conical frustum total surface area

To calculate the conical frustum surface area, slant height, and volume go through the simple steps provided below.

Conical Frustum Slant Height:

• Process 1:
  • Get radius of both circles and the height of the frustum.
  • Subtract those radii and square the value.
  • Add a square of height to the result.
  • Apply square root to the obtained answer to find slant height.
• Process 2:
  • Find lateral surface area, radii of the circles.
  • Add radius of circles, multiply sum with π.
  • Divide lateral surface area by the output to obtain slant height.
• Process 3:
  • Observe total surface area, 2 radii of the frustum.
  • Divide the total surface area by π.
  • Subtract the squares of the top, bottom circle radius, result.
  • Divide the obtained value by the sum of radii to get slant height.

Conical Frustum Top Surface Area:

• Make a note of the radius of the top circle.
• Square the radius and multiply π with it to check the top surface area.

Conical Frustum Volume:

• Check out the height, radius of the top and bottom circles of the frustum.
• Square the top, bottom circles radius and find the product of two radii.
• Add those squares and product.
• Divide the height by 3, multiply the result with π.
• Multiply the values obtained from step 3, step 4 to check volume.

Conical Frustum Base Surface Area:

• Note down the frustum base circle radius.
• Square the radius, multiply π with it to find the base surface area.

Conical Frustum Total Surface Area:

• Get the height, radius of two circles from the question.
• Calculate the lateral, base, top surface areas.
• Add those values to check the total surface area.

Conical Frustum Lateral Surface Area:

• Process 1:
  • Check slant height, radii of the top, bottom frustum.
  • Add those radii and multiply it with π, slant height to check lateral surface area.
• Process 2:
  • Get radii, the height of the frustum from the question.
  • Compute the slant height using the known details.
  • Multiply the obtained slant height with the sum of radii and π.

Conical Frustum Height:

• Process 1:
  • Check out the slant height, radius of the circles.
  • Subtract those circles radius from one another and square the value.
  • Subtract the result of the above step from the slant height square.
  • Apply square root to the answer to find frustum height.
• Process 2:
  • Make a note of the volume, radii of two circles.
  • Square each radius and multiply the radii.
  • Add up those values and multiply with π.
  • Divide thrice of volume by the result to check the height.

Example 1: If the conical frustum total surface area is 1850 m², diameters of the top, base circles are 20 m, 24 m. Find the frustum volume, height, slant height, base surface area, top surface area, and lateral surface area?

Solution:

Given that,

Conical Frustum total Surface Area A = 1850 m²

Diameter of top circle d = 20 m

Radius of top circle r = d / 2 = 20 / 2 = 10 m

Diameter of bottom circle D = 24 m

Radius of bottom circle R = D / 2 = 24 / 2 = 12 m

Slant height formula is s = [A/π - r² - R²] / (r + R)

s = [1850 /3.14 - 10² - 12²] / (10 + 12)

= [589.17 - 100 - 144] / (22)

= [589.17 - 244] / 22

= 345.17 / 22 = 15.689 m

Height h = √(s² - (r - R)²)

By substituting the values

h = √(15.689² - (10 - 12)²)

= √(246.164 - (-2)²)

= √(246.164 - 4)

= √242.164 = 15.56 m

Volume formula is

Volume V = (1/3) * π * √(s² - (r - R)²) * (r² + R² + (r * R))

V = (1/3) * π * √(15.689² - (10 - 12)²) * (10² + 12² + (10 * 12))

= (1/3) * π * √(246.164 - (-2)²) * (100 + 144 + (120))

= (1/3) * π * √(246.164 - 4) * (364)

= (1/3) * π * √242.164 * 364

= 121.33 * π * 15.56

= 1,887.94π

= 1,887.94 * 3.14 = 5,928.152 m³

Top Surface Area of frustum formula is T = πr²

T = π * 10²

= 100π = 100 * 3.14 = 314 m²

Base Surface Area B = πR²

B = π * 12²

= π * 144 = 144 * 3.14 = 452.16 m²

Lateral Surface Area L = π * (r + R) * s

L = π * (10 + 12) * 15.689

= π * 22 * 15.689 = 345.158π

= 345.158 * 3.14 = 1,083.796 m²

∴ Conical frustum lateral surface area is 1,083.796 m², top surface area is 314 m², base surface area is 452.16 m², total surface area is 1850 m², top circle radius is 10 m, base circle radius 12 m, volume is 5,928.152 m³, height is 15.56 m, and slant height is 15.689 m.

Example 2: Solve conical frustum height, total surface area, and volume. If the frustum radii are 7 cm, 15 cm and slant height is 13.6 cm.

Solution:

Given that,

Conical frustum top radius is r = 7 cm

Base radius is R = 15 cm

Slant height s = 13.6 cm

Height of the conical frustum formula is

h = √(s² - (r - R)²)

h = √(13.6² - (7 - 15)²)

= √(185 - (-8)²)

= √(185 - 64)

= √(121) = 11

Volume V = (1/3) * π * h * (r² + R² + (r * R))

V = (1/3) * π * 11 * (7² + 15² + (7 * 15))

= (1/3) * π * 11 * (49 + 225 + 85)

= (1/3) * π * 11 * 359

= (1/3) * π * 3949 = 1,316.33π

= 1,316.33 * 3.14 = 4,133.28 cm³

Lateral Surface Area L = π * (r + R) * √(r - R)² + h²)

L = π * (r + R) * √(r - R)² + h²)

= π * (7 + 15) * √(7 - 15)² + 11²)

= π * (22) * √(-8)² + 11²)

= π * 22 * √64 + 121)

= π * 22 * √185

= π * 22 * 13.6 = 299.23 * π

= 299.23 * 3.14

= 939.58 cm²

Top Surface Area T = πr²

T = π * 7²

= 49π = 49 * 3.14

= 153.86 cm²

Base Surface Area B = πR²

B = π * 15²

= 225π = 225 * 3.14

= 706.5 cm²

Total Surface Area A = π * (r² + R² + (r + R) * s)

= π * (7² + 15² + (7 + 15) * 13.6)

= π * (49 + 225 + 22 * 13.6)

= π * (49 + 225 + 299.2)

= π * 573.2 = 573.2 * 3.14 = 1,799.84 cm²

∴ Conical frustum raddi are 7 cm, 15 cm, slant height is 13.6 cm, height is 11 cm, volume is 4,133.28 cm³, top surface area is 153.86 cm², lateral surface area is 939.58 cm², base surface area is 706.5 cm², and total surface area is 1,799.84 cm².

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1. What is the formula of conical frustum?

The conical frustum volume, surface area formulas in terms of radius and height are as follows:

Volume V = = (1/3) * π * height * (topradius² + baseradius² + (topradius * baseradius))

Total Surface Area = π * [ topradius² + baseradius² + (topradius + baseradius) * √((topradius - baseradius)² + height²) ]

2. What is the slant height of a frustum?

The slant height of a conical frustum is the distance measured along the lateral face from the base to the top. The formula to calculate the slant height is √((topradius - baseradius)² + height²)

3. What is a conical frustum?

A conical frustum is the part of a cone when it is cut by a plane into two parts. the upper part which is having an apex remains the same but the lower part is called a frustum.

4. What is the lateral surface area of a conical frustum?

The lateral surface area is nothing but the curved surface area. It is obtained by multiplying π, slant height, and difference of radii. Formula is π * slantheight (baseradius - topradius).