Area Axiom: Every polygonal region has an area and is measured in square units.
Congruent Area Axiom: If and are two congruent triangles, then . ie. two congruent regions have equal area.
Rectangle Area Axiom: If is a rectangular region such that units and units, then square units.
Parallelograms on the same base and between the same parallels
Theorem 1: A diagonal of a parallelogram divides it into two triangles of same area
In this case
Theorem 2: Parallelograms on the same base and between the same parallels are equal in area.
Theorem 3: Area of a parallelogram is the product of its base and the corresponding altitude.
Let the two adjacent sides of the parallelogram be and . Area
Triangle Area Axiom
Theorem 4: The area of a triangle is half the product of any of its sides and the corresponding altitude.
denotes the sides of the Triangle. Then:
Area . This is known as Heron’s Theorem.
Theorem 5: If a triangle and parallelogram are on the same base and between the same parallels, the area of the triangle is equal to the half of the parallelogram.
Trapezium Area Axiom
Theorem 6: The area of a trapezium is half the product of its heights and the sum of parallel sides.
A trapezium is a quadrilateral two of whose sides are parallel. A trapezium whose non-parallel sides are equal is known as an isosceles trapezium.
Let and be the parallel sides and be the distance between the parallel sides.
Triangles on the same base and between the same parallels
Theorem 7: Triangles on the same base and the same parallels have the same area.
Theorem 8: Triangles having equal areas and having one side of one of the triangles equal to one side of the other triangle, have their corresponding altitudes the same.
Theorem 9: Two triangles having the same bases (or equal bases) and equal area lie between the same parallels.