📐 Cone Surface Area Calculator 2026
Calculate lateral area, base area, and total surface area of a cone
Enter Cone Dimensions
r = radius, h = height, l =
slant height
📊 Results
Slant Height (l)
13 cm
Base Area
78.54 cm²
Lateral Area
204.2 cm²
Total Surface Area
282.74 cm²
📝 Step-by-Step Solution
Step 1: l = √(r² + h²) = √(5² + 12²) = √169 = 13 cm
Step 2: Base Area = πr² = π × 5² = 78.54 cm²
Step 3: Lateral Area = πrl = π × 5 × 13 = 204.20 cm²
Step 4: Total = Base + Lateral = 78.54 + 204.20 = 282.74 cm²
📐 Cone Surface Area Formulas
Slant Height: l = √(r² + h²)
Base Area: Abase = πr²
Lateral Surface Area: Alateral = πrl
Total Surface Area: Atotal = πr² + πrl = πr(r + l)
Understanding Cone Geometry
📐 Slant Height
l = √(r² + h²). The distance from apex to edge of base. Forms hypotenuse of right triangle with radius and height. Essential for lateral area.
🔵 Base Area
A = πr². Circular base of the cone. Same formula as circle area. Only included in "total" surface area, not "lateral."
📄 Lateral Surface Area
A = πrl. The curved surface only (excluding base). Opens into a sector of a circle. Used for material needed to make cone sides.
📦 Total Surface Area
A = πr(r + l). Base + lateral combined. Use for total material needed to completely enclose the cone.
Frequently Asked Questions
What is the surface area of a cone?
Total surface area = πr(r + l). Sum of base area (πr²) and
lateral/curved area (πrl). Depends on radius and slant height.
What is the formula for lateral surface area of a cone?
Lateral Area = πrl. Where r = radius and l = slant height. This is
the curved surface only, excluding the circular base.
How do I find the slant height of a cone?
l = √(r² + h²). Use Pythagorean theorem. Slant height is the
hypotenuse of the right triangle formed by radius, height, and slant.
What is the difference between lateral and total surface area?
Lateral = curved surface only (πrl). Total = lateral +
base (πrl + πr²). Use lateral for open cones; total for closed.
How do I calculate base area of a cone?
Base Area = πr². Same as area of a circle. The base is always a
perfect circle in a right circular cone.
Can I find surface area with only radius and height?
Yes! First find slant height: l = √(r² + h²). Then use the surface
area formulas. Height and slant height are related.
What if I know slant height but not height?
You can still calculate surface area directly: πr(r + l). To find
height: h = √(l² - r²).
What is the curved surface area of a cone?
Same as lateral surface area: πrl. "Curved surface" and "lateral
surface" are interchangeable terms for the cone's side surface.
How is cone surface area derived?
The lateral surface unrolls into a circular sector. Arc length =
2πr (base circumference). Radius of sector = l (slant height). Sector area = πrl.
What units should I use for surface area?
Square units. If radius is in cm, area is in cm². Keep all inputs
in same unit. Common: cm², m², in², ft².
How do I find the surface area of a cone without the base?
Use only lateral area: πrl. This is appropriate for open cones
like party hats or funnels without bottom.
What is a right circular cone?
Cone with apex directly above center of circular base. Height is
perpendicular to base. Most common type; formulas here apply to this type.
How do I find the radius from surface area?
Rearrange formula: A = πr(r + l). This is quadratic in r. Use
quadratic formula or calculate iteratively. Need to know slant height.
What is the relationship between cone and cylinder surface area?
Cone lateral area = half cylinder lateral area (same r, h). Cone:
πrl. Cylinder: 2πrh. If l = h, cone has half.
How do I calculate surface area of a truncated cone?
Frustum area = π(r₁ + r₂)l + πr₁² + πr₂². l = slant height of
frustum. Add both circular ends plus lateral surface.
What real-world objects are cones?
Ice cream cones, traffic cones, party hats, funnels, volcano shapes,
speaker cones, roof spires. Surface area helps calculate material needed.
How accurate is this calculator?
Uses precise mathematical formulas with π to full precision.
Results are accurate for all practical applications.