Skip to content

Formulas: geometry, grade 7

Angles

  • The sum of angles – the ray BD\displaystyle BD runs inside ∠ABC\displaystyle \angle ABC
    ∠ABC\displaystyle \angle ABC =\displaystyle = ∠ABD+∠DBC\displaystyle \angle ABD+\angle DBC
  • The difference of angles
    ∠DBC\displaystyle \angle DBC =\displaystyle = ∠ABC−∠ABD\displaystyle \angle ABC-\angle ABD
  • The bisector BD\displaystyle BD cuts ∠ABC\displaystyle \angle ABC into two equal angles
    ∠ABD=∠DBC=∠ABC2\displaystyle \angle ABD=\angle DBC=\frac{\angle ABC}{2}
  • Adjacent angles
    α+β=180∘\displaystyle \alpha+\beta=180^{\circ}
  • Vertical angles are equal
    α=γ\displaystyle \alpha=\gamma
  • The angles around a point
    α+β+γ+δ=360∘\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}

The angles of a triangle and a quadrilateral

  • The angles of a triangle
    α+β+γ=180∘\displaystyle \alpha+\beta+\gamma=180^{\circ}
  • The two acute angles of a right triangle
    α+β=90∘\displaystyle \alpha+\beta=90^{\circ}
  • In an isosceles triangle the base angles are equal
    ∠B=∠C\displaystyle \angle B=\angle CAB=AC\displaystyle AB=AC
  • In an equilateral triangle all the angles are equal
    α=β=γ=60∘\displaystyle \alpha=\beta=\gamma=60^{\circ}
  • The angles of a quadrilateral
    ∠A+∠B+∠C+∠D=360∘\displaystyle \angle A+\angle B+\angle C+\angle D=360^{\circ}

The perimeter and area of a rectangle and a square

  • The area of a rectangle – a\displaystyle a, b\displaystyle b two sides that meet at a vertex
    S\displaystyle S =\displaystyle = a⋅b\displaystyle a\cdot b
  • The perimeter of a rectangle
    P\displaystyle P =\displaystyle = 2⋅(a+b)\displaystyle 2\cdot(a+b)
  • The area of a square – a\displaystyle a the side
    S\displaystyle S =\displaystyle = a2\displaystyle a^2
  • The perimeter of a square
    P\displaystyle P =\displaystyle = 4⋅a\displaystyle 4\cdot a

The areas of polygons

  • 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 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 right triangle – a\displaystyle a, b\displaystyle b the legs
    S\displaystyle S =\displaystyle = a⋅b2\displaystyle \frac{a\cdot b}{2}∠C=90∘\displaystyle \angle C=90^{\circ}
  • 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 box and the cube

  • The surface area of a box – a\displaystyle a the length, b\displaystyle b the width, c\displaystyle c the height
    S\displaystyle S =\displaystyle = 2(ab+bc+ac)\displaystyle 2(ab+bc+ac)
  • The surface area of a cube – a\displaystyle a the edge
    S\displaystyle S =\displaystyle = 6a2\displaystyle 6a^2
  • The volume of a box
    V\displaystyle V =\displaystyle = a⋅b⋅c\displaystyle a\cdot b\cdot c
  • The volume of a cube
    V\displaystyle V =\displaystyle = a3\displaystyle a^3