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First, what is apothem?
Apothem: The length of the line joining the center of a regular polygon to the midpoint of one of its sides.
So, we have a regular triangle, or an equilateral triangle (very important fact that we need to use all the way of our work), where all three sides are equal, and all three angles are equal, each equals 60 degrees.
1. Draw the equilateral triangle ABC (label it in a counterclockwise direction starting with letter A, length BC is horizontal).
2. From A, and B draw the medians AD and BE, which we also know that are perpendicular to the sides BC and AC and bisect the vertex angles A and B, each part equals 30 degrees.
3. Label with O the point of intersection of these medians.
4. Look at the right triangles ODB and OEA. We know that angles B and A equal 30 degrees, and angles D and E equal 90 degrees. So that sides OD and OE, which equal 6 as they are apothems, are one half of the length measure of the hypotenuses OB and OA, as sides opposite to an angle of 30 degrees. Thus, these hypotenuses OB and OA equal 12. (or you can use the fact that the apothem is 1/3 of the length of the median)
5. So the height AD of the triangle ABC is 18 (6 + 12). In order to find the area of the triangle we need to know what length measure has the base BC.
6. In the right triangle BEC, the length of BC is approximately 20.78 (cos 30 degrees = 18/BC).
7. Since we know the length of the base BC, 20.78, and the length of the height, 18, we are able to find the area of the triangle ABC which is approximately 187 square unit ( A = (bh)/2, A = (20.78 x 18)/2 = 187.02)
8. Area of the triangle can be obtained by one of these two formulae:
A = 1/2 a*p where a is the apothem and p is the perimeter.
A = 3 a^2*sqrt(3)
9. Applying the first formula: A = 1/2*6 *62.34 = 187.02
10. Applying the second formula: A = 3*6^2*3^1/2 = 187.06
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What is the area of an equilateral triangle with an apothem of 5.8 cm?
Area = 8.06 cm2.
If one side of an equilateral triangle is 12Cm find the Apothem?
the perimeter is 36 and the area is 144
What is the area of a regular hexagon with apothem length of 24 inches?
For a regular hexagon, half the side length can be calculated from the apothem via trigonometry: half_side_length = apothem x tan 30° Then: area = apothem x 1/2 x perimeter = apothem x 1/2 x side_length x 6 = apothem x half_side_length x 6 = 24 in x (24 in x tan 30°) x 6 ≈ 1995 sq in
What is the length of the apothem of a regular hexagon with 10 cm sides?
The hexagon will consist of 6 equilateral triangles of 3 equal sides of 10 cm and the apothem will divide the triangle into 2 right angle triangles with a base of 5 and an hypotenuse of 10 and so by using Pythagoras' theorem the height of the triangle which is the apothem works out as 5 times square root of 3 or about 8.66 cm rounded to 2 decimal places.
What is the total surface area of a square pyramid with an apothem of 8 and a height of 8.54 and the squares length and width of 6?
138.48
What is the area of a regular hexagon with a side of 4 and an apothem of 3 46?
Easy. Since the side is the base and the apothem is the height of the triangle, multiply them and divide by two to get the area of the triangle. 3 * 3.46 = 10.38 /2 = 5.19. Then multiply by 6 to get the area of the hexagon. 5.19 * 6 = 31.14. You multiply by 6 because you can fit 6 regular triangles in a regular hexagon. We've already found the area of one regular triangle in the hexagon.
What is the area of an equilateral triangle with an apothem of 5.8 cm?
Area = 8.06 cm2.
How do you find the area of a square and triangle when they are out together?
Area of a triangle is A=1/2*apothem*Permiter orA=1/2*B*HSquare is A=L*W
If one side of an equilateral triangle is 12Cm find the Apothem?
the perimeter is 36 and the area is 144
What is the area of a regular hexagon with apothem length of 24 inches?
For a regular hexagon, half the side length can be calculated from the apothem via trigonometry: half_side_length = apothem x tan 30° Then: area = apothem x 1/2 x perimeter = apothem x 1/2 x side_length x 6 = apothem x half_side_length x 6 = 24 in x (24 in x tan 30°) x 6 ≈ 1995 sq in
Find the area of a regular 7-gon with a side length of 12?
You can find the area of any regular polygon with this equation: 1/2ap where a is the apothem and p is the perimeter.To find the perimeter, multiply 7 times 12 to get 84.To find the apothem, divide 360 (the amount of degrees around the center of the polygon) by 7 (the amount of sides you have) to get about 51.43. This gives you the measure of the angle of one triangle connected to 2 adjacent vertices. You have to make this a right triangle so that you can use trigonometry to find the apothem, so divide that angle by 2 to get 25.72. You know the side opposite of this angle is 6 because it is half the amount of the side (remember we divided this triangle by 2). To find the apothem at this point, use trigonometry. tan25.72 = 6/x. The apothem is 12.46.So, plug all of this into your calculator; a = 1/2(84)(12.46). You find the area to be about 523.3.
What is the area of a regular hexagon if a side is 30?
The area is about 2338.27 square units, from the formulaA = 3/2 (sqrt 3) s2 or about 2.598 s2--Let's draw a segment from the center of the hexagon to the middle of a side. This segment is called the apothem. Then use the 30-60-90 triangle rule. If half of a side is 15, that means the apothem is 15√3.If we divide the hexagon into equilateral triangles, we get 6 equilateral triangles.So if we find the area for one of these triangles and multiply it by 6, we get the area of the hexagon. The area of a triangle is found by 1/2(b*h). The apothem is your height for the triangles. So plug the numbers in: 1/2(30*15√3). Solve: 1/2(779.4228) = 389.7114. This is the area of one triangle. Now we multiply by 6, and this becomes: 2338.2686
What is the area of dudecagon?
The formula to find the area of a dodecagon (12-sided polygon) is (apothem x perimeter)/2.
How do you find the perimeter of a regular polygon with the apothem and area?
Perimeter = 2*Area/Apothem.
What is the area of a regular pentagon with a side length of ten units?
A regular pentagon has five (5) equilateral triangles within it. Find the area of each triangle (1/2bh where b is the base of the triangle or the length of a side of the pentagon, and h is the height of the triangle or the apothem of the pentagon) and multiply the area of the triangle times five (5).
How do you find an apothem?
The formula for finding an apothem is s = 1/2 aP. S is the area, a is the apothem, and P is the perimeter.
What is the area of a triangle with a base of 6 units and a height of 6 units?
The area of triangle is : 18.0
