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Last updated on September 17, 2025
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.
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.
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 ).
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 )
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).
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.
To ensure accuracy when calculating the area of a frustum, consider these tips and tricks:
Calculating the area of a frustum can lead to errors. Here are some common mistakes and how to avoid them:
A frustum of a cone has radii 8 m and 4 m, and a slant height of 10 m. What is the surface area?
We will find the surface area as \( 384\pi \, \text{m}^2 \).
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 ).
What is the surface area of a frustum if the radii are 6 cm and 3 cm, and the slant height is 8 cm?
We will find the surface area as \( 174\pi \, \text{cm}^2 \).
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 ).
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.
We find the other radius as 1 m.
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} ).
Find the surface area of a frustum with a slant height of 12 cm, where the radii are 9 cm and 2 cm.
We will find the area as \( 312\pi \, \text{cm}^2 \).
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 ).
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.
We will find the area as \( 525\pi \, \text{m}^2 \).
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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