Perimeter, Circumference, & Area

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Perimeter

The perimeter of a particular shape is the total length of its sides.

  • For a triangle:

P=l_a+l_b+l_c The perimeter is equal to the length of side a, l_a, plus the length of side b, l_b, plus the length of side c, l_c.

  • For a square:

P=4 l The perimeter is equal to 4 times the length (l) of a side.

  • For a rectangle:

P=2 (b+h) The perimeter is equal to 2 times the sum of the base times the height.

  • For regular polygons

P=n l The perimeter is equal to the number of sides (n) times the length (l) of a side.

Circles do not have sides made of line segments like polygons do but they do have a perimeter known as a circumference. C=2*\pi*r The circumference is equal to 2 times pi times the radius (r).

Areas

Area of Circles

The method for finding the area of a circle is

Area = \pi r^2

Where π is a constant roughly equal to 3.14 and r is the radius of the circle; a line drawn from any point on the circle to its center.

Area of Triangles

If one of the sides of the triangle is chosen as a base, then a height for the triangle and that particular base can be defined. The height is a line segment perpendicular to the base or the line formed by extending the base and the endpoints of the height are the corner point not on the base and a point on the base or line extending the base. Let B = the length of the side chosen as the base. Let
h = the distance between the endpoints of the height segment which is perpendicular to the base. Then the area of the triangle is given by:

Area = \frac{B \times h}{2}

This method of calculating the area is good if the value of a base and its corresponding height in the triangle is easily determined. This is particularly true if the triangle is a right triangle, and the lengths of the two sides sharing the 90o angle can be determined.

Area of Rectangle

The area calculation of a rectangle is simple and easy to understand. One of the sides is chosen as the base, with a length b. An adjacent side is then the height, with a length h, because in a rectangle the adjacent sides are perpendicular to the side chosen as the base. The rectangle's area is given by:

A = b \cdot h

Sometimes, the baselength may be referred to as the length of the rectangle, l, and the height as the width of the rectangle, w. Then the area formula becomes:

A = l \cdot w

Regardless of the labels used for the sides, it is apparent that the two formulas are equivalent.

Of course, the area of a square with sides having length s would be:

A = s^2

Area of Parallelograms

The area of a parallelogram can be determined using the equation for the area of a rectangle. The formula is:

A = b \cdot h

A is the area of a parallelogram. b is the base. h is the height.

The height is a perpendicular line segment that connects one of the vertices to its opposite side (the base).

Area of Rhombus

Remember in a rombus all sides are equal in length.

A= \frac{d_1 \cdot d_2}{2} d_1 and d_2 represent the diagonals.

Area of Trapezoids

The area of a trapezoid is derived from taking the arithmetic mean of its two parallel sides to form a rectangle of equal area.

A = \frac{\left(b_1 + b_2\right) \cdot h}{2}

Where b_1 and b_2 are the lengths of the two parallel bases.

Areas of other Quadrilaterals

The areas of other quadrilaterals are slightly more complex to calculate, but can still be found if the quadrilateral is well-defined. For example, a quadrilateral can be divided into two triangles, or some combination of triangles and rectangles. The areas of the constituent polygons can be found and added up with arithmetic.

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