Cone Calculator
Calculate cone volume, surface area, lateral area, slant height, height, and radius from any two known dimensions. Solve for the height of a cone from slant height or volume — with complete step-by-step working for every calculation.
Enter any two known values — leave the rest empty:
Find height, radius, or slant height of a cone from other known values — step by step.
Common cone dimensions — 10 proportions showing all properties. Values in cm.
| r (cm) | h (cm) | l = √(r²+h²) | V = ⅓πr²h | LA = πrl | SA = πr(r+l) |
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Cone Formulas — Volume, Surface Area & Slant Height
This cone calculator finds every cone property from any two known dimensions: volume, lateral surface area, total surface area, base area, slant height, height, radius, and diameter. The key geometric insight is that the perpendicular height h, base radius r, and slant height l form a right triangle — making l = √(r² + h²) a direct application of the Pythagorean theorem.
| Property | Symbol | Formula | Notes |
|---|---|---|---|
| Volume | V | (1/3)πr²h | Always ⅓ of cylinder |
| Slant height | l | √(r² + h²) | Pythagorean theorem |
| Lateral surface area | LA | πrl | Curved surface only |
| Total surface area | SA | πr(r + l) | Lateral + base circle |
| Base area | A | πr² | Circle at the base |
| Diameter | d | 2r | Twice the radius |
Key triangle: r (horizontal base), h (vertical height to apex), l (slant height along the cone surface) form a right triangle with l as the hypotenuse. So l = √(r² + h²), h = √(l² − r²), and r = √(l² − h²).
How to Find the Slant Height of a Cone
The slant height of a cone is the distance measured along the surface from the apex down to any point on the edge of the base circle. It is different from the perpendicular height h (which goes straight down from apex to the center of the base).
Three methods to find the slant height of a cone:
- From r and h: l = √(r² + h²) — the standard slant height formula
- From LA and r: l = LA / (πr) — rearrange the lateral area formula LA = πrl
- From SA and r: SA = πr(r + l) → l = SA/(πr) − r
Example — Slant height: r = 3 cm, h = 4 cm
- l = √(r² + h²) = √(3² + 4²) = √(9 + 16) = √25 = 5 cm
- This is the classic 3-4-5 right triangle ✓
- The slant height l = 5 cm is the length along the cone surface, not the vertical height h = 4 cm
Common mistake: Using the slant height l instead of the perpendicular height h in the cone volume formula V = (1/3)πr²h. The volume always uses h — the vertical distance from apex to base center. Using l gives a wrong answer that is approximately 25% too large for typical cones.
How to Find the Height of a Cone — Step-by-Step
Finding the height of a cone requires knowing at least one other dimension. There are two main methods — from the slant height and radius, or from the volume and radius.
Method 1 — Height from slant height and radius: h = √(l² − r²)
Example A — How to find the height of a cone with slant height: r=5, l=13
- h = √(l² − r²)
- = √(13² − 5²) = √(169 − 25) = √144
- h = 12 cm (5-12-13 Pythagorean triple ✓)
Example B — h from slant height: r=3, l=5
- h = √(5² − 3²) = √(25 − 9) = √16 = 4 cm (3-4-5 triple ✓)
Example C — h from slant height: r=6, l=10
- h = √(10² − 6²) = √(100 − 36) = √64 = 8 cm
Method 2 — Height from volume and radius: h = 3V / (πr²)
Example D — Height of a cone from volume: V=200, r=4
- h = 3V / (πr²) = 3 × 200 / (π × 4²)
- = 600 / (π × 16) = 600 / 50.2655
- h ≈ 11.937 cm
Why Does the Cone Volume Formula Have 1/3?
The cone volume formula V = (1/3)πr²h contains the 1/3 factor because a cone occupies exactly one-third of the volume of a cylinder with the same base radius r and the same height h. You can verify this experimentally: fill a hollow cone with water and pour it into a cylinder of equal base and height — it takes exactly three fills to fill the cylinder.
Intuition: Compare V_cone = (1/3)πr²h with V_cylinder = πr²h. The ratio is exactly 1:3. The formal proof uses calculus — integrating the areas of circular cross-sections from h=0 to h=H, each cross-section having radius (r×y/H) at height y, giving area π(r×y/H)², and integrating gives (1/3)πr²H.
Example — Never forget the 1/3!
- ❌ Wrong: V = π × 3² × 4 = 36π ≈ 113.1 cm³
- ✅ Correct: V = (1/3) × π × 3² × 4 = (1/3) × 36π = 12π ≈ 37.7 cm³
- The wrong answer is exactly 3× too large — the most common cone calculation error
Cone Worked Examples — Step-by-Step Problems
1. r=3cm, h=4cm — full solution
- l = √(9+16) = √25 = 5 cm
- V = (1/3)×π×9×4 = 12π ≈ 37.699 cm³
- LA = π×3×5 = 15π ≈ 47.124 cm²
- SA = π×3×(3+5) = 24π ≈ 75.398 cm²
- Base area = π×9 = 9π ≈ 28.274 cm²
2. r=5cm, h=12cm — 5-12-13 triple
- l = √(25+144) = √169 = 13 cm
- V = (1/3)×π×25×12 = 100π ≈ 314.159 cm³
- LA = π×5×13 = 65π ≈ 204.204 cm²
- SA = π×5×18 = 90π ≈ 282.743 cm²
3. Find height from l=13, r=5 (reverse)
- h = √(13²−5²) = √(169−25) = √144 = 12 cm
4. Find radius from h=8, l=10 (reverse)
- r = √(l²−h²) = √(100−64) = √36 = 6 cm
5. Find height from V=200cm³, r=4cm
- h = 3V/(πr²) = 3×200/(π×16) = 600/50.265 = 11.937 cm
6. Ice cream cone: r=3cm, h=10cm
- l = √(9+100) = √109 ≈ 10.440 cm
- V = (1/3)×π×9×10 = 30π ≈ 94.248 cm³
- LA = π×3×10.440 ≈ 98.44 cm² (waffle cone surface)
7. Given SA=200cm², r=5 — find h
- SA = πr(r+l) → 200 = π×5×(5+l) → 5+l = 200/(5π) = 12.732 → l = 7.732
- h = √(l²−r²) = √(59.78−25) = √34.78 ≈ 5.897 cm
8. r=7cm, h=24cm — large cone
- l = √(49+576) = √625 = 25 cm (7-24-25 triple!)
- V = (1/3)×π×49×24 = 392π ≈ 1231.5 cm³
- LA = π×7×25 = 175π ≈ 549.8 cm²
- SA = π×7×32 = 224π ≈ 703.7 cm²
Common Mistakes in Cone Calculations
Mistake 1 — Using slant height in the volume formula
- ❌ Wrong: V = (1/3)×π×r²×l — using slant height l instead of perpendicular height h
- ✅ Correct: V = (1/3)×π×r²×h — always use the perpendicular height in the volume formula
Mistake 2 — Forgetting the 1/3 in the volume formula
- ❌ Wrong: V = π×r²×h = 36π (3× too large for r=3, h=4)
- ✅ Correct: V = (1/3)×π×r²×h = 12π ≈ 37.7 cm³
Mistake 3 — Using diameter instead of radius
- If diameter d = 6 cm, then r = 3 cm. Always halve the diameter before plugging into any cone formula. Using d instead of r gives an area/volume 4× too large.
Mistake 4 — Forgetting the base for total surface area
- ❌ Lateral area only: LA = πrl (open cone — no base)
- ✅ Total surface area: SA = πrl + πr² = πr(r+l) (closed cone — includes base circle)
Mistake 5 — Wrong slant height formula: l² = r + h instead of l = √(r²+h²)
- ❌ Wrong: l = r + h = 3 + 4 = 7 (completely wrong)
- ✅ Correct: l = √(r²+h²) = √(9+16) = √25 = 5 — must square r and h before adding