Find Areas of Right and Other Triangles

In this lesson:

• Cut a rectangle in half to find a right triangle's area
• Enclose any triangle in a rectangle and halve
• Watch out for the slanted-side trap

What You Will Be Able to Do

By the end, you will:

1. Halve a rectangle to find a right triangle's area
2. Identify base and altitude of any triangle
3. Apply to acute, right, obtuse
4. Avoid the slanted-side trap
5. Confirm an area by counting unit squares

A Rectangle You Already Know

A rectangle on a grid has length and width .

Count the unit squares — does your count match?

Cut the Rectangle Along the Diagonal

Two congruent right triangles. Each has half the rectangle's area.

The Right Triangle Is Half a Rectangle

The two legs of a right triangle are perpendicular — they are the base and the height.

The comes from the cut, not from a memorized rule.

Try Two Right Triangles, Same Area

Legs and :

Legs and :

Different shapes can have the same area; the formula doesn't care which leg is "base."

A Right Triangle With Fractional Legs

Legs and :

The formula holds for fractional measurements — no special case needed.

Watch Out: Don't Forget the One-Half

Which equals , the area we counted on the grid?

• A.
• B.

Choice A finds the rectangle's area, not the triangle's.

Check-In: Find the Area of a Right Triangle

A right triangle has legs and .

1. Which leg is the base? Which is the height?
2. Compute the area using .

Pause and write your answer before advancing.

From Right Triangles to Any Triangle

A right triangle's legs are perpendicular by definition.

For an acute or obtuse triangle, the perpendicular distance from a vertex to the opposite side is the altitude — and that is the height we use.

Same formula, with one extra construction step.

The Altitude as Perpendicular Distance

The altitude is perpendicular to the base — not the slanted side.

Enclose Any Triangle in a Rectangle

Rectangle area . The two shaded outer pieces are right triangles.

Why It Halves: Two Outer Right Triangles

The target triangle and the two outer pieces share the rectangle equally.

The Formula Works for Any Triangle

For any triangle:

• The base is any side you choose.
• The height is the perpendicular distance from the opposite vertex to that side.

Worked Example for an Acute Triangle

Acute triangle with base , altitude :

Multiply base by height first, then halve. Or halve first — either order works.

Worked Example for an Obtuse Triangle

Base , altitude :

Watch Out for the Slanted Side

A triangle has base , slanted side , perpendicular altitude .

• A. ← uses the slant
• B. ← uses the altitude

The slant is the longest line in the picture — and the wrong one.

Practice: Three Triangles, Three Choices of Height

Find the area of each:

1. Right triangle with legs and
2. Acute triangle with base , altitude
3. Obtuse triangle with base , altitude (altitude outside)

Show for each.

Answers and Common Errors to Avoid

Wrong answer 28 on Q1 → forgot the .

Wrong answer 40 on Q3 → multiplied without halving.

Key Takeaways for Triangle Area

• Right triangle = half its enclosing rectangle (legs are base and height)
• Any triangle = half its rectangle (altitude is the height)
• Formula:
• The slant is never the height; the perpendicular altitude is

Coming Up: Parallelograms and Trapezoids

You can now find the area of any triangle.

In Lesson 2, you will:

• Find the area of a parallelogram by sliding a triangle
• Derive the trapezoid formula two different ways
• Decompose irregular polygons into known shapes