Height of a Cone Calculator
Enter any two known values for a right circular cone and get the height plus every other dimension. You can solve from radius and slant height, from radius and volume, or from height and radius directly. All outputs update instantly as you type.
What is the height of a cone?
The height of a right circular cone is the perpendicular distance from the centre of the circular base straight up to the apex. It is sometimes called the altitude. Unlike the slant height, which runs along the outer surface, the vertical height meets the base at a right angle. The three key lengths of a cone - radius (r), height (h), and slant height (l) - are related by the Pythagorean theorem: l^2 = r^2 + h^2. Knowing any two lets you find the third, and from there you can calculate volume, surface area, and the cone's apex and base angles.
How to find the height of a cone
There are several common routes to the height. If you know the base radius and the slant height, use h = sqrt(l^2 - r^2). If you know the base radius and the volume, rearrange the cone volume formula (V = (1/3) x pi x r^2 x h) to get h = 3V / (pi x r^2). If you know the base radius and the lateral surface area (L = pi x r x l), first recover the slant height as l = L / (pi x r) and then apply the Pythagorean theorem. Finally, if you know the volume and the height, you can recover the radius and then derive everything else. This calculator handles all five combinations.
Cone volume, surface area, and angles
Once the height is known, the rest of the geometry follows directly. The volume is V = (1/3) x pi x r^2 x h, exactly one-third the volume of the cylinder with the same base and height. The lateral (curved) surface area is L = pi x r x l, and the total surface area adds the base circle: A = pi x r x (r + l). The half-angle at the apex (the angle between the central axis and any slant side) is arctan(r / h), while the base angle (between the slant side and the base plane) is arctan(h / r). These angles matter in engineering, woodworking, and optics applications.
Practical uses
Cone geometry comes up in many fields. In construction and manufacturing, it governs the design of funnels, silos, rocket nose cones, and drill bits. In food science, knowing cone volume helps calculate ice-cream cone capacity or funnel flow rates. In acoustics, horn speakers and megaphones are conical; the half-angle determines how sound disperses. Packaging engineers use cone dimensions to minimise material while hitting a target volume. Surveyors use the relationship between slant height and true height to measure the heights of mountains and towers from a distance. The formulas here apply to any right circular cone regardless of scale.
Common cone formulas
| To find | Known values | Formula |
|---|---|---|
| Height (h) | r and l | h = sqrt(l^2 - r^2) |
| Height (h) | r and V | h = 3V / (pi x r^2) |
| Height (h) | r and L | h = sqrt((L / (pi x r))^2 - r^2) |
| Radius (r) | h and V | r = sqrt(3V / (pi x h)) |
| Slant height (l) | r and h | l = sqrt(r^2 + h^2) |
| Volume (V) | r and h | V = (1/3) x pi x r^2 x h |
| Lateral area (L) | r and l | L = pi x r x l |
| Total area (A) | r and l | A = pi x r x (r + l) |
All formulas assume a right circular cone with base radius r, height h, and slant height l.
Frequently asked questions
What is the difference between height and slant height?
The height is the straight vertical line from the centre of the base to the apex, meeting the base at 90 degrees. The slant height is the distance from the apex to the rim of the base measured along the outer surface of the cone. They are connected by the Pythagorean theorem: l^2 = r^2 + h^2, so the slant height is always longer than the vertical height.
How do I find the height of a cone if I only know the volume and radius?
Start from the volume formula V = (1/3) x pi x r^2 x h and rearrange: h = 3V / (pi x r^2). For example, if V = 524 cm^3 and r = 5 cm, then h = 3 x 524 / (pi x 25) = 1572 / 78.54 = approximately 20.01 cm.
Can I calculate the height from the slant height and radius alone?
Yes. Because the height, radius, and slant height form a right triangle, you can use the Pythagorean theorem: h = sqrt(l^2 - r^2). For instance, with a slant height of 13 cm and a base radius of 5 cm, the height is sqrt(169 - 25) = sqrt(144) = 12 cm.
Why does a cone hold one-third the volume of a cylinder?
A cone with base radius r and height h has volume (1/3) x pi x r^2 x h, while the enclosing cylinder has volume pi x r^2 x h. The ratio is exactly 1:3. This result can be proved by calculus (integrating thin circular cross-sections from base to tip) or by Cavalieri's principle combined with a geometric decomposition of a prism into three equal pyramids.
What if the slant height I enter is smaller than the radius?
That is geometrically impossible: the slant height is always the hypotenuse of the right triangle formed by the height and radius, so it must be strictly greater than the radius. The calculator will show no result and you should check your measurements.
How do the apex angle and base angle relate to each other?
The half-angle at the apex (theta = arctan(r / h)) and the base angle (beta = arctan(h / r)) are complementary: they always add up to 90 degrees. A tall, narrow cone has a small apex half-angle and a steep base angle, while a wide, flat cone has a large apex half-angle and a shallow base angle.