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Grade 7

Plane figures practice: sectors, transformations and constructions

For sectors, ask what fraction of the whole circle you have. For constructions, understand why each step works.

Basic, Standard, Advanced: Grade 7 · Term 3

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Arc length and sector area

Formulas (radius rr, central angle a∘a^\circ)
ℓ=2πr×a360,S=πr2×a360=12ℓr\ell = 2\pi r \times \frac{a}{360},\qquad S = \pi r^2 \times \frac{a}{360} = \frac{1}{2}\ell r

Example: radius 66 cm and angle 120∘120^\circ give arc length 4π4\pi cm and area 12π12\pi cm2^2.

Shaded areas

Write the shaded region as sums and differences of squares, sectors and triangles. Leave π\pi in the answer, for example 32π−6432\pi - 64.

Transformations

  • Translation: every point moves the same distance in the same direction.
  • Reflection in the yy-axis changes the sign of the xx-coordinate.
  • Rotation about the origin: 180∘180^\circ takes (a,b)(a, b) to (−a,−b)(-a, -b); 90∘90^\circ anticlockwise takes (a,b)(a, b) to (−b,a)(-b, a).

Basic constructions

  • Perpendicular bisector: equal circles centred at both ends; join the two intersections. It is the set of points equidistant from both ends.
  • Angle bisector: a circle centred at the vertex meets the sides; equal circles from those points meet on the bisector. It is the set of points equidistant from both sides.
  • Perpendicular from a point: a circle centred at the point meets the line twice; equal circles from those points give a second point on the perpendicular.