How do you calculate the volume of a cone?

Volume of a cone = (1/3) x pi x r^2 x h, where r is the base radius and h is the height. Example: a cone with radius 4 cm and height 9 cm has volume = (1/3) x 3.14159 x 16 x 9 = 150.8 cm^3. Note that a cone has exactly one-third the volume of a cylinder with the same base and height.

What is the slant height of a cone?

The slant height (l) is the distance from the apex (tip) of the cone to any point on the edge of the base circle, measured along the surface. It differs from the vertical height (h). Formula: l = sqrt(r^2 + h^2) using the Pythagorean theorem. Example: a cone with radius 3 cm and height 4 cm has slant height = sqrt(9 + 16) = sqrt(25) = 5 cm.

How do you find the surface area of a cone?

Total surface area of a cone = pi x r x l + pi x r^2, where r is the radius and l is the slant height. The first term (pi x r x l) is the lateral surface area (the curved side). The second term (pi x r^2) is the base circle area. Example: cone with radius 3 cm and slant height 5 cm: lateral area = 3.14159 x 3 x 5 = 47.12 cm^2. Base area = 3.14159 x 9 = 28.27 cm^2. Total = 75.4 cm^2.

What is the difference between a cone and a pyramid?

Both have a pointed apex and taper from a base to a point. The difference is the base shape: a cone has a circular base, while a pyramid has a polygonal base (triangle, square, etc.). Volume formulas are analogous: cone V = (1/3) x base area x height; pyramid V = (1/3) x base area x height. A cone can be thought of as a pyramid with an infinite number of faces making up a smooth circular base.

Where are cones found in everyday life?

Cones appear in many real-world contexts: ice cream cones, traffic cones, party hats, funnels, volcanoes, and the nose cones of rockets. In mathematics, conic sections (circles, ellipses, parabolas, hyperbolas) are all formed by slicing a cone at different angles, making the cone one of the most geometrically significant 3D shapes.

Cone Calculator

Calculate volume, total surface area, lateral area, and slant height of a cone.

🔺 What is a Cone?

A cone is a three-dimensional geometric shape that tapers smoothly from a flat circular base to a point called the apex (or vertex). The axis of a right circular cone is the straight line from the apex perpendicular to the base's centre. This is the most common type of cone and the one this calculator computes.

Cones appear throughout engineering, architecture, and everyday life - ice cream cones, party hats, traffic cones, funnels, and the nozzles of rockets are all cone-shaped. In manufacturing and construction, computing a cone's volume, surface area, and slant height is essential for material estimation, structural analysis, and packaging design.

Given just the base radius and vertical height of a cone, all other properties can be derived: the slant height via the Pythagorean theorem, the lateral (curved) surface area, the base area, and the total surface area.

📐 Cone Formulas

Slant Height: l = √(r² + h²)

Volume: V = (1/3) × π × r² × h

Lateral Area: A_L = π × r × l

Total Surface Area: A_T = πr(r + l)

r = base radius

h = vertical height (perpendicular from apex to base)

l = slant height (distance along the surface from base edge to apex)

π ≈ 3.14159265

The slant height l is calculated first using the Pythagorean theorem - the radius, height, and slant height form a right triangle. Lateral area is the area of the curved side only (excluding the base). Total surface area adds the base circle area (πr²) to the lateral area, giving the total exterior surface.

📖 How to Use This Calculator

Steps

Enter the base radius - the distance from the centre of the base circle to its edge.

Enter the vertical height - the perpendicular distance from the base to the apex.

Click Calculate to get volume, surface area, lateral area, slant height, and base area.

💡 Example Calculations

Example 1 - Traffic Cone (r = 15 cm, h = 45 cm)

Base radius = 15 cm, Height = 45 cm

Slant height l = √(15² + 45²) = √(225 + 2025) = √2250 = 47.43 cm

Volume = (1/3) × π × 15² × 45 = (1/3) × π × 225 × 45 = 10,602.9 cm³

Lateral Area = π × 15 × 47.43 = 2,236.2 cm²

Total Surface Area = π × 15 × (15 + 47.43) = 2,942.7 cm²

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Example 2 - Ice Cream Cone (r = 3 cm, h = 10 cm)

Base radius = 3 cm, Height = 10 cm

Slant height l = √(9 + 100) = √109 = 10.44 cm

Volume = (1/3) × π × 9 × 10 = 94.25 cm³

Total Surface Area = π × 3 × (3 + 10.44) = 126.7 cm²

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❓ Frequently Asked Questions

What is the difference between height and slant height?+

The height (h) is the vertical distance from the apex straight down to the centre of the base - measured at a 90° angle to the base. The slant height (l) is the distance along the surface from the apex to any point on the base circle's edge. Slant height is always greater than vertical height (l > h), and the three quantities form a right triangle: l² = r² + h².

What is the difference between slant height and vertical height of a cone?+

Vertical height (h) is the perpendicular distance from apex to base center. Slant height (l) is the distance from apex to base edge along the surface. They relate by: l = sqrt(r squared + h squared) via the Pythagorean theorem. Lateral surface area uses slant height; volume uses vertical height.

How do you find the volume of a truncated cone (frustum)?+

Volume of frustum = (pi x h / 3)(R squared + Rr + r squared), where R = large base radius, r = small base radius, h = height. A frustum is a cone with the top cut off - common in buckets and paper cups. To calculate: find the full cone volume and subtract the removed top cone volume.

What is the net of a cone?+

The net (unfolded surface) of a cone consists of a circular base and a sector (pie slice) of a larger circle for the lateral surface. The sector radius equals the slant height, and the arc length equals the base circumference. This is used in manufacturing cone-shaped objects from flat sheet materials like paper or metal.

Why is the volume of a cone one-third of a cylinder?+

This is a fundamental result from calculus (integration) and can also be shown experimentally: exactly three cones with the same base and height fill one cylinder. Intuitively, the cone tapers from a full circular base to a point, so on average it occupies one-third of the space that a cylinder of the same base and height would occupy.

How do I find the radius if I only know the volume and height?+

Rearrange the volume formula: V = (1/3)πr²h → r² = 3V / (πh) → r = √(3V / (πh)). For example, if V = 500 cm³ and h = 15 cm: r = √(3 × 500 / (π × 15)) = √(1500 / 47.12) = √31.83 = 5.64 cm.

What is lateral surface area used for in practice?+

Lateral surface area tells you the amount of material needed to cover just the curved side of a cone, without the base. This is useful when making a cone-shaped hat, funnel, or nozzle where the base is open. If you need to cover the entire exterior including the base, use the total surface area instead.

What is the formula for the volume of a cone?+

Volume of a cone = (1/3) x pi x r^2 x h, where r is base radius and h is perpendicular height. A cone holds exactly 1/3 the volume of a cylinder with the same base and height. Example: cone with radius 5 cm and height 12 cm: V = (1/3) x 3.14159 x 25 x 12 = 314.16 cm^3.

What is the lateral surface area of a cone?+

Lateral surface area = pi x r x l, where l is the slant height = sqrt(r^2 + h^2). Total surface area = pi x r x l + pi x r^2 (lateral + base). Example: cone with r = 3 cm, h = 4 cm: slant height = 5 cm, lateral area = pi x 3 x 5 = 47.12 cm^2, base area = pi x 9 = 28.27 cm^2, total = 75.39 cm^2.

What are real-world examples of cone shapes?+

Cones appear in traffic cones (truncated cones), ice cream cones, party hats, funnel shapes, volcanic mountains, and geometric solid models in engineering. The frustum (truncated cone) is used in buckets, cups, and structural transitions. Cone calculations are used in civil engineering (earth mounds, stockpiles), food packaging, and optics (light cones).

How is a cone different from a pyramid?+

A cone has a circular base and a curved lateral surface. A pyramid has a polygonal base (triangle, square, etc.) and flat triangular faces. Both have volume = (1/3) x base area x height. The cone is essentially a pyramid with an infinite-sided polygonal base. For the same base area and height, both hold the same volume.