<Ul> <Li> A triangle: 1 2 B h (\ displaystyle (\ tfrac (1) (2)) Bh) (where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known . If the lengths of the three sides are known then Heron's formula can be used: s (s − a) (s − b) (s − c) (\ displaystyle (\ sqrt (s (s-a) (s-b) (s-c)))) where a, b, c are the sides of the triangle, and s = 1 2 (a + b + c) (\ displaystyle s = (\ tfrac (1) (2)) (a + b + c)) is half of its perimeter . If an angle and its two included sides are given, the area is 1 2 a b sin ⁡ (C) (\ displaystyle (\ tfrac (1) (2)) ab \ sin (C)) where C is the given angle and a and b are its included sides . If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of 1 2 (x 1 y 2 + x 2 y 3 + x 3 y 1 − x 2 y 1 − x 3 y 2 − x 1 y 3) (\ displaystyle (\ tfrac (1) (2)) (x_ (1) y_ (2) + x_ (2) y_ (3) + x_ (3) y_ (1) - x_ (2) y_ (1) - x_ (3) y_ (2) - x_ (1) y_ (3))). This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points (x, y), (x, y), and (x, y). The shoelace formula can also be used to find the areas of other polygons when their vertices are known . Another approach for a coordinate triangle is to use calculus to find the area . </Li> <Li> A simple polygon constructed on a grid of equal - distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points: i + b 2 − 1 (\ displaystyle i+ (\ frac (b) (2)) - 1), where i is the number of grid points inside the polygon and b is the number of boundary points . This result is known as Pick's theorem . </Li> </Ul> <Li> A triangle: 1 2 B h (\ displaystyle (\ tfrac (1) (2)) Bh) (where B is any side, and h is the distance from the line on which B lies to the other vertex of the triangle). This formula can be used if the height h is known . If the lengths of the three sides are known then Heron's formula can be used: s (s − a) (s − b) (s − c) (\ displaystyle (\ sqrt (s (s-a) (s-b) (s-c)))) where a, b, c are the sides of the triangle, and s = 1 2 (a + b + c) (\ displaystyle s = (\ tfrac (1) (2)) (a + b + c)) is half of its perimeter . If an angle and its two included sides are given, the area is 1 2 a b sin ⁡ (C) (\ displaystyle (\ tfrac (1) (2)) ab \ sin (C)) where C is the given angle and a and b are its included sides . If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of 1 2 (x 1 y 2 + x 2 y 3 + x 3 y 1 − x 2 y 1 − x 3 y 2 − x 1 y 3) (\ displaystyle (\ tfrac (1) (2)) (x_ (1) y_ (2) + x_ (2) y_ (3) + x_ (3) y_ (1) - x_ (2) y_ (1) - x_ (3) y_ (2) - x_ (1) y_ (3))). This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points (x, y), (x, y), and (x, y). The shoelace formula can also be used to find the areas of other polygons when their vertices are known . Another approach for a coordinate triangle is to use calculus to find the area . </Li> <Li> A simple polygon constructed on a grid of equal - distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points: i + b 2 − 1 (\ displaystyle i+ (\ frac (b) (2)) - 1), where i is the number of grid points inside the polygon and b is the number of boundary points . This result is known as Pick's theorem . </Li> <Ul> <Li> The area between a positive - valued curve and the horizontal axis, measured between two values a and b (b is defined as the larger of the two values) on the horizontal axis, is given by the integral from a to b of the function that represents the curve: </Li> </Ul>

Where does the area of a rectangle come from