Geometry & Measurement formulas

All 20 formulas in this topic

Pythagorean Theorema2+b2=c2a^2 + b^2 = c^2Find the length of any side of a right triangle given the other two sides, or check whether a triangle is a right triangle.Area of a TriangleA=12bhA = \frac{1}{2}bhCalculates the area enclosed by a triangle given its base and perpendicular height; essential for geometry, SAT, and ACT math problems involving triangles.Heron's FormulaA=s(s−a)(s−b)(s−c),s=a+b+c2A = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \frac{a+b+c}{2}Heron's Formula finds the area of a triangle using only its three side lengths, making it useful when the height is unknown or hard to measure.Area of a CircleA=πr2A = \pi r^2The area enclosed by a circle, calculated using its radius; needed for geometry and real-world problems involving circles like pools, wheels, or pizza.Circumference of a CircleC=2πr=πdC = 2\pi r = \pi dThe circumference of a circle is the total distance around its perimeter, calculated using either the radius or diameter.Area of a RectangleA=ℓwA = \ell wMultiply the length and width of a rectangle to find the total area of flat space it covers, measured in square units.Area of a TrapezoidA=12(b1+b2)hA = \frac{1}{2}(b_1 + b_2)hCalculates the exact area of any trapezoid when you know the lengths of both parallel bases and the perpendicular height between them.Area of a ParallelogramA=bhA = bhFind the area of any parallelogram by multiplying its base by the perpendicular height—a formula that works even when sides aren't perpendicular.Area of a Regular PolygonA=12apA = \frac{1}{2}apFind the area of any regular polygon by multiplying half the apothem by the perimeter—the quickest route when you know those two measurements.Sum of Interior AnglesS=(n−2)⋅180∘S = (n - 2)\cdot 180^\circDetermines the sum of all interior angles in a polygon based on the number of sides; critical for solving angle problems in geometry.Volume of a CylinderV=πr2hV = \pi r^2 hCalculates the volume of a cylinder—the three-dimensional space inside—used for any cylindrical container or object in math and engineering.Volume of a ConeV=13πr2hV = \frac{1}{3}\pi r^2 hCalculate the space inside a cone by multiplying one-third of the base area by the height; essential for geometry and SAT problems.Volume of a SphereV=43πr3V = \frac{4}{3}\pi r^3Calculates the amount of space inside a sphere using its radius; essential for finding capacity of spherical objects in geometry and physics.Surface Area of a SphereS=4πr2S = 4\pi r^2Surface area tells you the total area covering a sphere—useful when you need to wrap, paint, or analyze a ball, planet, or dome.Volume of a Rectangular PrismV=ℓwhV = \ell w hCalculates the space inside a rectangular box by multiplying length, width, and height; essential for storage, shipping, and construction problems.Volume of a PyramidV=13BhV = \frac{1}{3}BhFind the volume inside a pyramid by multiplying one-third of its base area by its height—the most direct way to measure any pyramid's capacity.Arc Lengths=rθ(θ in radians)s = r\theta \quad (\theta \text{ in radians})Arc Length tells you the distance along a circular curve between two points, defined by the radius and the central angle in radians.Area of a SectorA=12r2θA = \frac{1}{2}r^2\thetaCalculates the area of a circular sector enclosed by two radii and an arc when you know the radius and the central angle in radians.Similar Trianglesa1a2=b1b2=c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}When two triangles have the same shape but different sizes, their corresponding sides are proportional — use this to find missing side lengths.Special Right Triangles30-60-90:;1:3:245-45-90:;1:1:230\text{-}60\text{-}90: ; 1 : \sqrt{3} : 2 \qquad 45\text{-}45\text{-}90: ; 1 : 1 : \sqrt{2}Special Right Triangles give you the exact side ratios for 30-60-90 and 45-45-90 triangles, so you can find missing sides without the Pythagorean Theorem.

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