Geometry, Area Calculation

This topic focuses on methods for determining the size of a two-dimensional surface or region. Questions involve calculating the areas of various geometric shapes like triangles, quadrilaterals, circles, and more complex composite figures, often requiring application of specific formulas.

• The Parallelogram

  In the drawing, there is a parallelogram with its diagonals drawn and the midpoints of two of its sides connected to opposite vertices.
  Which area is larger: the shaded or the striped?

  

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation
  Sources:
  • Young Mathematician Olympiad, 2018-2019, Final, Grades 5-6 Question 7

• Difference of Square Areas

  Two squares are given as depicted in the drawing. The side length of the larger square is 9,
  the side length of the smaller square is 8. What is the difference between the area of the orange region and the area of the blue region?

  

  Topics:

  Arithmetic Geometry -> Area Calculation
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 3-4 Question 1

• Orange Star of David

  The area of the blue triangle is equal to 1. Calculate the area of the orange Star of David:

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage A, Grades 5-6 Question 3

• How Many Triangles - 2?

  How many triangles are in the picture?

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Stage B, Grades 5-6 Question 6

• Blue or Orange?

  In the diagram, there is a square ABCD and a parallelogram BCEF. Which area is larger: the blue or the orange?

  

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation
  Sources:
  • Young Mathematician Olympiad, 2019-2020, Final, Grades 5-6 Question 3

• Largest Perimeter

  A polygon with an area of 12 is drawn on grid paper, with all its sides passing through the grid lines. What is the largest possible perimeter of this polygon?

  Topics:

  Geometry -> Area Calculation Minimum and Maximum Problems / Optimization Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 3-4 Question 5

• The Orange Path

  In the illustration, there is an orange path surrounding a blue square. The area of the path is 44% of the area of the square.
  What is the width of the orange path as a percentage relative to the side length of the blue square?

  

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Word Problems Arithmetic -> Percentages
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 5-6 Question 3

• Area of the Shape

  In the image, the area of the semicircle is equal to 1. Find the area of the larger shape, given that all the curved lines in the image are quarter circles.

  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Circles
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 2

• Enlarging a Rectangle

  A rectangle is given in the plane. Is it possible that after each side of the rectangle is increased by 1 cm, the area increases by 1 square meter? Provide an example or prove that it is impossible.

  (If the rectangle is 1x5, it becomes 2x6 and no side can be 0)

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  Topics:

  Geometry -> Area Calculation Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Grossman Math Olympiad, 2017, Juniors Question 1

• Question

  Inside a square ABCD with side length 1, a point E is marked, and outside the square, a point F is marked, such that triangles ABE and DAF are equilateral. Calculate the area of the pentagon CBEFD.
  

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Symmetry Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Rotation
  Sources:
  • Beno Arbel Olympiad, 2017, Grade 8 Question 6