📐 Math Formulas
34 formulas in 10 sections. For each one you get the formula, what every symbol means, where it is used, an explanation in plain words and a worked example.
Sections: Square · Rectangle · Triangle · Parallelogram · Rhombus · Trapezium · Circle · Quadrilateral · Polygon · Path
P = 4 × a
| a | length of one side |
|---|
Where it is used: To find the total length of the boundary of a square — fencing, framing, border tiles.
In simple words: All four sides of a square are equal, so add one side four times (or multiply by 4) to walk right around it.
Example: a = 5 cm → P = 4 × 5 = 20 cm
A = a × a = a²
| a | length of one side |
|---|
Where it is used: To find the flat space inside a square — floor tiles, plots of land, chess boards.
In simple words: Multiply a side by itself to find how much surface fits inside the square.
Example: a = 5 cm → A = 5 × 5 = 25 cm²
d = a × √2
| a | side of the square |
|---|---|
| d | diagonal |
Where it is used: To find the corner-to-corner distance of a square — packing, screen sizes.
In simple words: The slanting line across a square is its side times about 1.414.
Example: a = 5 → d = 5√2 ≈ 7.07 cm
A = ½ × d²
| d | the diagonal |
|---|
Where it is used: To find a square's area directly from its diagonal.
In simple words: Half the diagonal squared gives the square's area.
Example: d=10 → A = ½×100 = 50 cm²
P = 2 × (l + b)
| l | length |
|---|---|
| b | breadth (width) |
Where it is used: To find the boundary length of a rectangle — fencing a field, a picture frame, a room's skirting.
In simple words: Add one long side and one short side, then double it (because there are two of each).
Example: l = 6, b = 4 → P = 2 × (6 + 4) = 20 cm
A = l × b
| l | length |
|---|---|
| b | breadth (width) |
Where it is used: To find the flat space inside a rectangle — floors, walls, fields, sheets of paper.
In simple words: Multiply the length by the width to find how much surface is inside.
Example: l = 6, b = 4 → A = 6 × 4 = 24 cm²
d = √(l² + b²)
| l | length |
|---|---|
| b | breadth |
| d | diagonal |
Where it is used: To find the corner-to-corner distance of a rectangle — TV/monitor sizes, framing.
In simple words: It's Pythagoras: the diagonal joins the length and width like the slanting side of a right triangle.
Example: l=3, b=4 → d = √(9+16) = 5
A = ½ × b × h
| b | base |
|---|---|
| h | height (perpendicular to base) |
Where it is used: To find the space inside any triangle — roofs, sails, triangular plots, in geometry problems.
In simple words: A triangle is half of a rectangle. So multiply the base by the height and take half.
Example: b = 10, h = 6 → A = ½ × 10 × 6 = 30 cm²
A = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
| a, b, c | the three sides |
|---|---|
| s | half the perimeter (semi-perimeter) |
Where it is used: To find the area of a triangle when all three sides are known but the height is not — surveying, land measurement.
In simple words: If you know only the three side lengths, this finds the area without needing the height.
Example: a=3, b=4, c=5, s=6 → A = √(6·3·2·1) = 6 cm²
P = a + b + c
| a, b, c | the three sides |
|---|
Where it is used: To find the boundary length of any triangle — fencing, framing, borders.
In simple words: Just add up the lengths of all three sides.
Example: sides 3, 4, 5 → P = 12 cm
A = (√3 ÷ 4) × a²
| a | the equal side length |
|---|
Where it is used: To find the area of a triangle whose three sides are all equal — tiles, road signs, designs.
In simple words: For a triangle with all sides equal, square a side and multiply by about 0.433.
Example: a = 4 → A = (√3/4)×16 ≈ 6.93 cm²
A = ½ × base × height (the two perpendicular sides)
| base, height | the two sides at the right angle |
|---|
Where it is used: To find the area of a right-angled triangle using its two shorter sides directly.
In simple words: The two sides that meet at the square corner are the base and height.
Example: legs 6 and 8 → ½×6×8 = 24 cm²
A = (b ÷ 4) × √(4a² − b²)
| a | the two equal sides |
|---|---|
| b | the base |
Where it is used: To find the area of a triangle with two equal sides.
In simple words: A ready formula for the area when two sides are equal and the base is known.
Example: a=5, b=6 → (6/4)√(100−36) = 12 cm²
(3,4,5), (5,12,13), (8,15,17), (7,24,25)
| — | whole-number right-triangle sides |
|---|
Where it is used: Handy side sets that satisfy a²+b²=c² — speed up right-triangle problems.
In simple words: These whole-number trios always make a perfect right triangle.
Example: 6,8,10 is (3,4,5) doubled
A = b × h
| b | base |
|---|---|
| h | height (perpendicular distance) |
Where it is used: To find the space inside a slanted four-sided shape with parallel sides — used in geometry and design.
In simple words: Like a pushed-over rectangle: multiply the base by the straight-up height (not the slanting side).
Example: b = 8, h = 5 → A = 8 × 5 = 40 cm²
A = ½ × d₁ × d₂
| d₁, d₂ | the two diagonals |
|---|
Where it is used: To find the area of a diamond-shaped figure using its diagonals — kites, tiles, patterns.
In simple words: Multiply the two crossing diagonals and take half.
Example: d₁ = 6, d₂ = 8 → A = ½ × 6 × 8 = 24 cm²
A = base × height
| base | one side |
|---|---|
| height | perpendicular distance |
Where it is used: An alternate area of a rhombus using a side and its height (like a parallelogram).
In simple words: A rhombus is a slanted square, so area is base times straight-up height.
Example: base 6, height 4 → 24 cm²
A = ½ × (a + b) × h
| a, b | the two parallel sides |
|---|---|
| h | distance between them (height) |
Where it is used: To find the area of a four-sided shape with one pair of parallel sides — plots of land, cross-sections of canals.
In simple words: Add the two parallel sides, take the average by halving, then multiply by the gap between them.
Example: a=10, b=6, h=4 → A = ½ × 16 × 4 = 32 cm²
C = 2 × π × r
| C | distance around the circle |
|---|---|
| π | pi ≈ 3.14 or 22/7 |
| r | radius |
Where it is used: To find the boundary length of a circle — wheels, rings, circular tracks and pipes.
In simple words: It's the distance you'd walk all the way around a round shape. About 6.28 times the radius.
Example: r = 7 → C = 2 × 22/7 × 7 = 44 cm
A = π × r²
| π | pi ≈ 3.14 or 22/7 |
|---|---|
| r | radius (centre to edge) |
Where it is used: To find the flat space inside a circle — round fields, plates, pizzas, circular halls.
In simple words: Measure from the middle to the edge, multiply that by itself, then by about 3.14.
Example: r = 7 → A = 22/7 × 7 × 7 = 154 cm²
d = 2 × r
| d | diameter (edge to edge through centre) |
|---|---|
| r | radius |
Where it is used: To relate the full width of a circle to its radius — used everywhere circles appear.
In simple words: The diameter is the full width across the middle, which is just twice the radius.
Example: r = 7 → d = 2 × 7 = 14 cm
A = ½ × π × r²
| r | radius |
|---|
Where it is used: To find the area of a half-circle — arches, half-round windows, protractors.
In simple words: Half a circle has half the area of the full circle.
Example: r = 7 → A = ½ × 22/7 × 49 = 77 cm²
Arc = (θ ÷ 360°) × 2πr
| θ | the central angle (degrees) |
|---|---|
| r | radius |
Where it is used: To find the length of part of a circle's edge — curved roads, pie-chart edges, gears.
In simple words: An arc is a slice of the full circle's boundary — its share of 360° of the circumference.
Example: θ=90°, r=7 → (90/360)×44 = 11 cm
A = (θ ÷ 360°) × πr²
| θ | the central angle (degrees) |
|---|---|
| r | radius |
Where it is used: To find the area of a 'pizza-slice' part of a circle — pie charts, fan-shaped fields.
In simple words: A sector is a slice of the circle — its share of 360° of the whole area.
Example: θ=90°, r=7 → (90/360)×154 = 38.5 cm²
P = πr + 2r = r(π + 2)
| r | radius |
|---|
Where it is used: To find the boundary of a half-circle — the curved arc plus the straight diameter.
In simple words: Add the curved half-edge and the flat diameter across the bottom.
Example: r=7 → P = 7(22/7+2) = 36 cm
A = ¼ × π × r²
| r | radius |
|---|
Where it is used: To find the area of a quarter-circle.
In simple words: A quadrant is one-fourth of a circle, so it has a quarter of the area.
Example: r=14 → ¼×22/7×196 = 154 cm²
A = π(R² − r²)
| R | outer radius |
|---|---|
| r | inner radius |
Where it is used: To find the area between two concentric circles — washers, tracks, pipe cross-sections.
In simple words: Subtract the small circle's area from the big one to get the ring in between.
Example: R=7, r=5 → 22/7×(49−25) ≈ 75.4 cm²
Chord = 2√(r² − d²)
| r | radius |
|---|---|
| d | distance of chord from centre |
Where it is used: To find the length of a chord from the radius and its distance from the centre.
In simple words: Uses Pythagoras between the radius and the chord's distance from the middle.
Example: r=5, d=3 → 2√(25−9) = 8
A = C² ÷ (4π)
| C | circumference |
|---|
Where it is used: To find a circle's area when only the distance around it is known.
In simple words: Skip finding the radius — get the area straight from the boundary length.
Example: C=44 → 44²/(4×22/7) = 154 cm²
A = ½ × d × (h₁ + h₂)
| d | a diagonal |
|---|---|
| h₁, h₂ | perpendiculars to it from the other corners |
Where it is used: To find the area of any four-sided figure by splitting it along a diagonal.
In simple words: Cut the shape into two triangles along a diagonal and add their areas.
Example: d=10, h₁=4, h₂=6 → ½×10×10 = 50 cm²
P = n × s
| n | number of sides |
|---|---|
| s | length of a side |
Where it is used: To find the boundary of a regular polygon with equal sides.
In simple words: All sides are equal, so multiply one side by the number of sides.
Example: Regular hexagon, s=5 → P = 30 cm
Interior angle = (n − 2) × 180° ÷ n
| n | number of sides |
|---|
Where it is used: To find one inside angle of a regular polygon — tiling and design.
In simple words: Share the total interior angle equally among all the corners.
Example: Hexagon → (4×180)/6 = 120°
Exterior angle = 360° ÷ n
| n | number of sides |
|---|
Where it is used: To find the exterior angle of a regular polygon; also gives n if the angle is known.
In simple words: The outside angles always add to 360°, shared equally among the corners.
Example: Hexagon → 360/6 = 60°
A = (L+2w)(B+2w) − L×B
| L, B | inner length and breadth |
|---|---|
| w | width of the path |
Where it is used: To find the area of a uniform path or border around a rectangular field or garden.
In simple words: Find the big outer rectangle's area and subtract the inner rectangle to leave the path.
Example: L=20,B=10,w=1 → 22×12 − 200 = 64 m²