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Measuring Two-Dimensional Figures

Objective

In this lesson you will reflect on connections between formulas for the area and perimeter of two-dimensional figures, such as circles and polygons.

Previously Covered:

A polygon is a closed figure with sides made up of line segments.
A regular polygon is a convex polygon in which all sides are equal in length and angles are equal in measure.
A quadrilateral is a four-sided polygon.
A parallelogram is a quadrilateral which has exactly two pairs of opposite parallel sides.
A trapezoid is a quadrilateral with exactly one pair of parallel sides. The perimeter of a figure is the distance around the edge of a figure.

How do you keep all the different area formulas straight?

The formulas for areas of polygons can be examined by their similarities.

The area of a rectangle can be found by determining the number of unit squares that will cover it. By unit squares, we mean squares that each have an area of 1 square unit.

The number of unit squares that cover a rectangle can be determined by multiplying the length of the base b by the height h of the rectangle. The area of a rectangle is A = bh.

$Area of a rectangle$

With a height of four units and base of five units, this rectangle has an area of 20 square units.

Because the area of a parallelogram can be easily rearranged to form a rectangle, the area formula is the same, A = bh.

$Area of a parallelogram$

The height of a parallelogram is the verticalheight, measured at a 90° angle to the base.

Because any pair of congruent triangles can be arranged into a parallelogram, the area of a triangle is half the area of a parallelogram, , where h is the altitude of the triangle.

$Area of a triangle$

Because any pair of congruent trapezoids can be arranged into a parallelogram, the area of a trapezoid is also based on the area of a parallelogram, , where b₁ and b₂ are the lengths of the two parallel sides, and h is the vertical distance between those two lengths.

$Area of a Trapezoid$

Any regular polygon can be subdivided into congruent triangles. The height of those triangles is called the apothem. The area of a regular polygon can be found using the formula , where p is the perimeter of the polygon, and a is the length of the apothem.

$Area of a polygon$

What about circles?

The distance around the edge of a circle is called the circumference. That distance is found using the formula , where r is the radius of the circle.

A circle can be decomposed into wedges. If the wedges are infinitely thin, they can be rearranged into a parallelogram with height of r and a base of half of the circumference, or ; thus, the area is .

$Circle broken into wedge$

Review of New Vocabulary and Concepts

The area of a rectangle is A = bh.
The area of a parallelogram is A = bh.
The area of a triangle is .
The area of a trapezoid is .
The area of a circle is .
The perimeter of a figure is the distance around that figure.
The perimeter of a circle is called the circumference and can be found by using the formula .

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