Applied Math: Two-Dimensional Shapes

Triangles
A triangle is a three-sided figure with the sum of its interior angles being  

The perimeter of any triangle is found by summing the three side lengths; .

For an equilateral triangle, this is the same as , where a is any side length, since all three sides are the same length.

The area of any triangle can be found by taking half the product of one side length, referred to as the base and often given the variable b, and the perpendicular distance from that side to the opposite vertex called the altitude or height and given the variable h.

In equation form that is . Another formula that works for any triangle is , where s is the semiperimeter:  , and a, b, and c are the lengths of the three sides.

Special cases include isosceles triangles: , where b is the unique side and a is the length of one of the two congruent sides, and equilateral triangles: , where a is the length of a side.
 

Rectangle: A quadrilateral with four right angles.

All rectangles are parallelograms and trapezoids, but not all parallelograms or trapezoids are rectangles.

The diagonals of a rectangle are congruent. Rectangles have 2 lines of symmetry (through each pair of opposing midpoints) and 180-degree rotational symmetry about the midpoint.

The area of a rectangle is found by the formula , where A is the area of the rectangle, l is the length (usually considered to be the longer side) and w is the width (usually considered to be the shorter side). The numbers for l and w are interchangeable.
The perimeter of a rectangle is found by the formula  or , where l is the length, and w is the width. It may be easier to add the length and width first and then double the result, as in the second formula.
 

Square: A quadrilateral with four right angles and four congruent sides. Squares satisfy the criteria of all other types of quadrilaterals. The diagonals of a square are congruent and perpendicular to each other.

Squares have 4 lines of symmetry (through each pair of opposing midpoints and along each of the diagonals) as well as 90-degree rotational symmetry about the midpoint.

The area of a square is found by using the formula , where s is the length of one side. The perimeter of a square is found by using the formula , where s is the length of one side.

Because all four sides are equal in a square, it is faster to multiply the length of one side by 4 than to add the same number four times. You could use the formulas for rectangles and get the same answer.

Circles
The center of a circle is the single point from which every point on the circle is equidistant.
The radius is a line segment that joins the center of the circle and any one point on the circle. All radii of a circle are equal. Circles that have the same center, but not the same length of radii are concentric.

The diameter is a line segment that passes through the center of the circle and has both endpoints on the circle.

The length of the diameter is exactly twice the length of the radius. Point O in the diagram below is the center of the circle, segments , , and  are radii, and segment is a diameter.
of a Circle


The area of a circle is found by the formula , where r is the length of the radius.

If the diameter of the circle is given, remember to divide it in half to get the length of the radius before proceeding.

The circumference of a circle is found by the formula
, where r is the radius.

Again, remember to convert the diameter if you are given that measure rather than the radius.

Practice
P1. Find the area and perimeter of a square with side length 2.5 cm.
P2. Calculate the area of a triangle with side lengths of 7 ft, 8 ft, and 9 ft.

Practice Solutions:
P1. (a) ;
P2. Given only side lengths, we can use the semi perimeter to the find the area based on the formula, , where s is the semiperimeter, :