A regular pentagon can be divided into five triangles. [2] X Research source Where the height of the triangle is known as the apothem. [3] X Research source Then, using the apothem, the area of a regular pentagon will be ½ x apothem x 5. [4] X Research source Don’t confuse the apothem with the radius, which touches a corner (vertex) instead of a midpoint. If you only know the side length and radius, skip down to the next method instead. We’ll use an example pentagon with side length 3 units and apothem 2 units.
In our example, area of triangle = ½ x 3 x 2 = 3 square units.
In our example, area of triangle = ½ x 3 x 2 = 3 square units.
In our example, area of triangle = ½ x 3 x 2 = 3 square units.
In our example, A(total pentagon) = 5 x A(triangle) = 5 x 3 = 15 square units.
In this example, we’ll use a pentagon with side length 7 units.
The base of the triangle is ½ the side of the pentagon. In our example, this is ½ x 7 = 3. 5 units. The angle at the pentagon’s center is always 36º. (Starting with a full 360º center, you could divide it into 10 of these smaller triangles. 360 ÷ 10 = 36, so the angle at one triangle is 36º. )
In a right-angle triangle, the tangent of an angle equals 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 pentagon’s side). The side adjacent to the 36º angle 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 = (about) 4. 8 units.
In a right-angle triangle, the tangent of an angle equals 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 pentagon’s side). The side adjacent to the 36º angle 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 = (about) 4. 8 units.
In our example, Area of small triangle = ½bh = ½(3. 5)(4. 8) = 8. 4 square units.
In our example, the area of the whole pentagon = 8. 4 x 10 = 84 square units.
Area of a regular pentagon = pa/2, where p = the perimeter and a = the apothem. [11] X Research source If you don’t know the perimeter, calculate it from the side length: p = 5s, where s is the side length.
Area of a regular pentagon = (5s2) / (4tan(36º)), where s = side length. tan(36º) = √(5-2√5). [13] X Research source So if your calculator doesn’t have a “tan” function, use the formula Area = (5s2) / (4√(5-2√5)).
Area of a regular pentagon = (5/2)r2sin(72º), where r is the radius.
