Branch of mathematics

Geometry

Plane figures and solids in space — areas, perimeters, angles and volumes.

Topics in this branch

Basic
Area of a rectangle
The area of a rectangle is the product of its sides: A = a · b. See where the formula comes from, how area differs from perimeter, how to find the area of a square, how to get a side back from the area, and why the answer is written in square centimetres.
Basic
Area and perimeter of a triangle
The area of a triangle is half the product of a base and its height: A = ½ · a · h. See where the half comes from, which height goes with which side, how to work out the perimeter, and why the right triangle is the easiest case of all.
Basic
Area and circumference of a circle
The area of a disc is A = πr², its circumference is C = 2πr. See what the number π actually is, how the radius relates to the diameter, why the answer nearly always has to be rounded, and how a circle differs from a disc.
Basic
Angles
Angles are measured in degrees: acute below 90°, right exactly 90°, obtuse above it. Meet the kinds of angle, complementary and supplementary pairs, and the single most useful rule in plane geometry — the angles of a triangle always add up to 180°.
Intermediate
The Pythagorean theorem
In a right triangle a² + b² = c². See why it works, how to find the hypotenuse and a leg, what Pythagorean triples are, and how the converse lets you check whether a corner really is square.
Intermediate
Volume of solids
Volume says how much fits inside a solid. See the formulas for a cuboid, a cylinder, a pyramid, a cone and a sphere, why a third appears in the pyramid and the cone, and where the cubic unit comes from.
Intermediate
Surface area of solids
Surface area is the area of a solid’s net — how much material it takes to cover it. See the formulas for a cuboid, a cylinder, a pyramid, a cone and a sphere, and why a cone and a pyramid are measured by the slant height rather than the height.
Intermediate
Volume units
Litres, millilitres, cubic centimetres and cubic metres — one quantity written several ways. See why volume units convert by a thousand rather than by ten, and where the identity 1 dm³ = 1 l comes from.

Branch formulas

Branch: Geometry

Area of a rectangle

  • Area of a rectangle

    A=a⋅bA = a \cdot b

    a and b are the lengths of two adjacent sides

  • Area of a square

    A=a2A = a^2

    a square is a rectangle with equal sides

  • Perimeter of a rectangle

    P=2a+2b=2(a+b)P = 2a + 2b = 2(a + b)

    the total length of all four sides

  • A side from the area

    a=Ab,b=Aaa = \frac{A}{b}, \quad b = \frac{A}{a}

    the area formula reversed — division instead of multiplication

  • Units of area

    1 m2=10 000 cm21\,\text{m}^2 = 10\,000\,\text{cm}^2

    the side grows 100×, the area 100 · 100×

Area and perimeter of a triangle

  • Area of a triangle

    A=12⋅a⋅hA = \frac{1}{2} \cdot a \cdot h

    a is the base, h is the height dropped onto that base

  • Perimeter of a triangle

    P=a+b+cP = a + b + c

    the sum of the three side lengths

  • Area of a right triangle

    A=12⋅a⋅bA = \frac{1}{2} \cdot a \cdot b

    the legs are each other’s base and height

  • Height from the area

    h=2Aah = \frac{2A}{a}

    the area formula, the other way round

  • The triangle inequality

    ∣a−b∣<c<a+b|a - b| < c < a + b

    not every three segments make a triangle

Area and circumference of a circle

  • Area of a circle

    A=πr2A = \pi r^2

    r is the radius

  • Circumference

    C=2πr=πdC = 2\pi r = \pi d

    d = 2r, so the two spellings mean the same thing

  • Diameter and radius

    d=2rd = 2r

    the diameter is twice the radius

  • The number π

    π=Cd≈3.14\pi = \frac{C}{d} \approx 3.14

    the same ratio for every circle — which is why it has a name

  • Radius from the area

    r=Aπr = \sqrt{\frac{A}{\pi}}

    the area formula, the other way round

Angles

  • Angle sum of a triangle

    α+β+γ=180∘\alpha + \beta + \gamma = 180^\circ

    always, for every triangle in the plane

  • Complementary angles

    α+β=90∘\alpha + \beta = 90^\circ

    together they make a right angle

  • Supplementary angles

    α+β=180∘\alpha + \beta = 180^\circ

    together they make a straight line

  • Acute angle

    0∘<α<90∘0^\circ < \alpha < 90^\circ

    a right angle is 90°, an obtuse one is 90° < α < 180°

  • The third angle of a triangle

    γ=180∘−α−β\gamma = 180^\circ - \alpha - \beta

    two angles are enough to find the third

The Pythagorean theorem

  • The Pythagorean theorem

    a2+b2=c2a^2 + b^2 = c^2

    a and b are the legs, c is the hypotenuse

  • The hypotenuse

    c=a2+b2c = \sqrt{a^2 + b^2}

    the longest side, opposite the right angle

  • A leg

    b=c2−a2b = \sqrt{c^2 - a^2}

    the theorem the other way round — subtraction, not addition

  • Diagonal of a square

    d=a2d = a\sqrt{2}

    the theorem applied to a triangle with legs a and a

  • The 3-4-5 triple

    32+42=523^2 + 4^2 = 5^2

    the next ones: 5-12-13, 8-15-17, 7-24-25

Volume of solids

  • Cuboid

    V=a⋅b⋅cV = a \cdot b \cdot c

    a, b, c are the three edges meeting at one vertex

  • Cube

    V=a3V = a^3

    a cuboid with all edges equal

  • Prism and cylinder

    V=Ab⋅HV = A_b \cdot H

    base area times height — one formula for both solids

  • Cylinder

    V=πr2hV = \pi r^2 h

    the base is a circle, so A_b = πr²

  • Pyramid and cone

    V=13⋅Ab⋅HV = \frac{1}{3} \cdot A_b \cdot H

    same base and height as the prism, but three times less inside

  • Cone

    V=13πr2hV = \frac{1}{3} \pi r^2 h

    one third of the cylinder on the same base and height

  • Sphere

    V=43πr3V = \frac{4}{3} \pi r^3

    the only formula here with the radius to the third power

Surface area of solids

  • The general rule

    A=Ab+AlA = A_b + A_l

    base area plus lateral area

  • Cuboid

    A=2(ab+bc+ac)A = 2(ab + bc + ac)

    three pairs of identical faces

  • Cube

    A=6a2A = 6a^2

    six identical squares

  • Cylinder

    A=2πr2+2πrh=2πr(r+h)A = 2\pi r^2 + 2\pi rh = 2\pi r(r + h)

    two circles and a rectangle with sides 2πr and h

  • Pyramid on a square base

    A=a2+2alA = a^2 + 2al

    l is the slant height of a face, not the height of the solid

  • Cone

    A=πr2+πrl=πr(r+l)A = \pi r^2 + \pi rl = \pi r(r + l)

    l is the slant height — from the apex to the rim of the base

  • Slant height of a cone

    l=r2+h2l = \sqrt{r^2 + h^2}

    from the Pythagorean theorem for the radius and the height

  • Sphere

    A=4πr2A = 4\pi r^2

    exactly four areas of a great circle

Volume units

  • Cubic centimetre

    1 cm3=1 ml1\,\text{cm}^3 = 1\,\text{ml}

    one volume, two names

  • A cubic decimetre is a litre

    1 dm3=1 l=1000 cm31\,\text{dm}^3 = 1\,\text{l} = 1000\,\text{cm}^3

    a cube of edge 10 cm

  • Litre and millilitre

    1 l=1000 ml1\,\text{l} = 1000\,\text{ml}

    the same thousand, written differently

  • Cubic metre

    1 m3=1000 dm3=1000 l1\,\text{m}^3 = 1000\,\text{dm}^3 = 1000\,\text{l}

    a cube of edge 1 m

  • Metres to centimetres

    1 m3=1 000 000 cm31\,\text{m}^3 = 1\,000\,000\,\text{cm}^3

    100³, because 1 m = 100 cm

  • The scaling rule

    k times the edge⇒k3 times the volumek \text{ times the edge} \Rightarrow k^3 \text{ times the volume}

    the length factor is raised to the third power

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Why we measure area in squares

Perimeter is counted in metres, area in square metres — and that single exponent is behind most of the mistakes people make in plane geometry. It is why 1 m² is not 100 cm², why two plots with identical area can need wildly different amounts of fencing, and why the Pythagorean theorem is really a statement about the areas of squares rather than the lengths of sides.

11 min