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Last updated on September 17, 2025

Area of Frustum

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The area in the context of a frustum refers to the surface area of a three-dimensional shape that resembles a cone or pyramid with the top portion cut off. Calculating the area of a frustum is useful in fields like architecture and engineering. In this section, we will explore how to find the area of a frustum.

Area of Frustum for US Students
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What is the Area of a Frustum?

A frustum is a three-dimensional geometric shape formed by slicing the top off a cone or pyramid. It has two parallel bases of different sizes.

 

The total surface area of a frustum includes the areas of the two circular bases and the curved surface connecting them.

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Area of the Frustum Formula

To find the surface area of a frustum of a cone, we use the formula: ( A = pi(R + r)l + pi R2 + pi r^2 ), where ( R ) and ( r ) are the radii of the two bases, and ( l ) is the slant height. Now let’s see how the formula is derived.

 

Derivation of the formula: The lateral surface area of the frustum is the area of the curved surface, which can be calculated as the average circumference of the two bases times the slant height: ( pi(R + r)l ). The total surface area also includes the areas of the two bases: ( pi R2 ) and ( pi r2 ). By adding these areas, we get the total surface area of the frustum: ( A = pi(R + r)l + pi R^2 + pi r^2 ).

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How to Find the Area of a Frustum?

We can find the area of the frustum using three approaches, focusing primarily on the formula involving the radii and slant height. The methods are:

Method Using Radii and Slant Height

Method Using the Height and Volume

 

Method Using the Slant Height and Base Areas

Let's discuss these methods in detail.

 

Method Using Radii and Slant Height

If the radii ( R ) and ( r ) and the slant height ( l ) are given, we find the surface area using the formula: ( A = pi(R + r)l + pi R^2 + pi r^2 ). For example, if ( R ) and ( r ) are 10 cm and 5 cm, and ( l ) is 13 cm, what is the area of the frustum? ( A = pi(10 + 5) times 13 + pi times 10^2 + pi times 5^2 ) = ( pi times 15 times 13 + 100pi + 25pi ) = ( 195pi + 125pi ) = ( 320pi , text{cm}^2 )

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Unit of Area of Frustum

The area of a frustum is measured in square units. The unit depends on the measurement system used: In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²). In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).

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Special Cases or Variations for the Area of Frustum

Frustums can vary depending on their dimensions and the shapes of their bases. Here are some special cases:

 

Case 1: Frustum of a Cone If the shape is a frustum of a cone, use the formula ( A = pi(R + r)l + pi R^2 + pi r^2 ).

 

Case 2: Frustum of a Pyramid If the frustum is a part of a pyramid, additional calculations for the base areas using their specific shapes are needed.

 

Case 3: Using Different Measurements If only height and lateral surface area are given, additional formulas or steps are needed to calculate the total area.

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Tips and Tricks for Area of Frustum

To ensure accuracy when calculating the area of a frustum, consider these tips and tricks: 

 

  • Always verify whether the shape is a frustum of a cone or a pyramid; this impacts the formula. 
     
  • Ensure that all measurements are in the same unit before applying the formula. 
     
  • Use the Pythagorean theorem to find the slant height if the vertical height and radii are known and needed for calculations.
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Common Mistakes and How to Avoid Them in Area of Frustum

Calculating the area of a frustum can lead to errors. Here are some common mistakes and how to avoid them:

Mistake 1

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Incorrectly Using the Radii

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Ensure you are using the correct values for ( R ) and ( r ), and not confusing them with diameters.

 

Use the formula ( A = pi(R + r)l + pi R^2 + pi r^2 ) accurately.

Mistake 2

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Mixing Units

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Ensure that all measurements (radii, slant height) are in the same unit to avoid incorrect area calculations.

Mistake 3

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Confusing Slant Height with Vertical Height

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The slant height is different from the vertical height.

 

Use the slant height for surface area calculations.

Mistake 4

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Forgetting to Include Both Base Areas

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Remember to add the areas of both the top and bottom bases when calculating the total surface area of the frustum.

Mistake 5

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Misapplying Formulas for Different Frustums

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Ensure you are using the correct formula for a frustum of a cone versus a frustum of a pyramid, as they can differ.

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Area of Frustum Examples

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Problem 1

A frustum of a cone has radii 8 m and 4 m, and a slant height of 10 m. What is the surface area?

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We will find the surface area as \( 384\pi \, \text{m}^2 \).

Explanation

Here, ( R = 8 , text{m}, r = 4 , text{m}, ) and ( l = 10 , text{m} ). The surface area ( A = pi(R + r)l + pi R^2 + pi r^2 ) = ( pi(8 + 4) times 10 + pi times 8^2 + pi times 4^2 ) = ( 120pi + 64pi + 16pi ) = ( 384pi , text{m}^2 ).

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Problem 2

What is the surface area of a frustum if the radii are 6 cm and 3 cm, and the slant height is 8 cm?

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We will find the surface area as \( 174\pi \, \text{cm}^2 \).

Explanation

Given ( R = 6 , text{cm}, r = 3 , text{cm}, ) and ( l = 8 , text{cm} ). The surface area ( A = pi(R + r)l + pi R^2 + pi r^2 ) = ( pi(6 + 3) times 8 + pi times 6^2 + pi times 3^2 ) = ( 72pi + 36pi + 9pi ) = ( 117pi , text{cm}^2 ).

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Problem 3

The surface area of a frustum is 120π m², with a slant height of 5 m. If one radius is 7 m, find the other radius.

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We find the other radius as 1 m.

Explanation

To find the other radius, use the formula ( A = pi(R + r)l + pi R^2 + pi r^2 ). Given ( R = 7 , text{m}, A = 120pi , text{m}^2, ) and ( l = 5 , text{m} ): 120π = π(7 + r) × 5 + π × 7² + π × r² 120 = 5(7 + r) + 49 + r² 71 = 35 + 5r + r² Solve for ( r ) to find ( r = 1 , text{m} ).

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Problem 4

Find the surface area of a frustum with a slant height of 12 cm, where the radii are 9 cm and 2 cm.

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We will find the area as \( 312\pi \, \text{cm}^2 \).

Explanation

The given radii are ( R = 9 , text{cm}, r = 2 ,, text{cm}, ) and slant height ( l = 12 , text{cm} ). The surface area is calculated as: ( A = pi(R + r)l + pi R^2 + pi r^2 ) = ( pi(9 + 2) times 12 + pi times 9^2 + pi times 2^2 ) = ( 132pi + 81pi + 4pi ) = ( 217pi , text{cm}^2 ).

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Problem 5

Help Anna find the surface area of a frustum with a slant height of 18 m, where the radii are 10 m and 5 m.

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We will find the area as \( 525\pi \, \text{m}^2 \).

Explanation

Anna needs to calculate the surface area with ( R = 10 , text{m}, r = 5 , text{m}, ) and ( l = 18 , text{m} ). The surface area is: ( A = pi(R + r)l + pi R^2 + pi r^2 ) = ( pi(10 + 5) times 18 + pi times 10^2 + pi times 5^2 ) = ( 270pi + 100pi + 25pi ) = ( 395pi , text{m}^2 ).

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FAQs on Area of Frustum

1.Can the surface area of a frustum be negative?

No, the surface area of a frustum cannot be negative. The area of any shape is always a positive value.

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2.How to find the surface area of a frustum if the radii and slant height are given?

Use the formula: \( A = \pi(R + r)l + \pi R^2 + \pi r^2 \) when radii and slant height are given.

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3.How to find the area of a frustum if only height is given?

You would need additional information, such as the radii of the bases or the slant height, to calculate the area.

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4.How is the volume of the frustum calculated?

The volume of a frustum is calculated using the formula: \( V = \frac{1}{3}\pi h(R^2 + Rr + r^2) \), where \( h \) is the height of the frustum.

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5.What is meant by the surface area of the frustum?

The surface area of the frustum is the total surface area that includes the lateral surface area and the areas of the two bases.

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Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

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Fun Fact

: She has songs for each table which helps her to remember the tables

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