Area and Perimeter of Triangles

Triangles — Formulas, Heights, and Coordinates

In this lesson:

• Apply area and perimeter formulas
• Find missing heights using the Pythagorean theorem
• Compute area from vertex coordinates

What You Will Learn Today

1. Compute the perimeter from three side lengths
2. Apply to right and general triangles, identifying the perpendicular height
3. Find area with no height given: Pythagorean theorem (isosceles, equilateral, shortcuts) or Heron's formula
4. Compute area from vertex coordinates: shoelace or rectangle decomposition

Perimeter Sums All Three Side Lengths

Perimeter = distance around the triangle

• Add the three sides — no special formula needed
• If a side is missing, find it first (e.g., Pythagorean theorem)

Example: Sides 7, 10, and 13

Area Requires the Perpendicular Height

The height must be perpendicular to the base. A triangle is half the rectangle with the same base and height.

Right Triangle Area Uses Both Legs

A right triangle has legs 6 cm and 8 cm.

Area: The legs are perpendicular, so use them directly.

Perimeter: Find the hypotenuse first.

General Triangle With Altitude Drawn Inside

A triangle has base 12 in and height 5 in. A slant side measures 7 in.

Not — the slant side is not the height!

Obtuse Triangle Altitude Falls Outside

Base 6 on the -axis; the altitude from lands at , so .

Slant side 5 as height would give 15 — wrong.

Quick Check on Perpendicular Height

A triangle has base 9, a slant side of 5, and a perpendicular height of 4.

What is the area?

• (A)
• (B)

Choose carefully — which number is the height?

Isosceles Altitude Bisects the Base Exactly

Drop an altitude from the apex — it bisects the base.

Isosceles Example With Sides Thirteen and Ten

Isosceles triangle: equal sides 13, base 10. Find area.

Step 1: Altitude bisects the base → half-base = 5

Step 2: Find height via Pythagorean theorem

Step 3: Compute area

Equilateral Height From Pythagorean Theorem

For an equilateral triangle with side :

Equilateral Triangle With Side Length Six

Equilateral triangle with . Find the area both ways.

Method 1: Find height first

Method 2: Use the shortcut directly

Both methods give

Heron's Formula, Checked on Five-Twelve-Thirteen

Three sides known, height hard to find:

Here is the semi-perimeter, not a side. Check: sides 5, 12, 13 give and .

Heron's Formula Example With Three Sides

Triangle with sides 7, 8, and 9. Find the area.

Step 1: Semi-perimeter

Step 2: Apply Heron's formula

Coordinate Triangle With an Axis-Aligned Base

Vertices: , ,

Base: Along → length =

Height: Vertical distance to →

Sometimes no formula is needed — just read the coordinates.

Shoelace Formula for Any Coordinate Triangle

Given vertices , , :

• Works for any triangle orientation
• The absolute value is essential — area is never negative

Shoelace Worked Example With Three Vertices

Vertices: , ,

Reverse the vertex order and becomes . Dropping the would report 27.

Rectangle Decomposition Checks the Shoelace

Bounding rectangle: . Corner triangles: .

Word Problems: Name Base and Height

• Garden, base 14 ft, height 9 ft:
• Sail, right triangle, legs 6 m and 8 m:
• Area 40 cm², base 10 cm: , so cm
• Same base 12 in, heights 5 in and 8 in: larger

Key Takeaways and Common Mistakes

• Perimeter:
• Area: , height perpendicular
• No height given: Pythagorean theorem, equilateral , Heron
• Coordinates: shoelace or rectangle decomposition

Watch out: slant side as height, missing , mismatched base and height, as height, shoelace without absolute value

Coming Up Next in Polygon Topics

Up next: Interior and exterior angle sums of polygons

• Angle sums for polygons with any number of sides
• Exterior angles and what they always total
• Finding unknown angles in a polygon