Cone Calculator
Calculate cone properties including volume, slant height, lateral surface area, and total surface area. Includes frustum (truncated cone) and unfolded pattern calculations.
V = (1/3)pir^2h | s = sqrt(r^2 + h^2)Cone Properties
Volume
261.7994 cm^3
3D Visualization
Cone Dimensions
Real-World Cone Examples
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All Cone Properties
Radius (r)
5 cm
Height (h)
10 cm
Slant Height (s)
11.1803 cm
s = sqrt(r^2 + h^2)
Volume (V)
261.7994 cm^3
V = (1/3)pir^2h
Lateral Surface Area
175.6204 cm^2
A = pirs
Base Area
78.5398 cm^2
A = pir^2
Total Surface Area
254.1602 cm^2
A = pir(s + r)
Apex Angle
53.13 degrees
2 x arctan(r/h)
Exact Values (in terms of pi)
Volume
83.3333pi cm^3
Lateral Surface Area
55.9017pi cm^2
Base Area
25pi cm^2
Unfolded Pattern (Flat Development)
When you unfold the lateral surface of a cone, it forms a sector of a circle. Use these measurements to create a cone from flat material.
Sector Radius
11.1803 cm
= slant height
Arc Length
31.4159 cm
= base circumference
Sector Angle
161 degrees
= (r/s) x 360
Cone Formulas
Basic Formulas
- Volume: V = (1/3)pir^2h
- Slant Height: s = sqrt(r^2 + h^2)
- Lateral Area: A = pirs
- Total Area: A = pir(s + r)
- Apex Angle: 2 x arctan(r/h)
Frustum Formulas
- V = (pih/3)(R^2 + Rr + r^2)
- s = sqrt(h^2 + (R-r)^2)
- Lateral: pi(R + r) x s
- Total: Lateral + piR^2 + pir^2
Key Relationship: A cone has exactly 1/3 the volume of a cylinder with the same base radius and height. This was proven by Archimedes over 2000 years ago.
Cone Properties
Volume
261.7994 cm^3
?How to Calculate Cone Properties
Cone formulas: Volume = (1/3) x pi x r^2 x h (one-third pi r squared times height). Slant height s = sqrt(r^2 + h^2). Lateral Surface Area = pi x r x s. Total Surface Area = pi x r x (s + r). A cone has exactly 1/3 the volume of a cylinder with the same base and height.
What is a Cone?
A cone is a three-dimensional geometric shape with a circular base that tapers to a single point called the apex (or vertex). The height is the perpendicular distance from base to apex. The slant height is the distance from any point on the base circle to the apex along the surface. Common examples include ice cream cones, traffic cones, party hats, and funnels.
Key Facts About Cones
- Volume = (1/3) x pi x r^2 x h (one-third of cylinder volume)
- Slant height s = sqrt(r^2 + h^2) using Pythagorean theorem
- Lateral Surface Area = pi x r x s (the curved side)
- Total Surface Area = pi x r x (s + r) (lateral + base)
- Apex angle = 2 x arctan(r / h)
- A cone is exactly 1/3 the volume of a cylinder with same base and height
- When unfolded, the lateral surface forms a sector of a circle
- The frustum (truncated cone) is what remains after cutting off the top
Quick Answer
Cone formulas: Volume = (1/3) x pi x r^2 x h (one-third pi r squared times height). Slant height s = sqrt(r^2 + h^2). Lateral Surface Area = pi x r x s. Total Surface Area = pi x r x (s + r). A cone has exactly 1/3 the volume of a cylinder with the same base and height.
Practice Cone Problems
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Practice with 4 problems to test your understanding.
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Frequently Asked Questions
The volume of a cone is V = (1/3)pir^2h, exactly one-third of a cylinder with the same base and height. For a cone with radius 4 and height 9, V = (1/3) x pi x 16 x 9 = 150.8 cubic units.
Slant height (s) is calculated using the Pythagorean theorem: s = sqrt(r^2 + h^2). The radius, height, and slant height form a right triangle, with slant height as the hypotenuse. For r=3 and h=4, s = sqrt(9 + 16) = sqrt(25) = 5.
Lateral surface area is the area of the curved side of the cone (not including the circular base). The formula is A = pirs, where s is slant height. When unfolded flat, this surface forms a sector of a circle.
Total surface area includes the lateral surface plus the circular base: SA = pir(s + r) = pirs + pir^2. The first term is the curved surface, the second is the base.
A frustum is the portion of a cone that remains after cutting off the top with a plane parallel to the base. It looks like a lampshade or bucket. Volume = (pih/3)(R^2 + Rr + r^2), where R is bottom radius, r is top radius.
When unfolded, a cone lateral surface forms a sector of a circle. The sector radius equals the slant height, and the arc length equals the base circumference (2pir). The sector angle = (r/s) x 360 degrees.
The apex angle is the angle at the tip of the cone, formed by two opposite slant heights. It equals 2 x arctan(r/h). A wider base or shorter height gives a larger apex angle.
This 1:3 ratio can be proven using calculus or by Cavalieri's principle. Historically, Archimedes proved it by weighing physical models. The cone can be thought of as a pyramid with a circular base, and all pyramids are 1/3 of their corresponding prisms.
Last updated: 2025-01-15
Cone Properties
Volume
261.7994 cm^3