Category	Icon	Key Results & Properties
Triangles	🔺	• Sum of angles is 180°. • Sum of any two sides > third side. • Pythagoras Theorem: $a^2 + b^2 = c^2$ (for right-angled triangles). • Centroid: Point where medians meet; divides medians in 2:1 ratio. • Median divides a triangle into two equal areas.
Quadrilaterals	▱	• Parallelogram diagonals bisect each other. • Square/Rhombus diagonals bisect at 90°. • Rectangle diagonals are equal. • Rectangle has the greatest area among parallelograms of given sides.

Shape	Icon	Area Formula	Perimeter / Other
Rectangle	▮	$\text{Length} \times \text{Breadth}$	$2(\text{Length} + \text{Breadth})$
Square	◼	$(\text{side})^2$ or $\frac{1}{2}(\text{diagonal})^2$	$4 \times \text{side}$
Triangle (General)	△	$\frac{1}{2} \times \text{Base} \times \text{Height}$	Heron’s: $\sqrt{s(s-a)(s-b)(s-c)}$
Equilateral Triangle	▽	$\frac{\sqrt{3}}{4} \times (\text{side})^2$	Inradius = $\frac{a}{2\sqrt{3}}$; Circumradius = $\frac{a}{\sqrt{3}}$
Circle	◯	$\pi R^2$	Circumference = $2\pi R$
Parallelogram	▱	$\text{Base} \times \text{Height}$	–
Rhombus	💠	$\frac{1}{2} \times (\text{Product of diagonals})$	–
Trapezium	⏢	$\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$	–
4 Walls of a Room	🏠	$2(\text{Length} + \text{Breadth}) \times \text{Height}$	–

Component	Icon	Formula
Length of Arc	弧	$\frac{2\pi R\theta}{360}$
Area of Sector	🍕	$\frac{1}{2}(\text{arc} \times R)$ or $\frac{\pi R^2\theta}{360}$
Semi-circle	◑	Area = $\frac{\pi R^2}{2}$

📐 Fundamental Concepts: Triangles & Quadrilaterals

Category	Icon	Key Results & Properties
Triangles	🔺	• Sum of angles is 180°. • Sum of any two sides > third side. • Pythagoras Theorem: $a^2 + b^2 = c^2$ (for right-angled triangles). • Centroid: Point where medians meet; divides medians in 2:1 ratio. • Median divides a triangle into two equal areas.
Quadrilaterals	▱	• Parallelogram diagonals bisect each other. • Square/Rhombus diagonals bisect at 90°. • Rectangle diagonals are equal. • Rectangle has the greatest area among parallelograms of given sides.

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📝 Important Area & Perimeter Formulas

Shape	Icon	Area Formula	Perimeter / Other
Rectangle	▮	$\text{Length} \times \text{Breadth}$	$2(\text{Length} + \text{Breadth})$
Square	◼	$(\text{side})^2$ or $\frac{1}{2}(\text{diagonal})^2$	$4 \times \text{side}$
Triangle (General)	△	$\frac{1}{2} \times \text{Base} \times \text{Height}$	Heron’s: $\sqrt{s(s-a)(s-b)(s-c)}$
Equilateral Triangle	▽	$\frac{\sqrt{3}}{4} \times (\text{side})^2$	Inradius = $\frac{a}{2\sqrt{3}}$; Circumradius = $\frac{a}{\sqrt{3}}$
Circle	◯	$\pi R^2$	Circumference = $2\pi R$
Parallelogram	▱	$\text{Base} \times \text{Height}$	–
Rhombus	💠	$\frac{1}{2} \times (\text{Product of diagonals})$	–
Trapezium	⏢	$\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$	–
4 Walls of a Room	🏠	$2(\text{Length} + \text{Breadth}) \times \text{Height}$	–

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🥧 Circles, Sectors, and Arcs

Component	Icon	Formula
Length of Arc	弧	$\frac{2\pi R\theta}{360}$
Area of Sector	🍕	$\frac{1}{2}(\text{arc} \times R)$ or $\frac{\pi R^2\theta}{360}$
Semi-circle	◑	Area = $\frac{\pi R^2}{2}$

Note: For a triangle, $s$ represents the semi-perimeter, calculated as $s = \frac{a+b+c}{2}$.

FUNDAMENTAL CONCEPTS

Results on Triangles:

1. Sum of the angles of a triangle is 180°.
2. The sum of any two sides of a triangle is greater than the third side.
3. Pythagoras Theorem: In a right-angled triangle, $(\text{Hypotenuse})^2 = (\text{Base})^2 + (\text{Height})^2$.
4. The line joining the mid-point of a side of a triangle to the opposite vertex is called the median.
5. The point where the three medians of a triangle meet, is called centroid. The centroid divided each of the medians in the ratio 2 : 1.
6. In an isosceles triangle, the altitude from the vertex bisects the base.
7. The median of a triangle divides it into two triangles of the same area.
8. The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of the given triangle.

Results on Quadrilaterals:

1. The diagonals of a parallelogram bisect each other.
2. Each diagonal of a parallelogram divides it into triangles of the same area.
3. The diagonals of a rectangle are equal and bisect each other.
4. The diagonals of a square are equal and bisect each other at right angles.
5. The diagonals of a rhombus are unequal and bisect each other at right angles.
6. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
7. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.

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IMPORTANT FORMULAE

1. Area of a rectangle = $(\text{Length} \times \text{Breadth})$.
   • $\text{Length} = \frac{\text{Area}}{\text{Breadth}}$ and $\text{Breadth} = \frac{\text{Area}}{\text{Length}}$.
   • Perimeter of a rectangle = $2(\text{Length} + \text{Breadth})$.
2. Area of a square = $(\text{side})^2 = \frac{1}{2}(\text{diagonal})^2$.
3. Area of 4 walls of a room = $2(\text{Length} + \text{Breadth}) \times \text{Height}$.
4. Triangle Formulas:
   • Area of a triangle = $\frac{1}{2} \times \text{Base} \times \text{Height}$.
   • Area of a triangle = $\sqrt{s(s – a)(s – b)(s – c)}$ where $a, b, c$ are the sides of the triangle and $s = \frac{1}{2}(a + b + c)$.
   • Area of an equilateral triangle = $\frac{\sqrt{3}}{4} \times (\text{side})^2$.
   • Radius of incircle of an equilateral triangle of side $a$ = $\frac{a}{2\sqrt{3}}$.
   • Radius of circumcircle of an equilateral triangle of side $a$ = $\frac{a}{\sqrt{3}}$.
   • Radius of incircle of a triangle of area $\Delta$ and semi-perimeter $s$: $r = \frac{\Delta}{s}$.
5. Other Quadrilaterals:
   • Area of parallelogram = $(\text{Base} \times \text{Height})$.
   • Area of a rhombus = $\frac{1}{2} \times (\text{Product of diagonals})$.
   • Area of a trapezium = $\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{distance between them}$.
6. Circle Formulas:
   • Area of a circle = $\pi R^2$, where $R$ is the radius.
   • Circumference of a circle = $2\pi R$.
   • Length of an arc = $\frac{2\pi R\theta}{360}$, where $\theta$ is the central angle.
   • Area of a sector = $\frac{1}{2}(\text{arc} \times R) = \frac{\pi R^2\theta}{360}$.
7. Semi-circle Formulas:
   • Circumference of a semi-circle = $\pi R$.
   • Area of semi-circle = $\frac{\pi R^2}{2}$.

Shape	Volume (V)	Surface Area (SA)	Key Properties
🧊 Cuboid	$l \times b \times h$	$2(lb + bh + lh)$	Diagonal: $\sqrt{l^2 + b^2 + h^2}$
🎲 Cube	$a^3$	$6a^2$	Diagonal: $\sqrt{3}a$
🧪 Cylinder	$\pi r^2 h$	$2\pi r(h + r)$	Curved Area: $2\pi rh$
🍦 Cone	$\frac{1}{3} \pi r^2 h$	$\pi r(l + r)$	Slant Height ($l$): $\sqrt{h^2 + r^2}$
🏀 Sphere	$\frac{4}{3} \pi r^3$	$4 \pi r^2$	$r$ is the only variable
🥣 Hemisphere	$\frac{2}{3} \pi r^3$	$3 \pi r^2$	Curved Area: $2 \pi r^2$

Quick Conversion Tips

• Capacity: $1 \text{ liter} = 1000 \text{ cm}^3$
• Pi ($\pi$): Usually taken as $\frac{22}{7}$ or $3.14$ for calculations.
• Units: Volume is always in cubic units (e.g., $\text{cm}^3$, $\text{m}^3$), while Surface Area is in square units (e.g., $\text{cm}^2$, $\text{m}^2$).

Shape	Formula Type	Formula Expression
1. CUBOID	1. Volume	$(l \times b \times h)$ cubic units.
(len = $l$, br = $b$, ht = $h$)	2. Surface area	$2(lb + bh + lh)$ sq. units.
	3. Diagonal	$\sqrt{l^2 + b^2 + h^2}$ units.
2. CUBE	1. Volume	$a^3$ cubic units.
(each edge = $a$)	2. Surface area	$6a^2$ sq. units.
	3. Diagonal	$\sqrt{3}a$ units.
3. CYLINDER	1. Volume	$(\pi r^2 h)$ cubic units.
(rad = $r$, ht = $h$)	2. Curved surface area	$(2 \pi rh)$ sq. units.
	3. Total surface area	$2 \pi r(h + r)$ sq. units.
4. CONE	1. Slant height ($l$)	$\sqrt{h^2 + r^2}$ units.
(rad = $r$, ht = $h$)	2. Volume	$\frac{1}{3} \pi r^2 h$ cubic units.
	3. Curved surface area	$(\pi rl)$ sq. units.
	4. Total surface area	$(\pi rl + \pi r^2)$ sq. units.
5. SPHERE	1. Volume	$\frac{4}{3} \pi r^3$ cubic units.
(rad = $r$)	2. Surface area	$(4 \pi r^2)$ sq. units.
6. HEMISPHERE	1. Volume	$\frac{2}{3} \pi r^3$ cubic units.
(rad = $r$)	2. Curved surface area	$(2 \pi r^2)$ sq. units.
	3. Total surface area	$(3 \pi r^2)$ sq. units.

Note: $1 \text{ litre} = 1000 \text{ cm}^3$

1. CUBOID

Let length = $l$, breadth = $b$ and height = $h$ units. Then,

1. Volume = $(l \times b \times h)$ cubic units.
2. Surface area = $2(lb + bh + lh)$ sq. units.
3. Diagonal = $\sqrt{l^2 + b^2 + h^2}$ units.

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2. CUBE

Let each edge of a cube be of length $a$. Then,

1. Volume = $a^3$ cubic units.
2. Surface area = $6a^2$ sq. units.
3. Diagonal = $\sqrt{3}a$ units.

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3. CYLINDER

Let radius of base = $r$ and Height (or length) = $h$. Then,

1. Volume = $(\pi r^2 h)$ cubic units.
2. Curved surface area = $(2\pi rh)$ sq. units.
3. Total surface area = $2\pi r(h + r)$ sq. units.

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4. CONE

Let radius of base = $r$ and Height = $h$. Then,

1. Slant height, $l = \sqrt{h^2 + r^2}$ units.
2. Volume = $\frac{1}{3}\pi r^2 h$ cubic units.
3. Curved surface area = $(\pi rl)$ sq. units.
4. Total surface area = $(\pi rl + \pi r^2)$ sq. units.

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5. SPHERE

Let the radius of the sphere be $r$. Then,

1. Volume = $\frac{4}{3}\pi r^3$ cubic units.
2. Surface area = $(4\pi r^2)$ sq. units.

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6. HEMISPHERE

Let the radius of a hemisphere be $r$. Then,

1. Volume = $\frac{2}{3}\pi r^3$ cubic units.
2. Curved surface area = $(2\pi r^2)$ sq. units.
3. Total surface area = $(3\pi r^2)$ sq. units.

Note: $1 \text{ litre} = 1000 \text{ cm}^3$.