Slant Height Formula for Cones and Pyramids

Key Points

• The slant height formula for a cone is $l = \sqrt{r^2 + h^2}$, where $r$ is the base radius and $h$ the vertical height.

• For a square pyramid the formula is $l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2}$, using half the base edge rather than the whole edge.

• Both versions are the Pythagorean theorem applied to a right triangle formed inside the solid.

• Slant height is always longer than vertical height, because the vertical height is one leg of that right triangle and the slant height is its hypotenuse.

• Slant height is what lateral surface area formulas require, which is why it must be found before the curved surface of a cone can be computed.

What Is the Slant Height Formula?

Slant height is the distance from the apex of a cone or pyramid down its sloping surface to the base edge. For a cone:

$$l = \sqrt{r^2 + h^2}$$

For a square pyramid, measuring to the midpoint of a base edge:

$$l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2}$$

Both follow from the Pythagorean theorem. MacTutor records that Pythagoras may have been the first to prove the theorem, though it was known to the Babylonians a thousand years earlier.

Why Pythagoras Produces the Slant Height

Slice a cone vertically through its apex and you expose a right triangle. Its vertical leg is the height $h$, its horizontal leg is the radius $r$, and its hypotenuse runs along the surface from apex to rim, which is exactly the slant height.

That is why the formula is a square root of a sum of squares rather than anything more exotic. The slant height is a hypotenuse, and a hypotenuse is always the longest side, so $l > h$ and $l > r$ in every cone.

The pyramid case changes only which horizontal distance is used. Measuring to the midpoint of a base edge means the horizontal leg is half the edge, $\frac{a}{2}$, not the whole edge.

Variable Key

Examples of the Slant Height Formula

Example 1

A cone has radius 3 and height 4. Find the slant height.

$$l = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$$

Final answer: $l = 5$

Example 2

A square pyramid has base edge 10 and height 12. Find the slant height using the full base edge.

The base edge is given as 10, so using it directly looks natural:

$$\sqrt{12^2 + 10^2} = \sqrt{144+100} = \sqrt{244} \approx 15.62$$

That is too large, and a sketch shows why. The right triangle runs from the apex down to the centre of the base and then out to the midpoint of an edge, so its horizontal leg reaches only halfway across, a distance of 5.

Using half the edge:

$$l = \sqrt{12^2 + 5^2} = \sqrt{144+25} = \sqrt{169} = 13$$

Final answer: $l = 13$

Example 3

A cone has radius 7 and height 24. Find the slant height.

$$l = \sqrt{49 + 576} = \sqrt{625} = 25$$

Final answer: $l = 25$

Example 4

A cone has slant height 13 and radius 5. Find its vertical height.

Rearranged, the formula gives $h = \sqrt{l^2 - r^2}$:

$$h = \sqrt{169 - 25} = \sqrt{144} = 12$$

Final answer: $h = 12$

Note the subtraction. Working back from the hypotenuse means subtracting, and a result under the root that comes out negative would mean the slant height was shorter than the radius, which is geometrically impossible.

Example 5

A cone has radius 6 and height 8. Find its curved surface area, taking $\pi \approx 3.14$.

Find the slant height first, since the lateral area formula $\pi r l$ requires it:

$$l = \sqrt{36+64} = \sqrt{100} = 10$$ $$\pi r l \approx 3.14 \times 6 \times 10 = 188.4$$

Final answer: about 188.4 square units

This is why slant height matters. The curved surface of a cone cannot be found from the vertical height directly.

Don't Confuse Slant Height With These Three

The apothem is the pyramid case's horizontal leg, which for a square base is exactly $\frac{a}{2}$. For a pyramid on a regular pentagon or hexagon the apothem is not half an edge, so the half-base shortcut applies only to squares.

Slant Height, Common Confusions Cleared Up

Mistake 1: Using the full base edge for a pyramid

Where it slips in: The base edge is the figure supplied in the question, and halving it is an extra step.

Don't do this: $l = \sqrt{h^2 + a^2}$

The correct way: $l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2}$. The horizontal leg reaches from the base centre to an edge midpoint, which is half the edge on a square base.

Students who sketch the internal right triangle before substituting anything almost never pick the wrong leg, because the drawing shows the triangle reaching only halfway across the base. A quick cross-section is worth more here than remembering two separate formulas.

Mistake 2: Treating slant height and vertical height as interchangeable

Where it slips in: Both are called heights, and a question may supply either.

Don't do this: Substitute $l$ where a volume formula wants $h$.

The correct way: Volume uses the vertical height, while lateral surface area uses the slant height. Since $l > h$ always, swapping them inflates a volume.

Mistake 3: Adding instead of subtracting when working backwards

Where it slips in: The forward formula adds two squares, so the reverse direction gets the same treatment.

Don't do this: $h = \sqrt{l^2 + r^2}$

The correct way: $h = \sqrt{l^2 - r^2}$. The slant height is the hypotenuse, so recovering a leg means subtracting the other leg's square.

Key Takeaways

• The slant height formula is $l = \sqrt{r^2 + h^2}$ for a cone and $l = \sqrt{h^2 + (a/2)^2}$ for a square pyramid.

• Both are the Pythagorean theorem applied to a right triangle exposed by a vertical cross-section.

• The slant height is a hypotenuse, so it always exceeds both the vertical height and the base radius.

• Volume uses vertical height while lateral surface area uses slant height, and the two are not interchangeable.

• To work through solid geometry with a teacher, explore a geometry tutor or math classes online.

Practice These Three Before Moving On

Find the slant height of a cone with radius 9 and height 12. Then find the slant height of a square pyramid with base edge 16 and height 15, and recover the vertical height of a cone with slant height 17 and radius 8.

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Read More

• Cone

• Surface Area Formula

• Volume Formula

• Hypotenuse

• Pythagorean Triples