# Triangles and Quadrilaterals

## Big Picture

Triangles and quadrilaterals are among the more basic and common polygons. Triangles always have interior angles sum to 180° while quadrilaterals always have interior angles sum to 360°.

## Key Terms

Perimeter: The distance around a shape.

Area: The amount of surface covered by a figure.

## Square

Perimeter = $$P_{square} = s + s + s + s = 4s$$

Postulate: The area of a square is the square of the length of its side.

• Area = $$A_{square} = s \cdot s = s^2$$

## Rectangle

Perimeter = $$p_{rectangle} = \color{blue}{b} \color{black}{+} \color{blue}{b} \color{black}{+} \color{red}{h} \color{black}{+} \color{red}{h} \color{black}{=} 2\color{blue}{b} \color{black}{+} 2\color{red}{h}$$

Theorem: The area of a rectangle is the product of its base and height.

• Area = $$A_{rectangle} = \color{blue}{b} \color{black}{\,\cdot \,} \color{red}{h} \color{black}{=} \color{blue}{b}\color{red}{h}$$

## Parallelogram

Either pair of parallel sides can be the bases of a parallelogram. The height is perpendicular to the base - the side is NOT the height!

• Area = $$A_{parallelogram} = \color{blue}{b}\color{red}{h}$$

The area of a parallelogram is the same as the area of a rectangle.

# Triangles and Quadrilaterals Cont.

## Triangle

Perimeter = $$P_{triangle} = a + b + c$$

Theorem: The area of a triangle is one half the product of the base and its corresponding height.

• Area = $$A_{triangle} = \frac{1}{2}bh$$

If a parallelogram is cut in half along a diagonal, there would be two congruent triangles. The area of the triangle, then, is half the area of the area of a parallelogram.

## Trapezoid

The height of a trapezoid is the perpendicular distance between its bases.

Theorem: The area of a trapezoid is one half the product of the height and the sum of the lengths of the bases.

• Area = $$A_{trapezoid} = \frac{1}{2}\color{red}{h} \color{black}{(b_1 + b_2)}$$

A trapezoid can be turned into a parallelogram with height h and base $$b_1 + b_2$$.

• The area for the parallelogram is $$\color{red}{h}\color{black}{(b_1 + b_2)}$$
• So the area of the trapezoid is half the parallelogram:

$$\frac{1}{2}\color{red}{h}\color{black}{(b_1 + b_2)}$$

## Rhombus and Kite

Both rhombuses (left) and kites (right) have perpendicular diagonals.

### Rhombus

Theorem: The area of a rhombus is half the product of the lengths of

$$A_{rhombus} = \frac{1}{2}d_1d_2$$

### Kite

Theorem: The area of a kite is half the product of the lengths of the diagonals.

• $$A_{kite} = \frac{1}{2}d_1d_2$$

The formulas for the areas of rhombus and kite are the same!

The areas for rhombus and kite can be found by creating two rectangles:

The area of the rectangles is: $$\frac{1}{2}d_1d_2$$