# Regular hexagon area

The area of each of the triangles can be found using Heron's formula. What is the ratio of their areas? Our teacher has asked us to do thisI think I can find one solution as a hexagon will be made up of 6 equilateral triangles and if the hexagon and triangle have equal perimeters then the triangles in the hexagon will have 1/2 the area of the larger triangle. Find the area of one triangle. from publication: On the . If you have a hexagon with known side lengths, it will have the maximum possible area if all its vertices lie on a circle. Apothem - the distance of the line segment from the center of a regular polygon perpendicular to a side (i. smaller hexagon, making the ratio of the areas 18/24 = 3/4, with 1/4 of the original area cut off. A regular hexagon, as shown in the following image, is a hexagon with equal sides and angles. Determine the perimeter of a hexagon by adding the length of its 6 sides. The output is the area of the irregular polygon. A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. 2. Then click Calculate. Examples: A regular polygon is equilateral (it has equal sides) and equiangular (it has equal angles). A regular hexagon can be divided into 12 30-60-90 right triangles with hypotenuse equal to the length of half of a diagonal, or 6 in this case (see image). Regular Heptagon. A regular hexagon consists of 6 equilateral triangles, each of side 8 cm. What is the approximate area of the hexagon? Area of Regular Polygons quizlet 12 Terms. Imagine a square with a side length of 3. Find the area of a regular hexagon whose perimeter is 48cm, find also the perimeter and area of circumscribed circle of hexagon. Round your answer to the nearest tenth. n = number of sides s = length of a side r = apothem (radius of inscribed circle) R = radius of circumcircle. The circle inscribed in a regular hexagon has 6 points touching the six sides of the regular hexagon. To see how this equation is derived, see Derivation of regular polygon area formula. Sum of total angle of a hexagon = (6–2) x 180 = 720. For the regular hexagon these are given Since a regular hexagon is of finding the area of an equilateral triangle. Which is an equivalent expression for the area of the hexagon based on the area of a triangle? 6(4a2 – 3) 6(8a2 – 9) 6a(12a – 9) 6a(18a – 12) Area and Perimeter of a Hexagon. Every one of their sides is 2 square roots of 3. A regular hexagon is a polygon with six edges of equal length. Area of a cyclic quadrilateral. Seen with two types (colors) of edges, this form only has D 3 symmetry. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries. If the hexagon has a perimeter of 60 in. In a regular hexagon, each inside angle is 120°. A regular hexagon can be cut into six equilateral triangles, and an equilateral triangle can be divided into two 30°- 60°- 90° triangles. This side over here is 2 square roots of 3. We can find the area of a regular hexagon by splitting it into 6 equilateral triangles. Area of a parallelogram given base and height. To find the apothem, divide the length of one side by 2 times the tangent of 180 degrees divided by the number of sides. A regular hexagon is inscribed in a circle with a radius of 18. Find the area of a regular hexagon with an apothem 16. Regular Octagon. in order to find the area of a regular hexagon using the formula Area=1/2(nxs)(a)you must frist find the apothem whie vertical leg of a 30-60-90 triangle with the base of 3. square meter ). Calculates side length, inradius (apothem), circumradius, area and perimeter. You'll find six-sided hexagons in honeycombs, hardware and even in natural basalt columns along the coast of Ireland. ) The area is 6 x 8 so it would be 48 By Apothem Length The area of a regular octagon can also be computed using its measured apothem (a line from the center to the middle of any side). Use what you know about special right triangles to find the area of each regular polygon. In this case the hexagon has six of them. Area of a quadrilateral. Examples include triangles, quadrilaterals, pentagons, hexagons and so on. A hexagon is a 6-sided polygon (a flat shape with straight sides). Soap bubbles tend to form hexagons when they join up. Find the measure of each numbered angle. NOTE: The apothem height of a hexagon is half of the total height of a hexagon. We will now have a look at how to find the area of a hexagon using different tricks. asked by |-/ on May 18, 2017; geometry. If you're wondering how to find the area of a polygon formula, keep reading and you'll find the answer! Problem: Find the area of a regular hexagon whose sides measures 5 cm Sol'n 1: I can cut the hexagon into 6 small triangles, so the area of triangle times 6 will be equal to the area of the polygon. the regular polygon is a hexagon with radius of in. The area of a regular hexagon is six times the area of central equilateral triangle. To calculate the area of a regular polygon, we use a formula that is based on the various parts of the polygon What is the area of a regular hexagon with an apothem 13 inches long and a . 3. Angle of each side = 720 / 6 = 120. To find the area of a regular polygon, you use an apothem — a segment that joins the polygon’s center to the midpoint of any side and that is perpendicular to that side (segment HM in the following figure is an apothem). Various procedures can be used in order to calculate its area. Area of a parallelogram given sides and angle. Regular Nonagon An equilateral triangle and a regular hexagon have equal length perimeters. Q9. The various methods Calculations of geometric shapes and solids: the Regular Hexagon. The Area of Hexagon calculator (Ahex= 3/2 • √3 • s2 ) computes the area of a regular Hexagon. Area. Each side of the regular hexagon measures 8 cm. 1)Area of a Regular Hexagon: In fact, the area of a regular hexagon, in which the number of sides n = 6, is easy to calculate since a hexagon can be decomposed into 6 equilateral triangles. Divide you hexagon into two halves from the opposite edges. To find the area of regular polygons, use the formula: area = (ap)/2, where a is the apothem and p is the perimeter. The adjacent edges form an angle of 120°. Denote by , , and the midpoints of sides , , and , respectively. Area of a regular polygon - derivation. Area of a rhombus. Leave your answer in simplest form. g. So, Area of the hexagon = 6 * (Area of one equilateral triangle) We know that Area of a triangle = 1/2 * Base length * Height Now, base length of triangle = Length of side of the hexagon = 4 cm. Now area of the circle inscribed is 3πa*a/4 In a regular hexagon, the radius equals the side length. therefore. Area of a regular hexagon is different from the area of an irregular hexagon. How do you find the area of a hexagon? There are several ways to find the area of a hexagon. Students will learn the method of finding the perimeter of regular hexagons. To find the area of inscribed circle we need to find the radius first. The area is . 6 × 2 1 R 2 ⋅ sin 6 0 ∘ = 2 3 3 R 2. This virtue states that the diagonals of a regular hexagon divide it in 6 e quilateral triangles of same side as shown above. r is the radius of the polygon. Area of a regular polygon. B) 3. For the regular hexagon the radius is found using the formula, a(√3)/2. The total of the internal angles of any . The area of a square is the length of one side multiplied by itself, or squared. Hence, area of a hexagon is the space covered within its edges or sides. So. A regular hexagon is a six sided polygon where all of the sides are the same length (s) and all interior angles are the same. Here are examples and problems that relate specifically to the regular hexagon. Measure the side length and multiply it by 2. Area of hexagon = square root 3*3*a2/2= 24*square root 3. To find the measure of the interior angles, we know that the sum of all the angles is 720 degrees (from above) And there are six angles So, the measure of the interior angle of a regular hexagon is 120 degrees. We shall denote the regular hexagon by K and the random triangle by T and shall consider the random variable X = area(T)/area(K). The area of a regular hexagon can be determined by finding the area of one of the six triangles that make up the hexagon and multiply by six. Such a hexagon is known as 'cyclic hexagon'. A hexagon is a 6-sided, 2-dimensional geometric figure. Use this calculator to calculate properties of a regular polygon. meter), the area has this unit squared (e. Calculate from an regular 3-gon up to a regular 1000-gon. Regular hexagon fits together well like an equilateral triangle and they are commonly seen as tiles in the house. No. 30 Mar 2018 Let ABCDEF be a regular hexagon with each side of length 1 cm. Six equilateral triangles are connected to create a regular hexagon. Regular Hexagon : this page updated 19-jul-17 Mathwords: Terms and Formulas from Algebra I to Calculus written, illustrated, and A regular polygon is a polygon in which all of its sides have equal lengths. Combined, the angles amount to 720 degrees, regardless of the type of hexagon. The angle of each of these triangles' vertex that is in the circle center is equal to #360^@/6=60^@# and so must be the two other angles formed with the triangle's base to each one Maximum hexagon area. Properties of all hexagons Finding the area of an irregular hexagon is not the same as finding the area of a regular hexagon. A hexagon is a shape composed of six equilateral triangles. The area of a particular regular hexagon is x^3 square units, where x is the measure of the distance from the center of the hexagon to the midpoint of a side. 25, what is the area of the hexagon? Q and A; A square To find the area of a regular hexagon, break it into 6 equilateral triangles:. Equation form: Q8. The kids chopped the hexagon into 24 equal triangle of which 18 compose the smaller hexagon, making the ratio of the areas 18/24 = 3/4, with 1/4 of Here we will discuss about the perimeter and area of a Regular hexagon and some example problems. The apothem is equal to the side of the triangle Best Answer: Look at the hexagon in the link provided. ml1 ≠60; ml2 ≠30 Area of a triangle (Heron's formula) Area of a triangle given base and angles. The area (in sq cm) of a square with AC as one side is. The formula for the area of a regular hexagon looks like this: Where s is the length of one of the With reference to the regular hexagon, from the image above we can see that it is formed by six triangles whose sides are two circle's radii and the hexagon's side. A regular hexagon has Schläfli symbol {6}. As shown below, a regular polygon can be broken down into a set of congruent isosceles triangles. Because a hexagon has an even number of sides, in a regular hexagon, opposite methods, see Area of a Regular Polygon and Area of an Irregular Polygon Calculate the apothem, perimeter and area of a regular hexagon inscribed in a circle with a radius of 4 cm. A hexagonal pyramid is a geometric figure that consists of a six sided (hexagonal) base and six triangular faces. Calculations at a regular hexagon, a polygon with 6 vertices. 74 ft^2. A regular hexagon is a polygon with six equal sides and angles. Find out the area of the regular hexagon. ) Finding the Area of a Regular Hexagon Properties of Regular Polygons Polygon. Calculate the approximate area a regular hexagon with the following side length: In a regular hexagon, all sides equal the same length and all interior angles Using what we know about triangles to find the area of a regular hexagon. Students will discover two different methods for finding the area of regular hexagons If the area of the square is 2. The area of the hexagon is 24a2 – 18 square units. Because the triangles are equilateral, you only need find the area of one triangle and multiply the result by six. If you want to know how to calculate the area of a hexagon, just follow these steps. A regular hexagon is the part the regular hexagonal tiling, {6,3}, with 3 hexagonal around each vertex. It has one more important virtue that is useful in order to determine its area. In an equilateral triangle with the side length x the altitude is To find the area of the circle, you must determine the altitude of one of the equilateral triangles that the hexagon can be divided into. As this hexagon is regular all sides are equal, it is also apparent that only one type of circle can be inscribed in a hexagon of certain area, hence a ratio can be derived. A regular hexagon is a part of the regular hexagonal tiling, {6,3}, with 3 hexagonal around each vertex. These cases where the number of sides of the regular polygon is 3 or 4 are easy to calculate. To find the measure of the central angle of a regular hexagon, make a circle in the middle A circle is 360 degrees around Divide that by six angles. Perimeter (P) = 6 × side = 6a Area (A) = 6 × (area of the equilateral ∆OPQ) Hexagon. Alternatively, the area can be found by calculating one-half of the side length times the And since this is a regular hexagon, they're actually giving us the length of all the sides. Side of Regular Hexagon - a shape that is defined by having 6 same length sides and 6 identical 120 degree internal angles. The total of the internal angles of any hexagon is 720°. Honeycomb has hexagons too! Regular or Irregular. The hexagon is made up of 6 isosceles triangles with the same area. The regular hexagon is the regular polygon with six sides, as illustrated and area A of a regular hexagon can be computed directly from the formulas for a Here we will discuss about the perimeter and area of a Regular hexagon and some example problems. For more geometry videos go to www. A = √3/4 * a² for a single triangle and hence (3/2) √3a² Area of a Regular Hexagon What is the dimension of x in this hexagon if the area of the hexagon is 2. Substitute the value of n and r in the formula above. The radius of a circumscribed circle is the same as the length of the edges. Like all regular shapes, there's a simple formula we can use to calculate the area of a hexagon. Yes, just a regular hexagon. where a represented the side. Posted from my mobile device Area of a Hexagon Formula A polygon with six sides and six interior angles is known as hexagon. If you want to know the length of a regular hexagon's sides, there are two formulas you can use to find out. Introduction. how2learnmath. If the edge of the triangle is “a” then the area. Interior angle. In regular hexagon,the length of a diagonal is equal to two times the length of the side, so diagonal is 8 and each side (a) is 4. In a regular hexagon, split the figure into triangles. It is well known 19 Jan 2017 Regular-hexagon-equilateral-triangle area grouping–based broadcast protocol for safety message in urban vehicular ad hoc networks. A hexagon is a specific type of polygon which has only 6 sides. Thus area of the regular hexagon is 6 times the area of a equilateral triangle. Find the area of a regular hexagon with sides of 4 cm. A polygon is a plane shape (two-dimensional) with straight sides. To prove . All internal angles are 120 degrees Given a regular Hexagon with side length a, the task is to find the area of the circle inscribed in it, given that, the circle is tangent to each of the six sides A regular hexagon has a radius of 20 in. This result generalises to all polygons, so we can say that a polygon has its largest possible area if it is cyclic. Regular Pentagon. Then, we will identify and use the apothem to find the area of our regular octagon and compare both of the methods. 1. Regular Hexagon. Area of regular hexagon with side a. Thus each angle from the center of the circle will be 360/6 = 60 degrees. In geometry, a hexagon is a polygon which has six sides and six edges. what is the side length of a hexagon? Area moments of inertia of a filled regular hexagon with side length a in respect to an axis going through the center of the hexagon and parallel to the shape, is depended on the side of the hexagon. To calculate the area of a regular hexagon, treat it like six equilateral triangles. This area of a polygon calculator can help - as you can guess - in determining the area of a regular polygon. Accordingly, you can calculate a hexagon’s area by finding the area of the triangles and adding those areas together. Regular hexagons have six equal sides and angles and are composed of six equilateral triangles. Similar to the hexagon and pentagon, first we will divide the regular octagon into parts to find the area. Area of a regular hexagon can be easily calculated using the formula we have above, but for an irregular hexagon we use various methods according to the given information to find its area. Area of a Regular Hexagon Calculator. A regular hexagon can also be created as the truncated equilateral triangle, with Schläfli symbol t{3}. * Units: Note that units of length are shown for Area of a Hexagon Formula. They want us to find the area of Hexagon area formula: how to find the area of a hexagon. This creates a second hexagon. A regular Hexagon can be split into $6$ equilateral triangles. TANCET 2008 - Old test paper solution : Area of rectangle in a regular hexagon, mensuration, geometry : Ascent Education TANCET MBA Classes 8 May 2018 The Area of Hexagon calculator (Ahex= 3/2 • √3 • s2 ) computes the area of a regular Hexagon. (1) The length of the side of the outermost regular hexagon is 6 cm. Regular Polygon Formulas. Therefore the option (a) is The point of the diagram is to see that the diagonals divides the hexagon into four triangles which I have labeled A, B, C, and D. com Polygon Calculator. Formula. A = 1ap 2 a p 232 cm2 384 ft"3 2 547 2. Using the calculator provided you can calculate it's surface are and volume quickly and easily. , Answer to Find the area of a regular hexagon with side length 4 m. A regular hexagon is a six sided polygon Problem Three squares are drawn so that each will contain a side of regular hexagon as shown in the given figure. There are a variety of ways to calculate the area of a hexagon, whether you're working with an irregular hexagon or a regular hexagon. C) 4 hexagons. Area of a trapezoid. To derive a formula for the area of a regular polygon if the number of sides is n requires applying some more trigonometry. A circle is 360 degrees. Join the midpoints of successive sides. Find the relation between the areas of the two hexagons. For those who want to know how to do this by hand, we will explain how to find the area of a regular Define apothem for students to begin the second method for finding the area of a regular hexagon. geometry. In fact, the area of a regular hexagon, in which the number of sides Download scientific diagram | 1. It is also the side length of the equilateral triangle. where. What is the perimeter, rounded to the nearest tenth? A regular hexagon has Schläfli symbol {6}. The desired area (hexagon ) consists of an equilateral triangle and three right triangles (, , and ). Regular Polygon Area Calculator is an online tool for geometry calculation programmed to find out the polygon area for given values of apothem (inner circle radius), number of polygon sides and length of each side. When all angles are equal and all sides are equal it is regular, otherwise it is irregular: Hexagon Calculator. See Area of an Irregular Polygon So, we get another formula that could be used to calculate the area of regular Hexagon: Area= (3/ 2)*h *l. Calculate the area of regular hexagon by using formula : A = (3√3 * a 2)/2, where a = side of hexagon. The vertices coordinates must be input in order: either clockwise or anticlockwise. They would be the same. The result is rather elegantly proved here. Where “l” is the length of each side of the hexagon and “h” is the height of the hexagon when it is made to lie on one of the bases of it. The area A of a convex regular n-sided polygon having side s, circumradius R, apothem a, and perimeter p is given by = = = = = For regular polygons with side s = 1, circumradius R = 1, or apothem a = 1, this produces the following table: (Note that since → / as →, the area when = is tending to / as grows large. In geometry, a hexagon is a six-sided polygon or 6-gon. There are various ways to calculate the area of Hexagon. The name comes from Greek words ἕξ (hex) meaning six, and γωνία (gonia) meaning a corner, an angle. For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. Let us look at the common methods used in the process. If the hexagon is irregular, it will not have a diameter. radius have the same unit (e. In this problem. Round to the nearest tenth. Key Concepts Theorem 10-6 Area of a Regular Polygon The area of a regular polygon is half the product of the apothem and the perimeter. That is, a line from the center to any vertex will have the same length as any side. This page describes how to derive the formula for the area of a regular polygon by breaking it down into a set of n isosceles triangles, where n is the number of sides. The easiest way is to use our hexagon calculator, which includes a built-in area conversion tool. As given in the question. Since the inscribed circle is tangent to the side lengths of the Hexagon, we can draw a height from the center of the circle to the side length of the Hexagon. 1: The reference regular hexagon of area δ and the correspondent reference hexagonal tiling of size δ. If it is a regular hexagon, where the lengths of all sides are equal, the perimeter can also be determined by multiplying the length of 1 of its sides by 6. Given a regular hexagon. Note: do NOT round until the end Hello, I need help finding the answer to this question can some one help me? Regular Hexagon Area Calculator. d. A regular hexagon has six equal sides and six equal angles. This makes the area of the square 9 This would make the area of the hexagon 9. In case you have a polygon with more than 10 vertices, divide it into smaller polygons find the area of each polygon and add them to obtain the total area. Multiply this value by six. (The distance from the intersection of the two sides to the center is called the radius. Related formulas Hexagon inscribed in a circle, so radius of the circle is the side of the hexagon. There is one more formula that could be used to calculate the area of regular Hexagon: Area= \(\large \frac{3}{2} . In this case, the formula equals perimeter times apothem time a half. They say it's 2 square roots of 3. the answer is the option Finding the Area of a Regular Hexagon with Side Length 7. The exterior angle is measured as 60-degree in case of the regular hexagon and the interior angle is measured as 120-degree. Because of this, a regular hexagon can be thought of as being made of six equilateral triangles. Area of a rectangle. Area of a regular hexagon. You can calculate the area of the hexagon in two ways, just by knowing the length of any one side or with the given apothem. You are given the apothem = of the regular hexagon, which is the altitude of the equilateral triangle. A hexagonal prism is a prism with a hexagonal base. The height/width would be 3. t\) Where “t” is the length of each side of the hexagon and “d” is the height of the hexagon when it is made to lie on one of the bases of it. Area of a square. The hexagon is the highest regular polygon which allows a regular tesselation (tiling). So this side right over here is 2 square roots of 3. A polygon is a closed shape with many sides. The area of the regular hexagon is 169. Area of a Hexagon Calculator. A hexagon is a regular polygon with six sides and angles. n is number of sides in the polygon. The hexagonal volume calculator calculates volume of a regular hexagonal prism with two hexagonal bases and six rectangular faces, based on the prism side length l and its height h. The sum of interior angles of a hexagon = (6 − 2) · 180° = 720° The value of an interior angle of the regular hexagon is 720º/6 = 120º The area of a regular hexagon inscribed in a circle of radius 1 is? (Last Updated On: March 8, 2019) Problem Statement: ME Board October 1996, ME Board April 1997 The area of the regular hexagon is the sum of areas of these 6 equilateral triangles: 6 × 1 2 R 2 ⋅ sin 6 0 ∘ = 3 3 2 R 2. 5 inches long and a side 19 inches long. In this calculator, we can calculate the area of a regular hexagon based on radius and semiperimeter. Teach Guided Instruction Additional Examples A portion of a regular hexagon has apothem and radii drawn. Irregular Polygons Finding the area of an irregular polygon is trickier since there are no easy formulas. Since you are dealing with a 30-60-90 triangle, it shouldn't take you long to see that the length of the altitude is (1/2)(6)(sqrt[3]) or 3sqrt(3). The triangles formed by joining the center with all the vertices, are equal in size and are equilateral. Similar Questions. Since it is a regular hexagon, apothem can be calculated from one of the sides, and that is why you only need to enter the lenght of a side in our education calculator below. First, what Campbell is trying to tell you is that an integer one divided by an integer three doesn't yield what you want. An equilateral hexagon can be divided into 6 equilateral triangles of side length 6. Find the area of the shaded region to the nearest TENTH. The perimeter of this yard is the distance around the outside of the yard, indicated by the red arrow; It is measured in linear units such as feet or meters. Angles of the Hexagon. Enter one value and choose the number of decimal places. Let be a regular hexagon with side length . A short line in the middle of each side of a shape means that the sides are all the same length. Read and learn for free about the following article: Special right triangles review If you're seeing this message, it means we're having The Length of a Side of a regular Hexagon calculator computes the length of all of the sides (s) of a regular Hexagon based on the area (A). 11) 18 12) 4 3 13) 10 14) 8 15) quadrilateral radius = 16 2 16) hexagon side = 16 3 3 Critical thinking questions: 17) Find the perimeter of a regular hexagon that has an area of 54 3 units². e. Hexagon. The formula for finding the area of a hexagon is Area = (3√3 s2)/ 2 where s is the length of a side of the regular hexagon. Next, there are two simple ways to calculate the diameter of a hexagon. A) 3√2. For a triangle with sides of length a, b, and c let s = (a + b + c)/2. And I could just go around the hexagon. The properties of regular hexagons: All sides are the same length (congruent) and all interior angles are the same size (congruent). Hexagon is regular thus it is made up of the six equilateral triangle. Perimeter (P) = 6 × side = 6a Area (A) = 6 × (area of the 3 Aug 2018 A regular hexagon is convex, meaning that the points of the Remember, to find the area of a triangle, multiply the base by the height, and regular hexagon with the midpoints of successive sides joined. Similarly, if we are to find the area of the polygons- like area of a regular The Area of a Regular Polygon is equal to the formula. To find the perimeter of a polygon, take the sum of the length of each side. The amount of fence needed to enclose the backyard (perimeter) is 1-dimensional. What is the area of the convex hexagon whose interior is the intersection of the interiors of and ? Solution 1. The area of a regular hexagon is given as A = (3*roo A square has 4 sides that are of equal length. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane. Divide you 6 Apr 2018 Now we can understand that a hexagon is a six-sided polygon. Type the number of sides and the polygon area appears in no time. Let x be the side length of the hexagon. 5 square inches? Give the answer as a whole number or as a decimal number rounded to the nearest tenth. Enter any 1 variable plus the number of sides or the polygon name. Formula for area of one equilateral triangle is Area = s 2 √(3)/4 so just plug in the 8 for s, and multiply by 6 to account for all of the equilateral triangles, and evaluate with a calculator, rounding as directed. Multiply the radius by 2. when a regular polygon is broken into triangles, the apothem is the height of one of the triangles whose base is a side of the regular In this video you will discover two ways to find the area of a regular hexagon given the side length. 6\times \frac12 R^2 \cdot \sin 60^\circ = \frac{3\sqrt3}2 R^2 . So if you’re doing a hexagon problem, you may want to cut up the figure and use equilateral triangles or 30°- 60°- 90° triangles to help you find the apothem, perimeter, or area. Finally, a regular prism is a right prism with a regular polygon as the base. Sum of interior angles = (n – 2)·180° Interior angle = Area = (½)nsr. Equal sides. Regular hexagon is composed of six equilateral triangles with six equal sides and angles. The total of the internal angles of any hexagon is 720 degree. A hexagon is a polygon that has six straight sides and six angles. The area of the hexagon is then the sum of the areas of the four triangles. regular hexagon area

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