Cone Slant Height
Calculator
Results
- Slant height
- 5
- Lateral surface area
- 47.123889
- Base area
- 28.274333
- Total surface area
- 75.398223
Results
| Slant height | 5 |
| Lateral surface area | 47.123889 |
| Base area | 28.274333 |
| Total surface area | 75.398223 |
formula-map diagram
- Slant height
- 5
- Lateral surface area
- 47.123889
- Base area
- 28.274333
- Total surface area
- 75.398223
Formula map
Formula
l = √(r² + h²), A_lat = π × r × l= 5
Note
Simplified model: these are the exact results of the stated idealised geometric formulas, assuming perfect shapes, exact inputs and consistent units. Real measurements carry error, and results such as the cyclic-quadrilateral area only hold when the figure truly satisfies the stated condition. Verify independently before relying on these figures.
More in Advanced geometry
See all →Frequently asked questions
What is slant height, and how is it different from a cone's vertical height?+
Slant height is the distance measured along the cone's curved lateral surface, from the apex down to a point on the base's edge, whereas vertical height is the straight perpendicular distance from apex to the center of the base. The two are related by the Pythagorean theorem, since they form a right triangle with the base radius.
What is the formula relating slant height, radius, and vertical height?+
Slant height l = √(r² + h²), where r is the base radius and h is the vertical height, directly from the Pythagorean theorem applied to the right triangle formed by the radius, height, and slant as hypotenuse. This is one of the most common applications of the Pythagorean theorem in solid geometry.
Why is slant height needed to calculate a cone's lateral surface area instead of vertical height?+
The lateral (curved) surface area formula, A = πrl, uses slant height because when the cone's curved surface is unrolled, it forms a sector of a circle whose radius is the slant height, not the vertical height. Using vertical height in that formula would give an incorrect surface area.
Does a taller cone always have a longer slant height than a shorter, wider one?+
Not necessarily — slant height depends on both radius and height combined via the Pythagorean relationship, so a short, wide cone can have a similar or even longer slant height than a tall, narrow one, depending on the specific values. Both dimensions matter, not height alone.
What happens to slant height if the cone's height is zero?+
If h = 0, the 'cone' flattens into a flat disk, and the formula correctly reduces to l = √r² = r, meaning the slant height equals the radius. This makes geometric sense since a flattened cone's apex sits right at the edge of the base.