A regular polygon is a polygon with all sides equal in length and all interior angles equal in measure. Common examples include equilateral triangles, squares, and regular hexagons. The sum of the interior angles of any polygon with n sides is given by (n − 2) × 180°, so a hexagon with 6 sides has interior angles summing to 720°. For a regular n-sided polygon, each individual interior angle is (n − 2) × 180° / n; a regular pentagon, for example, has interior angles of 108° each. The sum of the exterior angles of any convex polygon is always 360°, regardless of how many sides it has. A polygon with n sides has n(n − 3) / 2 diagonals, so a hexagon has 9 diagonals.
For regular polygons, a useful measurement is the apothem, the perpendicular distance from the center of the polygon to the midpoint of any side. The area of a regular polygon is A = ½ × p × a, where p is the perimeter and a is the apothem. The perimeter itself is simply P = n × s, where s is the length of one side and n is the number of sides. These formulas turn regular polygons into straightforward calculations.
Other common two-dimensional shapes have their own area formulas. A rectangle has area A = length × width and perimeter P = 2(length + width). A triangle has area A = ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex. A parallelogram has area A = base × height, with the height being the perpendicular distance between the two parallel bases, and perimeter P = 2(a + b). A trapezoid has area A = ½ × (b₁ + b₂) × h, where b₁ and b₂ are the lengths of the two parallel bases. A rhombus has area A = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of its diagonals. When only the three side lengths of a triangle are known, Heron's formula gives the area directly: A = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2 is the semi-perimeter.
Three-dimensional solids extend these ideas by adding height. A rectangular prism has volume V = length × width × height and surface area SA = 2(lw + lh + wh). A cylinder has volume V = πr²h and total surface area SA = 2πr² + 2πrh, where the first term represents the two circular bases and the second the curved lateral surface. A cone has volume V = ⅓πr²h, exactly one-third the volume of a cylinder with the same base and height, and total surface area SA = πr² + πrl, where l is the slant height given by l = √(r² + h²). A sphere has volume V = (4/3)πr³ and surface area SA = 4πr², which is four times the area of a great circle. A pyramid has volume V = ⅓ × base area × height, so a square pyramid with base side s has volume V = ⅓s²h. A triangular prism has volume V = base area × length, which simplifies to V = ½bh × l for a triangle with base b, height h, and prism length l. A hemisphere, being half a sphere, has volume V = (2/3)πr³ and total surface area SA = 3πr², combining the curved portion (2πr²) with the flat circular base (πr²).