1. Section 7.2 Perimeter and Area of Polygons
The perimeter of a polygon is the sum of the lengths of all sides of the polygon.
11/13/2018
Section 7.2 Nack

2. Formulas for Perimeter
Triangles
Scalene: P = a + b + c
Isosceles: P = b + 2s (base + 2 equal sides)
Equilateral: P = 3s (3 x sides)
Quadrilaterals
Quadrilateral: P = a + b + c + d (sum of the sides)
Rectangle: P = 2b + 2h or P = 2(b + h)
2 x (base + height)
Square (or rhombus): P = 4s (4 x sides)
Parallelogram: P = 2b + 2s or 2(b + s)
(2 bases + 2 sides)
11/13/2018
Section 7.2 Nack

3. Heron’s Formula for the Area of a Triangle
Semiperimeter: s = ½ (a + b + c)
Heron’s Formula:
A = s (s - a) (s - b) (s - c)
Example: Find the area of a triangle with sides 4, 13, 15.
s = ½ ( ) = 16
A = 16 (16 - 4) ( ) ( ) = 24 sq. units.
11/13/2018
Section 7.2 Nack

4. Brahmagupta’s Formula for the area of a cyclic* quadrilateral
Semiperimeter = ½(a + b + c + d)
Area = A =  (s - a) (s - b) (s - c) (s – d)
*cyclic quadrilateral can be inscribed in a circle
so that all 4 vertices lie on the circle.
11/13/2018
Section 7.2 Nack

5. Area of a Trapezoid
Theorem 7.2.4: The area A of a trapezoid whose bases have lengths b1 and b2 and whose altitude has length h is given by:
A = ½ h (b1 + b2 ) = ½ (b1 + b2)h The average of the bases times the height
Proof p. 350
Example 4 p. 350
11/13/2018
Section 7.2 Nack

6. Quadrilaterals with Perpendicular Diagonals
Theorem 7.2.4: The area of any quadrilateral with perpendicular diagonals of lengths d1 and d2 is given by A = ½ ( d1d2 ).
Corollary 7.2.5: The area A of a rhombus whose diagonals of lengths d1 and d2 is given by A = ½ ( d1d2 )
Corollary 7.26: The area A of a kite whose diagonals of lengths d1 and d2 is given by A = ½ ( d1d2 )
Proof: Draw lines parallel to the
diagonals to create a rectangle.
The area of the rectangle A = (d1d2 ).
Since the rectangle is twice the size
of the kite, the area of the kite
A = ½ ( d1d2 )
11/13/2018
Section 7.2 Nack

7. Areas of Similar Polygons
Theorem 7.2.7: The ratio of the areas of two similar triangles equals the square of the ratio of the lengths of any two corresponding sides:
Proof p. 353
Note: This theorem can be extended to any pair of similar polygons (squares, quadrilaterals, etc.)
11/13/2018
Section 7.2 Nack