Understanding Frustum 

Learn about frustums with simple examples and practice

What is a Frustum?

A frustum is what remains when you cut off the top of a cone or pyramid with a plane parallel to the base. Think of cutting the top off an ice cream cone!

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Types of Frustum

🔵 Frustum of a Cone

Has two circular bases of different sizes

🔶 Frustum of a Pyramid

Has two polygonal bases of different sizes

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Important Formulas

Volume of Frustum:
V = (1/3)πh(R² + Rr + r²)
Where: h = height, R = lower radius, r = upper radius


Curved Surface Area:
CSA = π(R + r)l
Where: l = slant height

Example 1: Volume Calculation

📝 Problem:

Find the volume of a frustum with height 12cm, lower radius 8cm, and upper radius 5cm.

✅ Solution:

Given: h = 12cm, R = 8cm, r = 5cm

V = (1/3)πh(R² + Rr + r²)

V = (1/3)π × 12 × (8² + 8×5 + 5²)

V = (1/3)π × 12 × (64 + 40 + 25)

V = (1/3)π × 12 × 129

V = 516π cm³ ≈ 1,621 cm³

🎮 Interactive Quiz

Which of these are frustums?

✅ Bucket (wide bottom, narrow top)

❌ Complete cone

✅ Lampshade

❌ Cylinder (same width throughout)

Real-Life Examples

🪣 Bucket Water containers	💡 Lampshade Light fixtures
🌪️ Funnel Pouring aids	🏛️ Buildings Architecture

Example 2: Surface Area

📝 Problem:

A frustum has radii 6cm and 4cm, with slant height 10cm. Find the curved surface area.

✅ Solution:

Given: R = 6cm, r = 4cm, l = 10cm

CSA = π(R + r)l

CSA = π(6 + 4) × 10

CSA = 100π cm²

CSA ≈ 314 cm²

🧠 Practice Problem

Try This:

Calculate the volume: h = 6cm, R = 7cm, r = 3cm

Answer: Use V = (1/3)π × 6 × (7² + 7×3 + 3²) = (1/3)π × 6 × 79 ≈ 497 cm³

✏️  Drawing StepsDraw the larger base (circle)
Draw the smaller base above it
Connect edges with straight lines
Add labels (h, R, r, l)

📚 Practice Problems


A frustum has height 15cm, lower radius 10cm, upper radius 6cm. Find its volume.

Calculate the curved surface area of a frustum with radii 8cm and 5cm, slant height 12cm.

A bucket shaped like a frustum has radii 20cm and 15cm with height 25cm. How much water can it hold?