TRIANGLES

Perimeter of a Triangle

If $a, \ b, \ c\$  are the lengths of the sides of any triangle. Then:

1. Perimeter of a triangle $= (a+b+c)$ units
2. Area of the triangle $= \sqrt{(s(s-a)(s-b)(s-c)} \ units.$

where $s=\frac{1}{2}(a+b+c)$.This is also known as  $Heron's \ Formula$

Area of a Triangle

Refer to the adjoining figure. If b is the base and h is the height, then

$Area\ of\ triangle = \frac{1}{2}(base \times height) = \frac{1}{2}(b \times h)$

Note: If you could take any side as the base, then the corresponding height is the

would be the length of the perpendicular to this side from the opposite vertex.

Area of Right Angled Triangle

Let the $\triangle ABC$, with $\angle B = 90^o$. Please refer to the adjoining figure.

$Area \ of \ triangle = \frac{1}{2}(BC \times AB)$

Area of an Equilateral triangle

In an equilateral triangle, all three sides are equal.

Let us say the side $= a$  unit.

1. Height of an equilateral triangle = $\frac{\sqrt{3}}{2}a$  units
2. Area of an equilateral triangle with side $a \ is\ \frac{\sqrt{3}}{2}a^2$ units

RECTANGLE & SQUARE

Perimeter and Area of Rectangle

If the sides of a rectangle are$l$ units and $b$units (refer to adjoining figure), then

1. Perimeter of rectangle $= 2(l+b)$ units
2. Area of rectangle $= (l \times b)$  sq. units
3. Diagonal of a rectangle $(d) = \sqrt{(l^2+b^2)}$ units

Perimeter and Area of a Square

If the sides of a square is $a$ units, then

1. Perimeter  $= 4a$ units
2. Area $= a^2$ units
3. Diagonal of a square $= a \sqrt{2}$ units

PARALLELOGRAM, RHOMBUS AND TRAPEZIUM

Area of a Parallelogram

Let ABCD be a Parallelogram with base b  and height h units. Let AC be the diagonal. Refer to the adjoining figure.

$Area = (b \times h)$ sq. units.

Area of a Rhombus

Please refer to the adjoin diagram. Let $d1 \ and\ d2$ are the diagonals of the Rhombus. We know that the diagonals intersect at right angles and bisect each other.

$Area\ of\ Rhombus= \frac{1}{2} \times (product\ of\ diagonals) = \frac{1}{2} \times (AC \times DB)$ sq. units

Area of a Trapezium

Please refer to the adjoining figure.  $AB \parallel DC$

$Area\ of\ Trapezium \ ABCD \\= \frac{1}{2} \times (sum \ of \ the \ parallel \ sides) \times (distance\ between\ them) \\= \frac{1}{2} \times (AB + DC) \times (h) \ sq. \ units$

CIRCLE

Circumference and Area of a Circle

Let the circle be of radius r

Circumference of the circle $= 2\pi r = \pi d$

Area of the circle $= \pi r^2$

Area of a Ring (shaded area)

Refer to the adjoining figure. The radius of the larger circle is R and that of the smaller circle is r.  Area of the ring is the shaded area.

$Area\ of\ the\ ring=\pi R^2 - \pi r^2 = \pi(R^2-r^2) = \pi(R+r)(R-r)$