Perimeter and Area

Perimeter: Measuring the Boundary

The perimeter of a shape is the total distance around its outside edge — just add up the lengths of all the sides. For any polygon, P = the sum of all side lengths.

Area Formulas

Area measures how much surface a shape covers, in square units. Here are the essential formulas for this lesson:

Rectangle

For a rectangle with length l and width w:

Triangle

For a triangle with base b and the corresponding height h (the perpendicular distance from the base to the opposite vertex):

Picture a triangle sitting on a horizontal base; the height is a straight vertical segment dropped from the top vertex down to that base line, meeting it at a right angle — even if that meeting point falls outside the triangle for an obtuse triangle.

Parallelogram

A parallelogram with base b and height h (the perpendicular distance between the two parallel bases) has area:

This makes sense because you can slice a triangular piece off one end of the parallelogram and shift it to the other end to form a rectangle of the same base and height.

Trapezoid

A trapezoid with parallel bases b_1 and b_2, and height h (the perpendicular distance between the bases), has area:

Think of this as averaging the two parallel sides and multiplying by the height — essentially treating the trapezoid like a rectangle with an "average width."

Example

A trapezoid has parallel sides of length 8 and 14, and a height of 5. Find its area.

1. Use the trapezoid formula: A = \frac{1}{2}(b_1+b_2)\times h = \frac{1}{2}(8+14)\times 5
2. Add the bases: 8+14=22
3. Multiply: A = \frac{1}{2}(22)(5) = \frac{1}{2}(110) = 55

The trapezoid's area is 55 square units.

Key Takeaways

• Perimeter: add up all side lengths.
• Rectangle area: A = l \times w.
• Triangle area: A = \frac{1}{2}bh.
• Parallelogram area: A = bh.
• Trapezoid area: A = \frac{1}{2}(b_1+b_2)h.
• In every formula, the height must be measured perpendicular to the base you're using.