Learning Mathematics

Pythagorean Theorem

The most fundamental relation in Euclidean geometry. The area of the square on the hypotenuse equals the sum of the areas on the two shorter sides.

a² + b² = c²

Adjust the two legs and the construction updates so the triangle and all three outward-facing squares remain visible at once.

a 15.0

b 15.0

c = √(a² + b²) ≈ 21.21

a² + b² = c² ≈ 225.00 + 225.00 = 450.00

Binomial Square

Squaring a sum or difference is not just squaring each term separately. The cross term appears because the product picks up two matching mixed pieces, which is why 2ab sits in the middle.

(a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b²

A geometric area model makes the identity easy to remember: one large square splits into two smaller squares and two congruent rectangles. The same pattern powers expansion, factoring, and completing the square.

Circle Area

The area enclosed by a circle depends on the square of its radius. As the radius grows, each extra layer wraps all the way around, and the accumulation of those rings produces the familiar πr² formula.

A = πr²

Because the radius is squared, scaling is non-linear: a circle with twice the radius covers four times as much area.

Circle Equation

The standard equation of a circle describes all points that lie exactly r units from a fixed center. It is the coordinate version of the geometric definition of a circle.

(x - h)² + (y - k)² = r²

Writing the expression in completed-square form makes the center and radius immediately visible, while expanding it connects the graph back to quadratic algebra.

Compound Interest

Compound interest turns repeated percentage growth into exponential growth. Each compounding period earns interest on the original principal and on the interest that has already accumulated.

A = P(1 + r⁄n)^nt

The formula makes timing explicit: the same nominal rate leads to different totals depending on how often interest is added back into the balance.

Cone Surface Area

A cone’s total surface area combines two pieces: the base circle and the curved sheet that runs from the rim to the apex. That curved part depends on the slant height, not directly on the vertical height.

A = πr² + πrl = πr(r + l)

This split is useful in practice because many fabrication problems care about whether the base is included, while others only use the lateral surface.

Cone Volume

The volume of a cone is one third of the volume of a cylinder with the same base radius and height. The factor of one third reflects how each horizontal cross-section shrinks as it moves toward the tip.

V = ⅓πr²h

This comparison makes the formula easy to remember and links cone volume to the broader family of pyramid-like solids.

Cylinder Volume

The volume of a cylinder is the area of its circular base multiplied by its height. Because the cross-section stays constant from bottom to top, the formula is a direct application of prism volume.

V = πr²h

The squared radius matters: doubling r makes the base four times larger, so the whole cylinder holds four times as much volume if height stays fixed.

Difference of Squares

The difference of two perfect squares factors into the product of a sum and a difference. It is one of the fastest ways to simplify expressions and solve equations without full expansion.

a² - b² = (a - b)(a + b)

The identity is closely tied to symmetry: the middle terms cancel when the conjugates are multiplied, leaving only a² - b².

Exponential Decay

Exponential decay models processes that shrink by a constant percentage over equal time intervals. Radioactive decay, capacitor discharge, and some cooling models all follow this pattern.

N(t) = N₀e^-λt

The key idea is that the quantity disappears faster when there is more of it, and slower as less remains. That proportionality is what creates the exponential curve.

Linear Equation

A linear equation is an equation whose variables appear only to the first power and are not multiplied together. In two variables, its graph is a straight line.

ax + by = c

Linear models matter because they are the simplest non-trivial relationships and because many more complicated models can be approximated linearly near a point.

Slope-Intercept Form

Slope-intercept form writes a line in the easiest form for quick graphing. The constant term tells where the line crosses the vertical axis, and the slope tells how steeply it rises or falls.

y = mx + b

This form makes linear behavior easy to read at a glance and is the usual entry point for connecting algebra with coordinate geometry.

Surface Area of Sphere

The surface area of a sphere depends only on its radius and grows like r². The factor of four is one of the elegant surprises of classical geometry.

A = 4πr²

It says that the curved outer skin of the sphere equals four copies of the area of the largest circle that fits inside it.

Triangle Area

The area of a triangle is half the product of a chosen base and its perpendicular height. The factor of one half appears because two congruent copies make a full parallelogram.

A = 1⁄2 bh

This makes the formula flexible: the same triangle can be measured with different bases as long as the corresponding perpendicular height is used each time.

Trig Angle Sum Identity

Angle-sum identities explain how sine and cosine behave when two angles are added. They let complex rotations be decomposed into simpler pieces and sit underneath many standard trigonometric transformations.

sin(A + B) = sin A cos B + cos A sin B cos(A + B) = cos A cos B - sin A sin B

These formulas are especially useful in wave analysis, coordinate rotation, and derivations of double-angle and subtraction identities.

Quadratic Formula

The quadratic formula gives the exact solutions of any quadratic equation. The discriminant controls the geometry: it decides whether the parabola cuts the axis twice, touches it once, or misses it entirely.

x = (-b ± √(b² - 4ac))⁄2a

This is why the graph of a quadratic and the algebra of its roots are tightly linked. Moving the curve up or down changes the discriminant and the visible intersections at the same time.

Distance Formula

The distance formula is the Pythagorean theorem written in coordinate form. Horizontal and vertical changes become the two legs of a right triangle, and the segment between points becomes the hypotenuse.

d = √((x₂ - x₁)² + (y₂ - y₁)²)

This lets geometry and algebra speak the same language: every straight-line distance in the plane can be computed from coordinate differences.

Arc Length

Arc length measures how much of a circle’s boundary is covered by a given angle. In radians, the relationship is especially clean: length is just radius times angle.

s = rθ

That formula works because radians already measure angle by comparing arc length to radius. Double the radius or double the angle and the arc length doubles too.

Sector Area

A sector is a slice of a circle cut out by two radii and an arc. Its area is proportional to the central angle, so widening the angle increases the slice by the same fraction of the full disk.

A = 1⁄2 r²θ

This is the area counterpart to arc length. In radians, both formulas keep the geometry direct and compact.

Derivative as Slope

The derivative at a point is the instantaneous rate of change there. Geometrically, it is the slope of the tangent line that just touches the curve at that point.

f′(x) = lim_h→0 (f(x + h) - f(x))⁄h

As the point moves, the tangent rotates with the curve, making change visible instead of abstract. Steeper tangent means larger derivative magnitude.

Integral as Area Under Curve

A definite integral accumulates the signed area under a curve across an interval. It turns local values into a total quantity by adding infinitely many thin slices.

∫_a^b f(x) dx

This is why integrals appear in area, displacement, mass, charge, and probability. They gather small contributions into one whole answer.