Geometry

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

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

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

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.

Parallelogram

Geometry

Triangles and Quadrilaterals Cont.

Triangle

Triangle perimeterTriangle area

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

Triangle The area of a trapezoidtrapezoid is half the parallelogram

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

\(A_{rhombus} = \frac{1}{2}d_1d_2\)

Rhombus

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:

Kite

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