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Formulas: geometry, grade 9

Angles and sides in a triangle

  • The sum of the angles of a triangle
    α+β+γ=180∘\displaystyle \alpha+\beta+\gamma=180^{\circ}
  • An exterior angle δ\displaystyle \delta equals the sum of the two interior angles not next to it
    δ\displaystyle \delta =\displaystyle = α+β\displaystyle \alpha+\beta
  • The larger side faces the larger angle, and the other way round
    a>b ⇔ α>β\displaystyle a>b\ \Leftrightarrow\ \alpha>\beta

The right triangle

  • Pythagoras’ theorem – a\displaystyle a, b\displaystyle b the legs, c\displaystyle c the hypotenuse
    a2+b2\displaystyle a^{2}+b^{2} =\displaystyle = c2\displaystyle c^{2}
  • The leg a\displaystyle a facing a 30∘\displaystyle 30^{\circ} angle is half the hypotenuse c\displaystyle c
    a\displaystyle a =\displaystyle = 12c\displaystyle \frac{1}{2}c
  • The median mc\displaystyle m_c to the hypotenuse is half the hypotenuse c\displaystyle c
    mc\displaystyle m_c =\displaystyle = 12c\displaystyle \frac{1}{2}c

Angles between parallel lines

  • Corresponding angles are equal, and so are alternate angles
    α=β\displaystyle \alpha=\beta
  • Co-interior angles
    α+β=180∘\displaystyle \alpha+\beta=180^{\circ}

Angles in a quadrilateral

  • The sum of the angles of a quadrilateral
    α+β+γ+δ=360∘\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}
  • In a trapezoid: the two angles beside a leg
    α+β=180∘\displaystyle \alpha+\beta=180^{\circ}

Areas

  • The area of a triangle – ha\displaystyle h_a the height to the side a\displaystyle a
    S\displaystyle S =\displaystyle = a⋅ha2\displaystyle \frac{a\cdot h_a}{2}
  • The area of a parallelogram – ha\displaystyle h_a the height to the side a\displaystyle a
    S\displaystyle S =\displaystyle = a⋅ha\displaystyle a\cdot h_a
  • The area of a rectangle
    S\displaystyle S =\displaystyle = a⋅b\displaystyle a\cdot b
  • The area of a square
    S\displaystyle S =\displaystyle = a2\displaystyle a^{2}
  • The area of a trapezoid – a\displaystyle a, b\displaystyle b the bases, h\displaystyle h the height
    S\displaystyle S =\displaystyle = (a+b)⋅h2\displaystyle \frac{(a+b)\cdot h}{2}
  • The area of a rhombus or a kite – d1\displaystyle d_1, d2\displaystyle d_2 the diagonals
    S\displaystyle S =\displaystyle = d1⋅d22\displaystyle \frac{d_1\cdot d_2}{2}
  • The ratio of the areas of similar triangles – k\displaystyle k the ratio of similarity
    S1S2\displaystyle \frac{S_1}{S_2} =\displaystyle = k2\displaystyle k^{2}