Area of Triangles, Special Quadrilaterals
The square is the simplest figure to calculate the area of (A = x2, where x is the length of one side), with the rectanglealmost as less important (A = lw, where l is length and w is width). The formula for the area of a parallelogram is nearly as simple (A = bh, where b is the length of the base and h is the height), particularly when you note that you can cut a triangle from one end and paste it onto the other end to form a rectangle.
Finding the actual height is normally the most difficult part. Once we have the area formula for a parallelogram, we note that given any triangle we can join two together to form a parallelogram. Make use of AIA to show the resulting sides are parallel.
This gives us the normal area formula for triangles: A = ½bh. This is frequently read as: "one-half base times height." Even though this is clearest for a right triangle, it is in reality a general result. Nevertheless, we ought to be sure h is an altitudetherefore perpendicular to the side we are making use as a base. You have a choice of three as to which side you make use of as base. This can be utilized to solve for an unknown altitude.
The next quadrilateral to be considered is the trapezoid, which can be split into two triangles and the areas summed. Therefore, the formula: A = ½(b1 + b2)h. You can as well cut triangles off two corners, put them on nearby corners to produce a rectangle. Therefore, part of the formula is vividly seen as the average of the two bases. In the end we will consider kite ABCD. In reality, the following formula applies to any quadrilateral with perpendicular diagonals (kites, rhombuses, squares) as can be shown by breaking it down into four right triangles: A=½AC×BD or "one-half the product of the lengths of the diagonals."
| Parallelogram with moved triangle | Triangles with altitude | Trapezoid as two triangles |
| Square: | A = l2, length squared |
| Rectangle: | A = l × w, length times width |
| Parallelogram: | A = b × h, base times height (since a parallelogram can be maneuvered into a rectangle.) |
| Triangle: | A = ½bh, where b is the base, and h is the height. (Two congruent triangles can be maneuvered to form a parallelogram.) |
| Trapezoid: | A = ½(b1 + b2)h. (A trapezoid can be divided into two triangles.) |
| Kite: | A = ½AC×BD, where AC and BD are the lengths of the diagonals (A kite can be divided into four right triangles.) |
| Regular polygon: | A = ½asn = ½ap, where a is the apothem, s is the length of each side, n is the number of sides, and p is the perimeter. |
Pythagorean Theorem
We already covered the Pythagorean Theorem in our previous lesson. Note there particularly the triangle for a 3-4-5 right triangle. Please review the information there. Note as well that the law of cosines is a generalization of the Pythagoras Theorem and can be applied to any triangle: c2 = a2 + b2 - 2abcos C.
(Do diagrams for 32=22+5 and 52=12+13, etc.)
A few of the favorite questions involve finding the precise area of regular hexagons or octagons! These make wide use of the special triangles, their side length ratios, multiplying binomials, and manipulating radicals. For instance, consider polygons with sides of length 2. The hexagon is then made up of six equilateral triangles with sides of length 2. The height will be . The area of each triangle will be ½bh=½(2)( )= . Thus the total area will be 6 . Questions that involves regular hexagons are everywhere on standardized tests and contests.
It is normally easiest to bound the octagon in a square and subtract off the four corner triangles. We can easily calculate the sides of these isosceles right triangles to be . Therefore, the total area is given by (2+2 )2-4(½). There is an alternate method that involves trigonometry. Divide the octagon into eight triangles by making use of the vertices and center. Every triangle is isosceles with base angles of 135º/2=67.5º. The tangent function gives the ratio of the opposite side to the adjacent side in a right triangle. Therefore, tan 67.5º gives the height of this triangle since when we bisect the triangle we obtain a right triangle with base of one. tan 67.5º= +1. This as well offers a method for an approximate answer for any regular n-gon.
Example:
A concrete driveway is to be 90' long, 9' wide, and 4" thick. Calculate the number of cubic yards of concrete needed, two ways, first by converting these numbers into yards and second by calculating the volume in cubic feet and then converting. Compare your results and then calculate the cost based on a 5.5 bag limestone mix, with 5# of fiberglass reinforcing fibers per yard, at N75/yard3.
Answer:
The driveway is 30 yards long, 3 yards wide, and 1/9 thick for a total of 30•3÷9=10 yards3 or a cost of N750.00 plus 6% sales tax or N45. Or, the driveway is 270 feet3, which at (3 ft/yd)3=27 feet3 per yard3 gives 10 yards3 as above. Mistakes like dividing by 9 instead of 27 are all very common.
Example:
My 10 acre corn field is a 1/4 mile long. How wide is it?
Answer:
Solve the proportion: 10/640 = 1320x/52802 for x = 330'. This would be a long 10 acres rather than a square of 10 acres. In this case, it was the W ½ of the W ½ of the NE ¼ of the NW ¼
Example:
A 24' long by 16' wide swimming pool angles down uniformly from shallow (3') to deep (10'). Find the area of the bottom and sides of the pool.
Answer:
The bottom forms the hypotenuse of a 7-24-25 right triangle, so the bottom is 16'×25'= 400 ft2. The ends are rectangles 3'×16' = 48 ft2 and 10'×16'=160 ft2. The sides are trapezoids with bases 3' and 10' and height 24' or 2•½•(3'+10')•24'=312 ft2. The total surface area is therefore: 920 ft2.
Example:
A triangle has sides with lengths 8, 15, and 17. What is the altitude to the longest side?
Answer:
We first note that 82+152=64+225=289=172 and therefore we have a right triangle. The area is then ½•8•15=60 units2. Any altitude can be used as a height though so A = 60=½•x•17. From which we discover: x = 60/17 or about 7.06.
Example:
An isosceles trapezoid has bases of 10" and 16" and a 36" perimeter. What is its area?
Answer:
We can calculate the length of the missing sides by subtracting the bases from the perimeter and dividing by two: (36"-10"-16")/2=5". Then we observe how (16"-10")/2=3" is the side of a triangle whose other side is the height and hypotenuse is this 5" side. We immediately recognize the 3-4-5 right triangle and can obtain the area as: ½(10"+16")•4"=52 in2.
A triangle which has all three of its sides equal in length.
Try this Drag the orange dots on each vertex to reshape the triangle. Notice it always remains an equilateral triangle. The sides AB, BC and AC always remain equal in length
An equilateral triangle is one in which all three sides are congruent (same length). Because it also has the property that all three interior angles are equal, it really the same thing as an equiangular triangle. See Equiangular triangles.
An equilateral triangle is simply a specific case of a regular polygon, in this case with 3 sides. All the facts and properties described for regular polygons apply to an equilateral triangle. See Regular Polygons
Properties
• All three angles of an equilateral triangle are always 60°. In the figure above, the angles ∠ABC, ∠CAB and ∠ACB are always the same. Since the angles are the same and the internal angles of any triangle always add to 180°, each is 60°.
• The area of an equilateral triangle can be calculated in the usual way, but in this special case of an equilateral triangle, it is also given by the formula:
• where S is the length of any one side. See Area of an equilateral triangle.
• With an equilateral triangle, the radius of the incircle is exactly half the radius of the circumcircle.
• Isosceles Triangle
• From Greek: isos - "equal" , skelos - "leg"
• A triangle which has two of its sides equal in length.
The word isosceles is pronounced "eye-sos-ell-ease" with the emphasis on the 'sos'. It is any triangle that has two sides the same length.
If all three sides are the same length it is called an equilateral triangle. Obviously all equilateral triangles also have all the properties of an isosceles triangle.
Properties
• The unequal side of an isosceles triangle is usually referred to as the 'base' of the triangle.
• The base angles of an isosceles triangle are always equal. In the figure above, the angles ∠ABC and ∠ACB are always the same.
• When the 3rd angle is a right angle, it is called a "right isosceles triangle".
• The altitude is a perpendicular distance from the base to the topmost vertex.
Area of a triangle
The number of square units it takes to exactly fill the interior of a triangle.
Most common method
Usually called "half of base times height", the area of a triangle is given by the formula below.
Area=2a⁄b
where
b is the length of the base a is the length of the corresponding altitude
You can choose any side to be the base. It need not be the one drawn at the bottom of the triangle. The altitude must be the one corresponding to the base you choose. The altitude is the line perpendicular to the selected base from the opposite vertex.
In the figure above, one side has been chosen as the base and its corresponding altitude is shown
| If you know: | Use this |
| Base and altitude | "Half base times height" method |
| All 3 sides | Heron's Formula |
| Two sides and included angle | Side-angle-side method |
| x,y coordinates of the vertices | Area of a triangle- by formula (Coordinate Geometry) Area of a triangle - box method (Coordinate Geometry) |
| The triangle is equilateral | Area of an equilateral triangle |
Area of an equilateral triangle
The area of an equilateral triangle (all sides congruent) can be found using the formula
where s is the length of one side of the triangle.
Area of an equilateral triangle
The area of an equilateral triangle (all sides congruent) can be found using the formula
where s is the length of one side of the triangle.
Try this Drag the orange dots on each vertex to reshape the triangle. The formula shown will recalculate the area using this method.
When you know all three sides of a triangle, the usual way to find the area is to use Heron's Formula. But in the case of equilateral triangles, where all three sides are the same length, there is a simpler formula:
where
s is the length of any side of the triangle.
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