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MathChapter 17: Triangles

Notes

Area of a Triangle Notes

Select perpendicular base-height pairs, calculate area and perimeter, and reason about equilateral and area-ratio relationships.

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Triangle area depends on a chosen base and its perpendicular height. The height can lie inside, on a side, or outside the triangle, but it must meet the base line at 90 degrees.

Any side can be a base

Three base-height choicesThree versions of the same scalene triangle show base BC with altitude from A, base AC with altitude from B, and base AB with altitude from C. Every altitude is perpendicular to its chosen base line.
Three base-height choicesThree versions of the same scalene triangle show base BC with altitude from A, base AC with altitude from B, and base AB with altitude from C. Every altitude is perpendicular to its chosen base line.base BC; height from Abase AC; height from Bbase AB; height from C
Triangle area
\[A=\frac12bh\]

The base and height must be perpendicular partners.

Base-height relationships
Chosen baseCorresponding heightArea expression
(BC)altitude from A(\tfrac12(BC)(h_A))
(AC)altitude from B(\tfrac12(AC)(h_B))
(AB)altitude from C(\tfrac12(AB)(h_C))
Worked example

Recover a missing height

A triangle has area 84 and base 14. Find the corresponding height.

  1. Equation

    (84=\tfrac12(14)h=7h).

  2. Solve

    (h=12).

The height is 12 units.

Equilateral-triangle area

Equilateral-triangle altitudeAn equilateral triangle with side a is split by an altitude into two 30-60-90 triangles; each base half is a over 2 and the height is a square root of 3 over 2.
Equilateral-triangle altitudeAn equilateral triangle with side a is split by an altitude into two 30-60-90 triangles; each base half is a over 2 and the height is a square root of 3 over 2.aaaa√3/2a/2a/2A = √3a²/4
Equilateral area
\[A=\frac{\sqrt3}{4}a^2\]

The altitude creates two 30-60-90 triangles with height (a\sqrt3/2).

Derivation

  1. The altitude bisects the base, giving short leg (a/2).

  2. The 30-60-90 ratio gives height (a\sqrt3/2).

  3. (A=\tfrac12(a)(a\sqrt3/2)=\sqrt3a^2/4).

Area ratios

Side scale factor 2 and area scale factor 4Two similar triangles have corresponding side lengths 3, 4, 5 and 6, 8, 10. The side scale factor is 2, so the area scale factor is 4.
Side scale factor 2 and area scale factor 4Two similar triangles have corresponding side lengths 3, 4, 5 and 6, 8, 10. The side scale factor is 2, so the area scale factor is 4.ABCDEF3456810side ×2; area ×4
Area-ratio rules
ConditionArea ratio
Similar triangles; side factor (k)(k^2)
Equal heightsbase ratio
Equal basesheight ratio
Equal heights and base ratioTwo triangles have the same perpendicular height h and bases b one and b two, so their area ratio equals their base ratio.
Equal heights and base ratioTwo triangles have the same perpendicular height h and bases b one and b two, so their area ratio equals their base ratio.hhb₁b₂A₁/A₂ = b₁/b₂
Equal base and height ratioTwo triangles share the same base b but have perpendicular heights h one and h two, so their area ratio equals their height ratio.
Equal base and height ratioTwo triangles share the same base b but have perpendicular heights h one and h two, so their area ratio equals their height ratio.h₁h₂common base bA₁/A₂ = h₁/h₂
Areas of squares on a right triangleSquares constructed on the two legs have areas a squared and b squared; the square on the hypotenuse has area c squared, with a squared plus b squared equal to c squared.
Areas of squares on a right triangleSquares constructed on the two legs have areas a squared and b squared; the square on the hypotenuse has area c squared, with a squared plus b squared equal to c squared.a²b²c²a² + b² = c²
Perimeter
\[P=AB+BC+CA\]
Worked example

Percent change

Base increases 20 percent while height decreases 25 percent. How does area change?

  1. Factors

    Base factor (1.20); height factor (0.75).

  2. Multiply

    Area factor (1.20(0.75)=0.90).

Area decreases by 10 percent.
Mini check

Square the similarity factor

Similar triangles have side ratio (2:5). What is their area ratio?

Show answer and explanation

Answer: (4:25).

Square both terms of the side ratio.

Key takeaways

Key takeaways

  • Use (A=\tfrac12bh) with a perpendicular pair.
  • Equilateral area is (\sqrt3a^2/4).
  • Similar-triangle area factors are squared side factors.
  • Equal-height areas follow base ratios; equal-base areas follow height ratios.
  • Percent changes multiply as factors rather than add.
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Apply the ideas with SAT-style questions, then reinforce key details with flashcards.