## Mensuration of Parallelograms

**Area of a parallelogram: **Recall that a parallelogram is a quadrilateral with both pairs of opposite sides parallel. Parallelograms come in the form of rectangles, rhombi and squares. When trying to determine the area of a parallelogram, it will be necessary to identify the **base** and the **height** of the parallelogram. The base of a parallelogram can be any side. The height of the parallelogram is the length of an altitude. The altitude of a parallelogram is a segment perpendicular to the base with endpoints on the base and the side opposite the base.

The **area** of a parallelogram is given by the product of the base and height. If a parallelogram has an area of 'A' square units, a base of 'b' units and a height of 'h' units, then **A = bh.**

**Perimeter of a parallelogram: **The perimeter of a parallelogram is the length of its boundary, which can be found by adding together the length of each side. Since each pair of opposite sides has the same length, the perimeter is twice the base length plus twice the side length or twice the sum of base length and side length.

If a parallelogram has a perimeter of 'P' units, a base of 'a' units and a side of 'b' units, then **P = 2a + 2b** or **P = 2(a + b).**

**Area of a rectangle: **A quadrilateral with four right angles is called a rectangle or A parallelogram each of whose angles measures 90° is called a rectangle. To find the area of a rectangle, multiply the length by the breadth. If a rectangle has an area of 'A' square units, a length of 'l' units and a breadth of 'b' units, then **A = lb.**

**Perimeter of a rectangle: **The perimeter of a rectangle is the length of its boundary, which can be found by adding together the length of each side. Since each pair of opposite sides has the same length, the perimeter is twice its width plus twice its breadth or twice the sum of width and breadth.

If a rectangle has a perimeter of 'P' units, a length of 'l' units and a breadth of 'b' units, then **P = 2l + 2b** or **P = 2(l + b).**

**Area of a square: **A square is a rectangle with four equal sides. To find the area of a square, multiply the length of one side by itself. If a square has an area of 'A' square units and length of a side 's' units, then **A = s × s = s ^{2}.**

**Perimeter of a square: **The perimeter of a square is the length of its boundary, which can be found by adding together the length of each side. Since all sides are equal in length, the perimeter is 4 times the length of a side.

If a square has a perimeter of 'P' units, and a side of length 's' units, then **P = 4s.**

## Mensuration of Triangles

**Area of a triangle: **Recall that a triangle is a closed figure bounded by three line segments. Similar to parallelograms, when trying to determine the area of a triangle, it will be necessary to identify the base and the height of the triangle. The base of a triangle can be any side. The height of the triangle is the length of the altitude drawn to the chosen base. The altitude of a triangle corresponding to any side is the perpendicular segment from the opposite vertex to that side.

The area of a triangle is one–half the product of the base and its corresponding height. If a triangle has an area of ‘A’ square units, a base of ‘b’ units and a corresponding height of ‘h’ units, then

**A = bh**.

**Perimeter of a triangle: **The perimeter of a triangle is the sum of the lengths of its sides. In the below figure, if 'P' is the perimeter and 'a', 'b', 'c' be the lengths of the sides of a ΔPQR, then

P = a + b + c units.

## Area of an equilateral triangle

Let ABC be an equilateral triangle with side **a**.

Therefore, AB = BC = AC = a.

The height of an equilateral triangle is the length of the perpendicular drawn from a vertex to its opposite side.

This height (AD) is given by =

(it is obtained by applying Pythagorean theorem to triangle ABD).

We know that, area of ^{le} |
= | × base × height. |

= | × BC × AD | |

= | × a × a | |

= | ||

∴ Area of equilateral Δ^{le} with side 'a' |
= |

## Area of a right-angled triangle

Let ABC be a right angled triangle with ∠B | = | 90° |

∴ Area of Δ ABC | = | × base × height |

= | × BC × AB | |

= | [Product of sides containing the right angle] | |

∴ Area of Δ ABC | = | × [Product of sides containing 90°] |

= | × b × h sq. units. |

In a right angled triangle, if base is equal to height(i.e, b = h)

then its becomes an isosceles right-angled triangle.

Its area is given by = b^{2} (or h^{2})

Let a, b, c be the three sides of a ΔABC.

Its perimeter is given by a + b + c.

We introduce a new parameter called semi-perimeter (meaning half perimeter) which is denoted by 's'.

∴

The area of ΔABC is given by

A =

This formula for calculating the area of any triangle, given its sides, is known as **Heron's formula**.