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

After this lesson, you will be able to:

1. Compute perimeter by summing all sides
2. Apply with the correct height
3. Find a missing height using the Pythagorean theorem
4. Use the equilateral shortcut
5. Compute area from coordinates with the shoelace formula

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.

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!

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

Quick Check on Equilateral Triangle Area

An equilateral triangle has side length 8.

What is its area?

Use either method — find the height first, or apply the shortcut.

Work it out before advancing...

Heron's Formula When Height Is Unknown

When all three sides are known but height is hard to find:

Note: The here is the semi-perimeter, not a side length.

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: , ,

ACT Practice Problems to Solve Now

Problem 1: Triangular garden, base 14 ft, height 9 ft. Find the area.

Problem 2: Area is 40 cm², base is 10 cm. Find the height.

Problem 3: Two triangles share base 12 in. Heights: 5 in and 8 in. Find the area difference.

Solve all three before advancing...

Solutions to the Practice Problems

Problem 1:

Problem 2:

Problem 3:

Key Takeaways and Common Mistakes

• Perimeter:
• Area: — height must be perpendicular
• Equilateral:
• Shoelace: Use absolute value

Watch out:

• Height is perpendicular, not slant
• Include
• is half the base, not height

Coming Up Next in Triangle Topics

Up next: Triangle similarity and congruence

• When are two triangles the same shape?
• AA, SAS, and SSS similarity criteria
• Using proportions to find missing sides