A pentagon is a polygon with five straight sides. Most problems you'll find in math class will include a regular pentagon with five equal sides. There are two general ways to find breadth, depending on the amount of information you have.
Step
Method 1 of 3: Finding Area of Side Length and Apothem
Step 1. Start with the side lengths and the apothem
This method can be used for regular pentagons with five equal sides. In addition to the side lengths, you will need the "appothem" of the pentagon. The apothem is a line from the center of the pentagon to one of the sides that intersects the side at a right angle of 90º.
- Don't confuse the apothem and the radius, which touches one of the vertices and not the midpoint. If you only know the length of the side and the radius, skip this method and move on to the next method.
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We will use the example of a pentagon with side length
Step 3. unit and apotem
Step 2. unit.
Step 2. Divide the pentagon into five triangles
Draw five lines from the center of the pentagon, leading to each vertex. Now you have five triangles.
Step 3. Find the area of one of the triangles
Each triangle has pedestal which is equal to the side of the pentagon. Each triangle also has tall which is equal to the apothem of the pentagon. (Remember, the height of a triangle extends from the vertex of the triangle to the opposite side, forming a right angle.) To find the area of any triangle, simply calculate x base x height.
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In our example, the area of the triangle = x 3 x 2 =
Step 3. unit squared.
Step 4. Multiply by five to find the total area
We have divided the pentagon into five equal triangles. To find the total area, simply multiply the area of one of the triangles by five.
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In our example, L(total pentagon) = 5 x L(triangle) = 5 x 3 =
Step 15. unit squared.
Method 2 of 3: Finding Area from Side Length
Step 1. Start with just the side lengths
This method only applies to regular pentagons that have five equal sides.
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In this example, we will use a pentagon with side length
Step 7. unit.
Step 2. Divide the pentagon into five triangles
Draw a line from the center of the pentagon to any vertex. Repeat this for all the corner points. Now you have five triangles, each of the same size.
Step 3. Divide the triangle in half
Draw a line from the center of the pentagon to the base of one of the triangles. This line should touch the base at a right angle of 90, dividing the triangle into two smaller equal triangles.
Step 4. Name one of the smaller triangles
We can already name one of the sides and one of the angles of the smaller triangle:
- pedestal triangle is of the length of the side of the pentagon. In our example, the length of the base is x 7 = 3.5 units.
- Big corner at the center of the pentagon is always 36º. (Starting at the 360 center, you can divide it into 10 of these smaller triangles. 360 10 = 36, so the angle in one of the triangles is 36º.)
Step 5. Calculate the height of the triangle. Tall of this triangle is the side that is perpendicular (forming a right angle) with the side of the pentagon, pointing towards the center. We can use basic trigonometry to find the length of this side:
- In a right triangle, tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side.
- The side opposite the 36º angle is the base of the triangle (half the side of the pentagon). The side adjacent to the angle 36º is the height of the triangle.
- tan(36º) = opposite / adjacent
- In our example, tan(36º) = 3.5 / height
- height x tan(36º) = 3, 5
- height = 3.5 / tan(36º)
- height = (approximately) 4, 8 unit.
Step 6. Find the area of the triangle
The area of a triangle is base x height. (L = at). Now that you know the height, enter these values to find the area of your little triangle.
In our example, the area of the small triangle = at = (3, 5)(4, 8) = 8, 4 units squared
Step 7. Multiply to find the area of the pentagon
One of these smaller triangles is 1/10 of the area of the pentagon. To find the total area, multiply the area of the smaller triangle by 10.
In our example, the area of the whole pentagon = 8, 4 x 10 = 84 unit squared.
Method 3 of 3: Using Formulas
Step 1. Use the perimeter and apothem
The apothem is a line from the center of a pentagon that touches one side at a right angle. If you are given the length of the apothem, you can use this easy formula.
- Area of a regular pentagon = ka/2, where k = perimeter and a = apothem.
- If you don't know the perimeter, calculate the perimeter from the side length: k = 5s, where s is the side length.
Step 2. Use the side lengths
If you only know the side lengths, use the following formula:
- Area of regular pentagon = (5 s 2) / (4tan(36º)), where s = side length.
- tan(36º) = (5-2√5). So, if your calculator doesn't have a tan function, use the formula Area = (5 s 2) / (4√(5-2√5)).
Step 3. Choose a formula that uses only the radius
You can even find the area if you only know the radius. Use this formula:
Area of regular pentagon = (5/2) r 2sin(72º), where r is the radius.
Tips
- The examples given here use rounded values to make calculations easier. If you measure the actual polygon with the given side lengths, you will get slightly different results for the other lengths and areas.
- If possible, use the geometric method and the formula method, and compare the results to make sure you have the correct answer. You might get a slightly different answer if you enter the formula all at once (since you won't be rounding off when you're doing the calculation), but the answer should be pretty much the same.
- An irregular pentagon, or a pentagon with unequal sides, is more difficult to learn. The best approach is usually to divide the pentagon into triangles, and add up the area of each triangle. You may also need to draw the larger shape around the pentagon, calculate its area, and subtract the area of the outside of the pentagon.
- The formulas are derived from geometric means, almost the same as those described here. Notice if you can figure out how to get the formulas. The radius formula is more difficult to derive than the other formulas (hint: you'll need a double or double angle identity).
