Geometry is a branch of mathematics that deals with the properties and relationships of shapes, sizes, and spatial figures. It is an essential topic in the 11+ exam as it helps students develop spatial awareness, logical reasoning, and problem-solving skills.

In geometry, students learn about various shapes, including:

  • 2D Shapes: These are flat shapes that have two dimensions (length and width). Examples include triangles, squares, rectangles, circles, and polygons.
  • 3D Shapes: These have three dimensions (length, width, and height). Examples include cubes, spheres, cylinders, cones, and pyramids.

Key Concepts in Geometry

1. Properties of Shapes

Each shape has specific properties, including the number of sides, angles, and symmetry. For example:

  • A triangle has three sides and three angles.
  • A square has four equal sides and four right angles.

2. Perimeter and Area

  • Perimeter is the total distance around a 2D shape. It is found by adding the lengths of all sides.
  • Area measures the space inside a shape. Different formulas apply to different shapes:
  • Rectangle: \text{Area} = \text{length} \times \text{width}
  • Triangle: \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
  • Circle: \text{Area} = \pi r^2 (where r is the radius)

3. Volume and Surface Area

  • Volume measures the space inside a 3D shape. Different formulas apply to different shapes:
  • Cube: \text{Volume} = \text{side}^3
  • Cylinder: \text{Volume} = \pi r^2 h (where h is the height)
  • Surface Area measures the total area of the surface of a 3D shape.

4. Angles

  • Angles are formed by two rays originating from a common point (the vertex). Common types of angles include:
  • Acute: less than 90°
  • Right: exactly 90°
  • Obtuse: more than 90° but less than 180°
  • Straight: exactly 180°

5. Transformations

Transformations involve moving shapes in different ways, including translations (slides), rotations (turns), and reflections (flips).

Practice Questions on Geometry

Easy Level

  1. What is the perimeter of a rectangle with a length of 5 cm and a width of 3 cm?
  2. Calculate the area of a square with a side length of 4 cm.
  3. What is the area of a triangle with a base of 6 cm and a height of 3 cm?
  4. How many sides does a hexagon have?
  5. What is the name of a shape with four equal sides and four right angles?
  6. Calculate the area of a circle with a radius of 3 cm (use \pi \approx 3.14).
  7. What is the sum of the angles in a triangle?
  8. Identify the type of angle that measures 120°.
  9. What is the perimeter of a square with a side length of 5 cm?
  10. How many degrees are in a straight angle?
  11. What is the volume of a cube with a side length of 2 cm?
  12. Calculate the area of a rectangle with a length of 8 cm and a width of 5 cm.
  13. How many degrees are in a right angle?
  14. What is the name of a 3D shape with a circular base and a point at the top?
  15. Identify the shape with three sides.
  16. Calculate the area of a rectangle with a length of 10 cm and a width of 2 cm.
  17. What type of triangle has two equal sides?
  18. What is the perimeter of a triangle with sides measuring 3 cm, 4 cm, and 5 cm?
  19. How many lines of symmetry does a square have?
  20. What is the volume of a cylinder with a radius of 3 cm and a height of 5 cm (use \pi \approx 3.14)?

Medium Level

  1. Calculate the area of a parallelogram with a base of 10 cm and a height of 5 cm.
  2. What is the circumference of a circle with a radius of 4 cm (use \pi \approx 3.14)?
  3. Find the volume of a rectangular prism with dimensions 4 cm, 5 cm, and 6 cm.
  4. What is the surface area of a cube with a side length of 3 cm?
  5. Calculate the area of a triangle with a base of 8 cm and a height of 4 cm.
  6. How many sides does a decagon have?
  7. What is the LCM of the angles in a quadrilateral?
  8. Calculate the perimeter of a rectangle with a length of 12 cm and a width of 5 cm.
  9. Find the measure of each angle in an equilateral triangle.
  10. How many faces does a cube have?
  11. What is the area of a circle with a diameter of 10 cm (use \pi \approx 3.14)?
  12. Identify the type of triangle with angles measuring 60°, 60°, and 60°.
  13. What is the sum of the interior angles of a hexagon?
  14. Calculate the surface area of a cylinder with a radius of 2 cm and a height of 5 cm (use \pi \approx 3.14).
  15. Find the area of a trapezium with bases of 10 cm and 6 cm, and a height of 4 cm.
  16. What is the volume of a cone with a radius of 3 cm and a height of 4 cm (use \pi \approx 3.14)?
  17. Identify the 3D shape that has one circular face and a vertex.
  18. What is the length of the diagonal of a rectangle with a length of 6 cm and a width of 8 cm?
  19. Calculate the area of a rhombus with diagonals measuring 10 cm and 6 cm.
  20. Find the angle between the hour and minute hands of a clock at 3:00.

Hard Level

  1. Calculate the area of a trapezium with bases of 12 cm and 8 cm, and a height of 5 cm.
  2. Find the volume of a sphere with a radius of 3 cm (use \pi \approx 3.14).
  3. What is the surface area of a rectangular prism with dimensions 3 cm, 4 cm, and 5 cm?
  4. Calculate the circumference of a circle with a diameter of 12 cm (use \pi \approx 3.14).
  5. Find the area of a triangle with sides measuring 7 cm, 8 cm, and 9 cm using Heron’s formula.
  6. What is the length of the diagonal of a square with a side length of 5 cm?
  7. Calculate the volume of a pyramid with a base area of 20 cm² and a height of 9 cm.
  8. What is the area of a sector of a circle with a radius of 10 cm and a central angle of 60° (use \pi \approx 3.14)?
  9. Find the angle measures in a right-angled triangle with sides measuring 6 cm and 8 cm.
  10. Calculate the area of a regular pentagon with a side length of 6 cm.
  11. What is the volume of a cylinder with a radius of 4 cm and a height of 10 cm (use \pi \approx 3.14)?
  12. Calculate the surface area of a cone with a radius of 3 cm and a slant height of 5 cm (use \pi \approx 3.14).
  13. Find the area of a kite with diagonals measuring 8 cm and 12 cm.
  14. What is the length of the arc in a circle with a radius of 5 cm and a central angle of 90° (use \pi \approx 3.14)?
  15. Calculate the area of a parallelogram with adjacent sides of length 10 cm and 6 cm, and an included angle of 60°.
  16. Find the volume of a prism with a triangular base with sides of 3 cm, 4 cm, and 5 cm, and a height of 10 cm.
  17. What is the angle of elevation from a point on the ground to the top of a tree that is 12 m tall if you are standing 16 m away from the base of the tree?
  18. Calculate the perimeter of a hexagon with a side length of 4 cm.
  19. Find the area of a sector of a circle with a radius of 7 cm and a central angle of 120° (use \pi \approx 3.14).
  20. Calculate the surface area of a sphere with a radius of 4 cm (use \pi \approx 3.14).

Answers and Explanations

Easy Level

  1. Perimeter of rectangle = 2 \times (5 + 3) = 16 \text{ cm}
  2. Area of square = 4 \times 4 = 16 \text{ cm}^2
  3. Area of triangle = \frac{1}{2} \times 6 \times 3 = 9 \text{ cm}^2
  4. Hexagon has 6 sides.
  5. A square.
  6. Area of circle = \pi r^2 = 3.14 \times 3^2 \approx 28.26 \text{ cm}^2
  7. The sum of angles in a triangle is 180°.
  8. Obtuse angle.
  9. Perimeter of square = 4 \times 5 = 20 \text{ cm}
  10. 180°.
  11. Volume of cube = 2^3 = 8 \text{ cm}^3
  12. Area of rectangle = 8 \times 5 = 40 \text{ cm}^2
  13. 90°.
  14. A cone.
  15. Triangle has 3 sides.
  16. Area of rectangle = 10 \times 2 = 20 \text{ cm}^2
  17. The triangle is isosceles.
  18. The perimeter of a triangle = 3 + 4 + 5 = 12 \text{ cm}
  19. 4 lines of symmetry.
  20. Volume of cylinder = \pi r^2 h = 3.14 \times 3^2 \times 5 \approx 141.3 \text{ cm}^3

Medium Level

  1. Area of parallelogram = \text{Base} \times \text{Height} = 10 \times 5 = 50 \text{ cm}^2
  2. Circumference = 2 \pi r = 2 \times 3.14 \times 4 \approx 25.12 \text{ cm}
  3. Volume = 4 \times 5 \times 6 = 120 \text{ cm}^3
  4. Surface area = 6^2 = 36 \text{ cm}^2
  5. Area = \frac{1}{2} \times 8 \times 4 = 16 \text{ cm}^2
  6. Decagon has 10 sides.
  7. The angles in a quadrilateral sum to 360°.
  8. Perimeter = 2(12 + 5) = 34 \text{ cm}
  9. Each angle = 60°.
  10. A cube has 6 faces.
  11. Area = \pi \left(\frac{10}{2}\right)^2 = \pi \times 25 = 78.5 \text{ cm}^2
  12. The triangle is equilateral.
  13. Sum of angles = (6 – 2) \times 180° = 720°
  14. Surface area = 2\pi r (r + h) = 2 \times 3.14 \times 2 (2 + 5) \approx 87.92 \text{ cm}^2
  15. Area = \frac{1}{2} \times (10 + 6) \times 4 = 32 \text{ cm}^2
  16. Volume = \frac{1}{3}\pi r^2 h = \frac{1}{3} \times 3.14 \times 3^2 \times 4 \approx 37.68 \text{ cm}^3
  17. A cylinder has a circular base.
  18. Diagonal = \sqrt{(6^2 + 8^2)} = \sqrt{100} = 10 \text{ cm}
  19. Area = \frac{1}{2} \times 10 \times 6 = 30 \text{ cm}^2
  20. Length of the arc = \frac{90}{360} \times 2 \pi r = \frac{1}{4} \times 2 \times 3.14 \times 5 = 3.93 \text{ cm}

Hard Level

  1. Area of trapezium = \frac{1}{2} \times (12 + 8) \times 5 = 50 \text{ cm}^2
  2. Volume of sphere = \frac{4}{3}\pi r^3 = \frac{4}{3} \times 3.14 \times 3^3 \approx 113.04 \text{ cm}^3
  3. Surface area = 2(3 \times 4 + 4 \times 5 + 5 \times 3) = 94 \text{ cm}^2
  4. Circumference = \pi d = 3.14 \times 12 \approx 37.68 \text{ cm}
  5. Area using Heron’s formula:
  • Semi-perimeter s = \frac{7+8+9}{2} = 12
  • Area = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \times 5 \times 4 \times 3} \approx 84 \text{ cm}^2
  1. Diagonal = s\sqrt{2} = 5\sqrt{2} \approx 7.07 \text{ cm}
  2. Volume of pyramid = \frac{1}{3} \times 20 \times 9 = 60 \text{ cm}^3
  3. Area of sector = \frac{60}{360} \times \pi r^2 = \frac{1}{6} \times 3.14 \times 10^2 \approx 52.36 \text{ cm}^2
  4. Using Pythagorean theorem: \text{hypotenuse} = \sqrt{6^2 + 8^2} = \sqrt{100} = 10 \text{ cm}
  5. Area = \frac{5}{4} \times \sqrt{5} \approx 14.72 \text{ cm}^2
  6. Volume = \pi r^2 h = 3.14 \times 4^2 \times 10 \approx 502.4 \text{ cm}^3
  7. Surface area = \pi r^2 + \pi r l = 3.14 \times 3^2 + 3.14 \times 3 \times 5 \approx 28.26 + 47.1 \approx 75.36 \text{ cm}^2
  8. Area of kite = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 8 \times 12 = 48 \text{ cm}^2
  9. Length of arc = \frac{90}{360} \times 2 \pi r = \frac{1}{4} \times 2 \times 3.14 \times 5 \approx 3.93 \text{ cm}
  10. Area = \text{Base} \times \text{Height} \sin(\theta) = 10 \times 6 \times \sin(60°) \approx 51.96 \text{ cm}^2
  11. Volume of triangular prism = \frac{1}{2} \times b \times h \times \text{height} = \frac{1}{2} \times 3 \times 4 \times 10 = 60 \text{ cm}^3
  12. Angle of elevation = \tan^{-1} \left(\frac{12}{16}\right) \approx 36.87°
  13. Perimeter = 6 \times 4 = 24 \text{ cm}
  14. Area of sector = \frac{120}{360} \times \pi r^2 = \frac{1}{3} \times 3.14 \times 7^2 \approx 51.73 \text{ cm}^2
  15. Surface area = 4 \pi r^2 = 4 \times 3.14 \times 4^2 \approx 201.06 \text{ cm}^2

These questions and answers provide a comprehensive overview of geometry concepts relevant to the 11+ exam, covering various difficulty levels and encouraging students to develop their understanding and problem-solving skills.