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📐 Truncated Cone Surface Area Calculator 2026

Calculate surface area of a frustum (cone with top cut off)

Enter Frustum Dimensions

h R r l
R = bottom radius, r = top radius
h = height, l = slant height

📊 Results

Slant Height (l)
7.21 cm
Top Area
50.27 cm²
Bottom Area
201.06 cm²
Lateral Area
271.43 cm²
Total Surface
522.76 cm²

📝 Step-by-Step Solution

Step 1: l = √[h² + (R-r)²] = √[6² + (8-4)²] = √52 = 7.21 cm
Step 2: Top Area = πr² = π × 4² = 50.27 cm²
Step 3: Bottom Area = πR² = π × 8² = 201.06 cm²
Step 4: Lateral = π(R + r)l = π(8 + 4) × 7.21 = 271.43 cm²
Step 5: Total = Top + Bottom + Lateral = 522.76 cm²

📐 Frustum Surface Area Formulas

Slant Height: l = √[h² + (R - r)²]
Top Area: Atop = πr²
Bottom Area: Abottom = πR²
Lateral Area: Alateral = π(R + r)l
Total Surface: Atotal = πr² + πR² + π(R + r)l

Understanding Frustum Geometry

📐 What is a Frustum?

A cone with its top cut off by a plane parallel to the base. Two circular faces (different sizes) connected by a curved surface. Also called truncated cone.

📏 Slant Height

l = √[h² + (R-r)²]. Distance along the curved surface from top edge to bottom edge. Not the same as height (h), which is vertical.

📄 Lateral Area

A = π(R + r)l. The curved surface area. Opens into a "ring sector." Depends on both radii and slant height.

📦 Total Surface Area

Both circles + lateral surface. A = πr² + πR² + π(R+r)l. Use for completely enclosed frustum.

Frequently Asked Questions

What is a truncated cone (frustum)?
A cone with its apex cut off by a plane parallel to the base. Has two circular faces of different sizes connected by a curved lateral surface.
What is the surface area formula for a frustum?
Total = πr² + πR² + π(R+r)l. Where R = large radius, r = small radius, l = slant height. Or simplified: π[r² + R² + (R+r)l].
How do I find the slant height of a frustum?
l = √[h² + (R-r)²]. Uses Pythagorean theorem. The "horizontal distance" is the difference in radii (R-r).
What is the lateral surface area of a frustum?
Alateral = π(R + r)l. The curved surface only, excluding both circular ends. Similar to cone formula but with two radii.
How is frustum lateral area derived?
Difference of two cone lateral areas. Full cone minus the small cone that was cut off. Simplifies to π(R+r)l formula.
What if the top radius is zero?
It becomes a regular cone. Formulas reduce to cone formulas: lateral = πRl, total = πR² + πRl.
How do I find frustum surface area from height?
First calculate slant height: l = √[h² + (R-r)²]. Then use surface area formulas. Height alone isn't enough — need both radii too.
What is the difference between frustum and cone formulas?
Frustum has two radii. Cone: πrl. Frustum: π(R+r)l. Frustum also has two circular areas instead of one.
Can the top radius be larger than bottom?
Yes, formulas still work. Just swap R and r definitions. The frustum would be upside-down relative to typical orientation.
How do I find the radius from surface area?
Complex — requires solving equations. Total area formula has multiple unknowns. Need to know at least one radius and either height or slant height.
What are real-world examples of frustums?
Lampshades, bucket shapes, flower pots, drinking cups, cooling towers, volcano calderas, traffic cone bases. Very common shape.
How do I calculate frustum volume?
V = (πh/3)(R² + Rr + r²). Different from surface area. Not covered by this tool but related formula.
What if both radii are equal?
It becomes a cylinder. R = r means no taper. Lateral area = 2πRh (cylinder formula). Slant height = height.
How do I find height from slant height?
h = √[l² - (R-r)²]. Rearrange the slant height formula. Need to know both radii and slant height.
What units should surface area be in?
Square units. If radii are in cm, area is cm². Keep all inputs in same unit before calculating.
Is a frustum always circular?
Technical: "frustum" applies to any cone. But this calculator assumes circular cross-sections (right circular cone frustum). Pyramids have similar shapes but are polygonal.
How accurate is this calculator?
Uses precise mathematical formulas with π to full precision. Accurate for all practical applications.