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Cone Calculator

Calculate cone volume, surface area, and slant height

Volume

314.16

Slant Height

13.00

Surface Area

282.74

Perpendicular height from base to apex

Volume

314.16

cubic units

V = (1/3)π(5.00)²(12.00)

Slant Height

13.00

units

l = √(5.00² + 12.00²) = 13.00

Surface Areas

Lateral Surface Area

204.20

πrl = π(5.00)(13.00)

Base Area

78.54

πr² = π(5.00)²

Total Surface Area

282.74

πr(r + l) = π(5.00)(5.00 + 13.00)

All Dimensions

Radius (r)5.00
Height (h)12.00
Slant Height (l)13.00
Base Diameter10.00
Base Circumference31.42

Formulas Used

Cone Volume

V = (1/3)πr²h

The volume of a cone is one-third the product of the base area and the height.

Where:

r= Radius of the circular base
h= Perpendicular height from base to apex
π= Pi, approximately 3.14159

Slant Height

l = √(r² + h²)

The slant height is the distance from the base edge to the apex, calculated using the Pythagorean theorem.

Where:

r= Radius of the circular base
h= Perpendicular height from base to apex
l= Slant height along the surface

Total Surface Area

SA = πr(r + l)

Total surface area includes the circular base (πr²) and the lateral surface (πrl).

Where:

r= Radius of the circular base
l= Slant height
πr²= Area of the circular base
πrl= Lateral (curved) surface area

Example Calculations

1Cone: Radius 5, Height 12

Inputs

Radius5
Height12

Result

Volume314.16
Slant Height13.00
Lateral Surface Area204.20
Base Area78.54
Total Surface Area282.74

Slant height = √(5² + 12²) = √169 = 13. Volume = (1/3)π(5)²(12) = 100π = 314.16. Lateral SA = π(5)(13) = 204.20. Base = π(25) = 78.54. Total SA = 78.54 + 204.20 = 282.74.

2Cone: Radius 3, Height 4

Inputs

Radius3
Height4

Result

Volume37.70
Slant Height5.00
Lateral Surface Area47.12
Base Area28.27
Total Surface Area75.40

Slant height = √(9 + 16) = √25 = 5. Volume = (1/3)π(9)(4) = 12π = 37.70. Lateral SA = π(3)(5) = 47.12. Base = π(9) = 28.27. Total SA = 28.27 + 47.12 = 75.40.

3Cone: Radius 10, Height 15

Inputs

Radius10
Height15

Result

Volume1570.80
Slant Height18.03
Lateral Surface Area566.32
Base Area314.16
Total Surface Area880.49

Slant height = √(100 + 225) = √325 = 18.03. Volume = (1/3)π(100)(15) = 500π = 1570.80. Lateral SA = π(10)(18.03) = 566.32. Base = π(100) = 314.16. Total SA = 314.16 + 566.32 = 880.49.

Frequently Asked Questions

Q

How do you calculate the volume of a cone?

Volume of a cone = (1/3) × π × r² × h, where r is the base radius and h is the perpendicular height. For example, a cone with radius 5 and height 12 has volume = (1/3) × π × 25 × 12 = 314.16 cubic units.

  • A cone is exactly 1/3 the volume of a cylinder with the same base and height
  • Radius 5, height 12: Volume = 314.16 cubic units
  • Radius 3, height 4: Volume = 37.70 cubic units
  • Doubling the radius quadruples the volume (r² factor)
  • A standard traffic cone (r≈3.5", h≈18") holds about 230 in³
RadiusHeightVolumeSlant Height
3437.705.00
512314.1613.00
710513.1312.21
10151,570.8018.03
Q

How do you find the slant height of a cone?

Slant height l = √(r² + h²), using the Pythagorean theorem where r is the radius and h is the perpendicular height. For radius 5 and height 12: l = √(25 + 144) = √169 = 13.

  • The slant height is the distance from the base edge to the apex along the surface
  • It forms the hypotenuse of a right triangle with r and h as legs
  • Radius 3, height 4: slant height = 5 (classic 3-4-5 right triangle)
  • Radius 5, height 12: slant height = 13 (classic 5-12-13 right triangle)
RadiusHeightSlant Height
345.00
51213.00
6810.00
101518.03
Q

What is the lateral surface area of a cone?

Lateral surface area = π × r × l, where r is the radius and l is the slant height. For radius 5 and slant height 13: lateral area = π × 5 × 13 = 204.20 square units. This excludes the circular base.

  • The lateral surface "unrolls" into a sector of a circle with radius equal to the slant height
  • Radius 5, slant 13: lateral area = 204.20 sq units
  • Add base area (πr²) to get total surface area
  • Useful for calculating material to wrap a cone or make a party hat
Q

What is the total surface area of a cone?

Total surface area = πr(r + l), which includes both the lateral surface (πrl) and the circular base (πr²). For radius 5 and slant height 13: SA = π × 5 × (5 + 13) = 282.74 square units.

  • Total SA = πr² + πrl = πr(r + l)
  • Radius 5, slant 13: total SA = 78.54 + 204.20 = 282.74 sq units
  • Radius 3, slant 5: total SA = 28.27 + 47.12 = 75.40 sq units
  • Use total SA for paint or coating calculations on a solid cone
  • For an open cone (like a funnel), use lateral area only
RadiusSlant HeightLateral AreaTotal SA
3547.1275.40
513204.20282.74
712.21268.40422.37
1018.03566.32880.49
Q

How is a cone related to a cylinder?

A cone with the same base radius and height as a cylinder has exactly one-third the volume. If a cylinder has volume πr²h, the cone has volume (1/3)πr²h. This is proven using integral calculus.

  • Cone volume = (1/3) × cylinder volume for same base and height
  • 3 cones of water fill exactly 1 cylinder of the same dimensions
  • The relationship was first proven by Archimedes around 250 BC
  • Similarly, a cone is 1/3 the volume of a prism with the same base area and height

Understanding Cone Calculations

A cone is a three-dimensional shape with a circular base that tapers to a single point called the apex. The key measurements are the base radius (r), the perpendicular height (h), and the slant height (l).

The slant height connects the edge of the base to the apex along the surface. It forms the hypotenuse of a right triangle with the radius and height as legs: l = √(r² + h²).

Cone calculations are essential in engineering (funnels, nozzles), construction (roof peaks, sand piles), and everyday objects like ice cream cones and traffic cones.

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Last Updated: Mar 11, 2026

This calculator is provided for informational and educational purposes only. Results are estimates and should not be considered professional financial, medical, legal, or other advice. Always consult a qualified professional before making important decisions. UseCalcPro is not responsible for any actions taken based on calculator results.

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