Cone Surface Area Calculator

A cone has a circular base and a curved surface that tapers to a single point called the apex. Calculating its surface area requires a third measurement beyond the radius and height: the slant height, which is the straight-line distance from the apex to the edge of the base. This calculator derives the slant height automatically from your radius and height using Pythagoras, then returns the lateral surface area (the curved part only), the total surface area (curved part plus circular base), and the slant height itself. The lateral surface area is pi times the radius times the slant height, which makes sense because if you cut down the slope and unroll the curved surface you get a flat sector of a circle. The total surface area adds the circular base, pi times the radius squared. Knowing which to use depends on the situation: a party hat or a conical tent has no floor, so you need only the lateral area; a closed conical hopper or a solid model needs the total. The calculator works for any positive radius and height. The slant height is derived, not entered, which removes a common source of confusion where people mix up vertical height and slant height. Use this tool for geometry homework, estimating how much material you need to make a conical shape, checking the surface a conical pile covers, or any practical problem involving a cone. The formulas and a step-by-step worked example with the default values are set out below.

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units
units
113.1
total surface area (square units)
Slant height5
Lateral SA62.83
Base area50.27

How it works

Slant height l = √(r² + h²), using Pythagoras on the right triangle formed by the radius, height and slope. Lateral surface area = π × r × l (the curved face). Base area = π × r² (the circular bottom). Total surface area = πr(l + r) = lateral SA + base area.

Worked example

For a cone with radius 4 and height 3: slant height = √(16 + 9) = √25 = 5 units. Lateral surface area = π × 4 × 5 = 20π ≈ 62.83 square units. Base area = π × 16 ≈ 50.27 square units. Total surface area = π × 4 × (5 + 4) = 36π ≈ 113.10 square units. These match the pre-filled defaults above.

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Frequently asked questions

What is the formula for the lateral surface area of a cone?

Lateral surface area = pi x radius x slant height. The slant height is the distance from the apex to the edge of the base, equal to the square root of (radius squared + height squared). For r=4 and h=3: slant = 5, lateral SA = pi x 4 x 5 = 62.83 square units.

How do you find the total surface area of a cone?

Add the lateral surface area to the circular base: total SA = pi x r x slant + pi x r squared = pi x r x (slant + r). For r=4 and slant=5: total SA = pi x 4 x 9 = 113.10 square units.

What is the slant height of a cone?

The slant height is the straight-line distance from the apex of the cone down the outside to any point on the base edge. It is longer than the vertical height and is found using Pythagoras: slant = sqrt(r squared + h squared). For r=4 and h=3: slant = sqrt(16+9) = sqrt(25) = 5.