Year 7 Maths: Bases of Three-Dimensional Figures

Introduction

In geometry, three-dimensional figures (3D shapes) have depth in addition to length and width. Examples of common 3D shapes include cubes, cylinders, pyramids, and cones. Each of these shapes has a base, which is an essential part of the figure and plays a key role in calculating properties such as volume and surface area.

The base of a 3D figure is typically a two-dimensional shape. For example, the base of a cylinder is a circle, while the base of a pyramid could be a triangle, square, or another polygon. Understanding how to identify and work with the base of a three-dimensional shape is a foundational concept in geometry, especially when calculating the volume and surface area of these figures.

Common 3D Shapes and Their Bases

• Cube: A cube has six square faces, any of which can be considered the base.
• Cylinder: A cylinder has two identical circular bases connected by a curved surface.
• Cone: A cone has a circular base and a vertex, forming a pointed shape.
• Pyramid: A pyramid has a polygonal base (such as a triangle or square) with triangular faces converging at a single point (apex).
• Prism: A prism has two identical polygonal bases connected by rectangular faces.

Key Concepts

1. Identifying the base: The base is usually the shape on which the figure stands or the identical shape that appears on both ends of the figure.
2. Properties of the base: The shape of the base determines the formula used to calculate the volume and surface area.
3. Perpendicular height: This is the distance from the base to the top or apex of the figure.

Easy Level Questions (20 Questions)

1. What is the shape of the base of a cube?
2. Identify the base of a cylinder.
3. What shape is the base of a cone?
4. If a pyramid has a triangular base, what is the shape of the base?
5. How many circular bases does a cylinder have?
6. What is the shape of the base of a square pyramid?
7. If a prism has a rectangular base, what is the shape of the base?
8. Name the base of a triangular prism.
9. What is the base of a cone?
10. How many bases does a cube have?
11. Identify the shape of the base of a hexagonal prism.
12. What is the shape of the base of a square-based pyramid?
13. How many sides does the base of a pentagonal prism have?
14. What type of base does a cylinder have?
15. What shape is the base of a cuboid?
16. Identify the shape of the base of a triangular pyramid.
17. What shape is the base of a rectangular prism?
18. How many triangular faces does a triangular pyramid have, excluding the base?
19. Identify the base of a circular cone.
20. If a prism has a pentagonal base, how many sides does the base have?

Medium Level Questions (20 Questions)

1. What is the area of the base of a square pyramid with a side length of ( 4 \, \text{cm} )?
2. Find the perimeter of the base of a rectangular prism with a base of dimensions ( 5 \, \text{cm} \times 3 \, \text{cm} ).
3. Calculate the area of the base of a cylinder with a radius of ( 7 \, \text{cm} ).
4. What is the shape of the base of a hexagonal pyramid?
5. Find the area of the base of a triangular prism with a base height of ( 6 \, \text{cm} ) and base width of ( 8 \, \text{cm} ).
6. Identify the number of edges on the base of a hexagonal prism.
7. What is the shape of the base of a rectangular-based pyramid?
8. Calculate the area of the base of a cone with a radius of ( 5 \, \text{cm} ).
9. Find the perimeter of the base of a pentagonal prism where each side of the pentagon is ( 4 \, \text{cm} ).
10. What is the total number of edges around the base of a triangular pyramid?
11. If a cylinder has a radius of ( 10 \, \text{cm} ), find the circumference of the base.
12. What is the volume of a cube with a base edge of ( 3 \, \text{cm} )?
13. Calculate the area of the base of a rectangular prism with a base of dimensions ( 4 \, \text{cm} \times 6 \, \text{cm} ).
14. How many vertices are on the base of a triangular pyramid?
15. If a prism has a hexagonal base, how many faces does the base have?
16. Find the area of the base of a square prism with side lengths of ( 9 \, \text{cm} ).
17. What is the shape of the base of a cylinder?
18. Calculate the area of the base of a triangular pyramid with a base height of ( 5 \, \text{cm} ) and base width of ( 6 \, \text{cm} ).
19. Find the number of vertices on the base of a square-based pyramid.
20. What is the area of the base of a cylinder with a diameter of ( 12 \, \text{cm} )?

Hard Level Questions (20 Questions)

1. Find the volume of a cylinder with a base radius of ( 6 \, \text{cm} ) and a height of ( 10 \, \text{cm} ).
2. Calculate the surface area of a cube with a base edge of ( 5 \, \text{cm} ).
3. If a hexagonal pyramid has a base area of ( 50 \, \text{cm}^2 ), what is the volume of the pyramid with a height of ( 12 \, \text{cm} )?
4. Find the surface area of a cone with a base radius of ( 7 \, \text{cm} ) and a slant height of ( 10 \, \text{cm} ).
5. A square-based pyramid has a base area of ( 36 \, \text{cm}^2 ). If its height is ( 8 \, \text{cm} ), what is the volume?
6. Calculate the lateral surface area of a cylinder with a base radius of ( 8 \, \text{cm} ) and height ( 15 \, \text{cm} ).
7. What is the surface area of a triangular prism with a base height of ( 4 \, \text{cm} ), base width of ( 6 \, \text{cm} ), and height of ( 10 \, \text{cm} )?
8. Find the total surface area of a pentagonal prism with a base side length of ( 5 \, \text{cm} ) and height ( 10 \, \text{cm} ).
9. Calculate the volume of a rectangular prism with a base of dimensions ( 4 \, \text{cm} \times 6 \, \text{cm} ) and a height of ( 8 \, \text{cm} ).
10. If a cube has a base edge of ( 7 \, \text{cm} ), find the total surface area.
11. Find the lateral surface area of a cone with a base radius of ( 9 \, \text{cm} ) and slant height of ( 12 \, \text{cm} ).
12. A triangular prism has a base area of ( 24 \, \text{cm}^2 ) and a height of ( 12 \, \text{cm} ). What is the volume?
13. Find the total surface area of a cube with a base edge of ( 6 \, \text{cm} ).
14. Calculate the volume of a cone with a base radius of ( 4 \, \text{cm} ) and a height of ( 9 \, \text{cm} ).
15. Find the surface area of a cylinder with a base radius of ( 10 \, \text{cm} ) and height of ( 20 \, \text{cm} ).
16. Calculate the volume of a hexagonal pyramid with a base area of ( 48 \, \text{cm}^2 ) and height ( 14 \, \text{cm} ).
17. Find the lateral surface area of a square-based pyramid with a base edge of ( 6 \, \text{cm} ) and slant height of ( 10 \, \text{cm} ).
18. If a triangular prism has a base height of ( 8 \, \text{cm} ) and base width of ( 5 \, \text{cm} ), calculate its base area and volume if the height is ( 15 \, \text{cm} ).
19. Calculate the surface area of a rectangular prism with a base of ( 5 \, \text{cm} \times 8 \, \text{cm} ) and height ( 10 \, \text{cm} ).
20. A cone has a base radius of ( 6 \, \text{cm} ) and height of ( 12 \, \text{cm

} ). Find its volume.

Answers and Explanations

Easy Level Answers

1. Square
2. Circle
3. Circle
4. Triangle
5. Two
6. Square
7. Rectangle
8. Triangle
9. Circle
10. Six
11. Hexagon
12. Square
13. Five
14. Circle
15. Rectangle
16. Triangle
17. Rectangle
18. Three
19. Circle
20. Five

Medium Level Answers

1. Area of base ( = 4 \times 4 = 16 \, \text{cm}^2 )
2. Perimeter of base ( = 2(5 + 3) = 16 \, \text{cm} )
3. Area of base ( = \pi \times 7^2 = 153.94 \, \text{cm}^2 )
4. Hexagon
5. Area of base ( = \frac{1}{2} \times 6 \times 8 = 24 \, \text{cm}^2 )
6. Six edges
7. Rectangle
8. Area of base ( = \pi \times 5^2 = 78.54 \, \text{cm}^2 )
9. Perimeter of base ( = 5 \times 4 = 20 \, \text{cm} )
10. Three edges
11. Circumference of base ( = 2\pi \times 10 = 62.83 \, \text{cm} )
12. Volume ( = 3^3 = 27 \, \text{cm}^3 )
13. Area of base ( = 4 \times 6 = 24 \, \text{cm}^2 )
14. Three vertices
15. Six faces
16. Area of base ( = 9 \times 9 = 81 \, \text{cm}^2 )
17. Circle
18. Area of base ( = \frac{1}{2} \times 5 \times 6 = 15 \, \text{cm}^2 )
19. Four vertices
20. Area of base ( = \pi \times 6^2 = 113.1 \, \text{cm}^2 )

Hard Level Answers

1. Volume ( = \pi \times 6^2 \times 10 = 1130.97 \, \text{cm}^3 )
2. Surface area ( = 6 \times 5^2 = 150 \, \text{cm}^2 )
3. Volume ( = \frac{1}{3} \times 50 \times 12 = 200 \, \text{cm}^3 )
4. Surface area ( = \pi \times 7 \times 10 = 219.91 \, \text{cm}^2 )
5. Volume ( = \frac{1}{3} \times 36 \times 8 = 96 \, \text{cm}^3 )
6. Lateral surface area ( = 2\pi \times 8 \times 15 = 754 \, \text{cm}^2 )
7. Surface area ( = 2 \times \frac{1}{2} \times 6 \times 4 + 2 \times 10 \times 6 = 152 \, \text{cm}^2 )
8. Surface area ( = 5 \times 5 \times 10 + 2 \times \frac{5 \times 10}{2} = 250 + 50 = 300 \, \text{cm}^2 )
9. Volume ( = 4 \times 6 \times 8 = 192 \, \text{cm}^3 )
10. Surface area ( = 6 \times 7^2 = 294 \, \text{cm}^2 )
11. Lateral surface area ( = \pi \times 9 \times 12 = 339.29 \, \text{cm}^2 )
12. Volume ( = 24 \times 12 = 288 \, \text{cm}^3 )
13. Surface area ( = 6 \times 6^2 = 216 \, \text{cm}^2 )
14. Volume ( = \frac{1}{3} \times \pi \times 4^2 \times 9 = 150.8 \, \text{cm}^3 )
15. Surface area ( = 2\pi \times 10 \times (10 + 20) = 1884 \, \text{cm}^2 )
16. Volume ( = \frac{1}{3} \times 48 \times 14 = 224 \, \text{cm}^3 )
17. Lateral surface area ( = 4 \times 6 \times 10 = 120 \, \text{cm}^2 )
18. Base area ( = \frac{1}{2} \times 5 \times 8 = 20 \, \text{cm}^2 ), Volume ( = 20 \times 15 = 300 \, \text{cm}^3 )
19. Surface area ( = 2 \times 5 \times 8 + 2 \times 5 \times 10 + 2 \times 8 \times 10 = 340 \, \text{cm}^2 )
20. Volume ( = \frac{1}{3} \times \pi \times 6^2 \times 12 = 452.39 \, \text{cm}^3 )

This set of exercises is designed to help Key Stage 3 students build a solid understanding of three-dimensional figures, their bases, and calculations related to volume and surface area.