Lesson 8 · 35 min

The Tail: Arrangement and First Sizing

The wing is designed for lift; the tail is designed for balance. It trims the wing's nose-down moment, keeps the aircraft pointed into the wind, and gives the pilot control in pitch and yaw. Its proper size needs the stability analysis of Module 11, but a first size comes from the same kind of statistics as everything else in this course: tail volume coefficients.

Learning objectives

State the three jobs of the tail: trim, stability and control.
Compare the main tail arrangements and say when each is used.
Choose tail aspect and taper ratios from statistics.
Size the horizontal and vertical tails from tail volume coefficients and tail arms.

What the tail does

Trim. A cambered wing pitches nose down (Lesson 2), and the center of gravity is rarely exactly at the wing's aerodynamic center. The horizontal tail supplies the balancing moment, usually with a small download.
Stability. If the nose pitches up, the tail's angle of attack rises and its extra lift pushes the nose back down; the vertical tail does the same in yaw, like a weathervane.
Control. The elevator and rudder, hinged on the tails, change their lift to pitch and yaw the aircraft, and to hold it straight with an engine failed (Module 2).

Each job depends on the tail's force times its distance from the center of gravity: a tail far aft can be small. That is why tail sizes are compared through a volume coefficient, an area times an arm.

Tail arrangements

Figure 8.1 The main tail arrangements, seen from behind (horizontal surfaces blue, vertical surfaces violet). Choose one to read what it is for and what it costs.

About seven aircraft in ten use the conventional tail: it is light, well understood, and works. The others solve a particular problem. The T-tail lifts the horizontal tail out of the wing's wake and the engines' exhaust, which suits aircraft with engines at the rear, like the business jet; the horizontal tail also acts as an end plate, making the fin more effective, but the fin must be heavier to carry it, and in a deep stall the horizontal tail can be blanketed by the wing's wake. A cruciform tail is the compromise. H-tails put the fins in the propeller slipstream or allow a shorter fin. V-tails replace three surfaces with two, at the cost of mixing the controls. A canard moves the horizontal surface ahead of the wing, where it lifts upward to trim; a tailless aircraft trims with a reflexed airfoil or with sweep and washout.

Tail geometry

Tails are smaller wings, chosen by the same rules but with different priorities. Their aspect ratios are lower than the wing's, which delays their stall: the tail must still work when the wing is near its maximum lift. They are usually swept about \(5^\circ\) more than the wing and made a little thinner, so they reach drag divergence and stall later than the wing does.

Tail aspect and taper ratios (source: Raymer, Table 4.3; check the values against the edition you use)
Aircraft	Horizontal \(A\)	Horizontal \(\lambda\)	Vertical \(A\)	Vertical \(\lambda\)

The aspect ratio of a vertical tail uses its height: \(A_{VT} = h^2/S_{VT}\). The fin of a T-tail is short and broad to carry the horizontal tail on top.

First sizing: tail volume coefficients

The horizontal tail's moment is compared with the wing's pitching moment, which scales with \(S_W\bar c_W\); the vertical tail's with the wing's yawing moment, which scales with \(S_Wb_W\). Their ratios are the tail volume coefficients:

Tail volume coefficients

\[ c_{HT} = \frac{L_{HT}\,S_{HT}}{\bar c_W\,S_W}, \qquad c_{VT} = \frac{L_{VT}\,S_{VT}}{b_W\,S_W} \] \[ S_{HT} = \frac{c_{HT}\,\bar c_W\,S_W}{L_{HT}}, \qquad S_{VT} = \frac{c_{VT}\,b_W\,S_W}{L_{VT}} \]

\(L\) is the tail arm, from the wing's quarter-MAC to the tail's. For a first layout: about 60% of the fuselage length for a single propeller engine in the nose, 50 to 55% with engines on the wing, 45 to 50% with engines at the rear.

Figure 8.2 lays these quantities out on the business jet, and on a typical layout of each class in the table below. The business jet's wing is fixed by Lesson 6: its area \(S_W\), its span \(b_W\), and its mean aerodynamic chord \(\bar c_W\), whose quarter point is the wing's aerodynamic center. Note that the horizontal tail is compared with the mean aerodynamic chord, not the mean geometric chord \(S_W/b_W\). Each arm runs from the wing's quarter-MAC point to the tail's own, measured along the fuselage.

Figure 8.2 An aircraft's wing and tails, to scale: a top view of one half and a side view, lined up so that the dashed projection lines carry each quarter-MAC point (dot) from one view to the other. Choose a type. Each is an illustrative layout of typical size, with a wing area, aspect ratio, taper, sweep and position, a fuselage and engines to match, and the tail arm its engine location suggests: about 60% of the fuselage with a propeller in the nose, 52% with engines on the wing, and 45 to 48% with engines at the rear or inside. The wing (teal) has its mean aerodynamic chord marked in orange; the horizontal tail (blue) and vertical tail (violet) are sized from the type's volume coefficients. Lengthen the arm and both tails shrink. Switch to a T-tail: the horizontal tail rides on top of the swept fin, its arm grows, and it needs less area. Example 8.1 is the business jet with one arm for both tails, as the conventional layout gives.

Tail volume coefficients (source: Raymer, Table 6.4; check the values against the edition you use)
Aircraft	Horizontal \(c_{HT}\)	Vertical \(c_{VT}\)

Example 8.1 — The business jet's tails

The business jet has no class of its own in the table; in size and speed it is closest to a jet trainer, so take \(c_{HT} = 0.70\) and \(c_{VT} = 0.06\). Its wing has \(S_W = 28\ \text{m}^2\), \(\bar c_W = 2.015\ \text{m}\), \(b_W = 14.97\ \text{m}\) (Lesson 6); its fuselage is \(14\ \text{m}\) long with the engines at the rear. Size both tails, and the height of a T-tail fin with \(A_{VT} = 1.0\).

Show solution

Tail arm: about 47% of the fuselage length, \(L = 0.47(14) = 6.58\ \text{m}\), for both tails at this stage.

\[ S_{HT} = \frac{0.70(2.015)(28)}{6.58} = 6.00\ \text{m}^2 \] \[ S_{VT} = \frac{0.06(14.97)(28)}{6.58} = 3.82\ \text{m}^2 \] \[ h = \sqrt{A_{VT}S_{VT}} = \sqrt{1.0(3.82)} = 1.95\ \text{m} \]

The horizontal tail is 21% of the wing area, the vertical tail 14%. Module 3's sketch had a horizontal tail of 6.0 m², exactly this, and a fin of 4.5 m², somewhat generous. With the jet-transport coefficients the tails would be 8.6 and 5.7 m²: the choice of class matters as much here as it did for the empty weight.

Figure 8.3 Tail areas for the business jet's wing against the tail arm, from the volume coefficients of the chosen class: a longer arm needs a smaller tail. The markers are at the chosen arm. A longer fuselage buys a smaller tail, but adds its own weight and wetted area; Module 6 finds the balance.

Check your understanding

Key takeaways

The tail trims, stabilizes and controls; its effect is its force times its arm.
Conventional tails are the default; T-tails suit rear engines, V-tails save a surface, canards and tailless designs trim differently.
Tails have lower aspect ratios than wings and about \(5^\circ\) more sweep, so they stall and diverge later.
\(S_{HT} = c_{HT}\bar c_WS_W/L_{HT}\), \(S_{VT} = c_{VT}b_WS_W/L_{VT}\), with \(L\) about 45 to 60% of the fuselage length.
Next: Module 5 chooses the thrust-to-weight ratio and wing loading that fix the real wing area.