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AFCAT Numerical Ability · Area and Perimeter

Triangles: Base and Height, Right Triangles and Heron's Formula

A triangle's area is half its base times its height; when the height is not given, Pythagoras or Heron's formula supplies it from the sides.

All Area and Perimeter notes

Why this matters

Most of the page is EASY, and the questions differ only in what they hand you: a base and height, two legs, or three sides. The marks go when the half is dropped, when a slant side is used as the height, or when Heron's formula is started from the full perimeter instead of half of it.

Concept 1 of 3: Half the base times the height

A triangle is half of a rectangle with the same base and height. Draw the rectangle round it and the triangle fills exactly half, whichever way it leans. Any side can be the base, as long as the height is measured at right angles to that side.

Definition

  • Area =12×base×height= \tfrac12 \times \text{base} \times \text{height}, the height perpendicular to the base.
  • In a right triangle the two legs are a base and its height.
  • The same area gives a height for every side: h=2Asideh = \dfrac{2A}{\text{side}}.
  • Perimeter is the sum of the three sides; an equilateral triangle of side aa has perimeter 3a3a.

Area of a triangle

A=12×b×h,h=2AbA = \tfrac12 \times b \times h,\qquad h = \dfrac{2A}{b}

Worked example

A triangle has a base of 14 cm and a height of 9 cm. Find its area, and the height to its 18 cm side.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Area and Perimeter · Triangles

What is the area of a right-angled triangle with legs 66 cm and 88 cm?

Keep the half

A base of 14 and a height of 9 give 63, not 126. The product alone is the rectangle round the triangle.

The height is not the slant side

In a triangle with a 10 cm slant side and a 7 cm height on a 12 cm base, the area is 12×12×7=42\tfrac12 \times 12 \times 7 = 42. The slant side is not used.
Drill 3 more on half the base times the height

Concept 2 of 3: Right triangles and Pythagoras

In a right triangle the square on the hypotenuse equals the squares on the two legs together. That single fact turns two sides into the third, and with the legs known the area is half their product.

Definition

  • Pythagoras: c2=a2+b2c^2 = a^2 + b^2, where cc is the side opposite the right angle.
  • Triples worth knowing: 3, 4, 5 · 5, 12, 13 · 8, 15, 17 · 9, 40, 41 · 20, 21, 29, and any multiple of them.
  • Isosceles right triangle with legs aa: hypotenuse a2a\sqrt 2, area a22\dfrac{a^2}{2}.
  • Legs from their difference and the area: (a+b)2=(a−b)2+4ab(a + b)^2 = (a - b)^2 + 4ab, where ab=2×areaab = 2 \times \text{area}.
  • Squares drawn on the three sides: the square on the hypotenuse is as big as the other two together.

Pythagoras

c2=a2+b2,(a+b)2=(a−b)2+4abc^2 = a^2 + b^2,\qquad (a + b)^2 = (a - b)^2 + 4ab

Worked example

The legs of a right triangle differ by 3 cm and its area is 54 cm². Find its perimeter.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Area and Perimeter · Triangles

The difference between the sides at right angles in a right angled triangle is 14 cm. The area of the triangle is 120 cm2120 \text{ cm}^2. The perimeter of the triangle is

The hypotenuse faces the right angle

With sides 9, 12 and 15, the legs are 9 and 12 and the area is 54. Taking 12 and 15 as the base and height gives 90, which is wrong: 15 is opposite the right angle.

Legs, not the hypotenuse, in the isosceles case

An isosceles right triangle with hypotenuse 424\sqrt 2 has legs of 4 and area 8. Squaring the hypotenuse and halving gives 16, twice too much.
Drill 3 more on right triangles and pythagoras

Concept 3 of 3: Heron's formula and the isosceles triangle

When only the three sides are known, Heron's formula gives the area without any height. For an isosceles triangle there is a shorter road: the height from the apex cuts the base in half, so Pythagoras gives it at once.

Definition

  • Heron: s=a+b+c2s = \dfrac{a + b + c}{2} (half the perimeter), then area =s(s−a)(s−b)(s−c)= \sqrt{s(s - a)(s - b)(s - c)}.
  • Before using it, test for a right triangle: if the sides are a triple, half the product of the legs is quicker.
  • Isosceles with equal sides aa and base bb: height =a2−(b2)2= \sqrt{a^2 - \left(\tfrac b2\right)^2}, area =12×b×= \tfrac12 \times b \times height.

Heron's formula

A=s(s−a)(s−b)(s−c),s=a+b+c2A = \sqrt{s(s - a)(s - b)(s - c)},\quad s = \dfrac{a + b + c}{2}

Worked example

Find the area of a triangle with sides 9 cm, 10 cm and 17 cm.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Area and Perimeter · Triangles

A triangle has sides 1313 cm, 1414 cm, and 1515 cm. Find its area using Heron's formula:

s is half the perimeter

For sides 9, 10 and 17, s=18s = 18. Starting from 36 makes every bracket wrong and the root of a non-square.

Halve the base for the height

With equal sides 26 and base 20, the height uses half the base: 676−100\sqrt{676 - 100}. Using the whole base gives 276\sqrt{276}, which is not the height.
Drill 3 more on heron's formula and the isosceles triangle

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Watch out for (6)

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