Since the total area does not change while you move the triangle, the entire area is just the base times the height. You can transpose this triangle from the left side to the right side to make a normal rectangle. Why does this work for parallelograms though? Consider the following illustration.ĭrawing a line perpendicular from the base to one of the terminal points of the side gives you a right triangle with one of the sides equal to the height. The formula for the area of a parallelogram is the same as the formula for the area of a square or rectangle. a rectangle), this formula simplifies down into the Pythagorean theorem a 2 + b 2 = c 2 Other Properties Of Parallelograms Area In the case of a parallelogram where the angles are right angles (i.e. So we have proven that the sum of the squares of the sides of a parallelogram is equal to the sum of the square of the diagonals. (CD) 2 + (AD) 2 + 2(CD)(AD)cos(∠ABC) = BD 2Īdding and simplifying.
The parallelogram law can be proven as such: Since parallelograms have congruent opposite sides (AB = DC and BC = DA), the parallelogram law can be rewritten as: Mathematically, for the following parallelogram: The parallelogram law states that the sum of the squares of the sides is equal to the sum of the squares of the diagonals. Opposite sides of parallelogram are congruentĪ segment bisector intersects line segment to make two congruent segments. These diagonals also bisect each other, meaning that they intersect at each other’s midpoints. Diagonals Bisect Each OtherĪ diagonal in a polygon is a straight line drawn between pairs of non-adjacent angles. Therefore, adjacent angles of a parallelogram are supplementary. , interior angles on the same side of the trans. ∠A supp ∠B ∠B supp ∠C ∠C supp ∠D ∠D supp ∠A Normally when people hear the word “parallelogram” they think of a rhomboid-a parallelogram with non-congruent pairs of opposite sides. A rhombus is a special kind of parallelogram in which all four sides are equal length. All squares and rectangles are parallelograms, they are just special parallelograms where all interior angles are right angles. This definition includes various kinds of shapes. Definition Of A ParallelogramĪt its simplest, a parallelogram is any quadrilateral with 2 pairs of parallel opposite sides. The three-dimensional analog of a parallelogram is called a parallelepiped. In physics, parallelograms are used to model individual force components and to describe vector addition. Parallelogram-shaped devices are used in engineering to lift heavy loads and build large structures. The unique properties of parallelograms make them very useful in geometry, engineering, and science. sum of squares of sides equals sum of squares of diagonals (parallelogram law).Some key properties of parallelograms include: In other words, a parallelogram is a 4-sided figure in which opposite pairs of sides lie parallel to each other. In geometry, a parallelogram is a convex quadrilateral polygon that is characterized by having 2 sets of parallel sides.