Functional Skills Maths L2: 2D and 3D Shapes

Introduction to Shapes

Shapes are fundamental in mathematics and describe the form of objects. We primarily deal with 2-dimensional (2D) shapes, which are flat and have length and width, and 3-dimensional (3D) shapes, which have length, width, and height (or depth).

Understanding their properties, such as sides, angles, faces, edges, and vertices, is crucial for calculating measurements like perimeter, area, volume, and surface area.

2D Shapes

2D shapes are flat plane figures. Key measurements include perimeter (distance around the shape) and area (space enclosed by the shape).

Circle

A perfectly round shape where all points on the edge are equidistant from the center.

Properties:

Has a center point.
Radius (r): Distance from the center to any point on the circumference.
Diameter (d): Distance across the circle through the center (d = 2r).
Circumference (C): The distance around the circle.

Formulas (π ≈ 3.14):

Circumference (Perimeter): C = πd or C = 2πr
Area: A = πr²

Square

A quadrilateral with four equal sides and four right angles (90°).

Properties:

4 equal sides.
4 right angles (90° each).
Opposite sides are parallel.

Formulas:

Perimeter: P = 4s (where s is side length)
Area: A = s²

Rectangle

A quadrilateral with four right angles (90°) and opposite sides equal in length.

Properties:

4 right angles (90° each).
Opposite sides are equal and parallel.

Formulas:

Perimeter: P = 2(l + w) or P = 2l + 2w
Area: A = l × w

Triangle

A polygon with three sides and three angles.

Properties:

Sum of interior angles is 180°.
Types include equilateral, isosceles, scalene, right-angled.

Formulas:

Perimeter: P = a + b + c (sum of side lengths)
Area: A = ½ × base × height

Parallelogram

A quadrilateral with two pairs of parallel sides.

Properties:

Opposite sides are equal in length.
Opposite sides are parallel.
Opposite angles are equal.

Formulas:

Perimeter: P = 2(a + b) (sum of adjacent side lengths)
Area: A = base × height

Trapezium

A quadrilateral with at least one pair of parallel sides.

Properties:

At least one pair of parallel sides (called bases).
The non-parallel sides are called legs.

Formulas:

Perimeter: P = a + b + c + d (sum of all side lengths)
Area: A = ½ × (a + b) × h (where a and b are parallel sides, h is height)

2D Shape Examples: Perimeter and Area

Circle Perimeter Example: Calculate the circumference of a circle with a radius of 5 cm. (Use π ≈ 3.14)

Answer: 31.4 cm

Workings:

Formula for Circumference: C = 2πr

Given radius (r) = 5 cm, π ≈ 3.14

C = 2 × 3.14 × 5

C = 10 × 3.14

C = 31.4 cm

Circle Area Example: Calculate the area of a circle with a radius of 5 cm. (Use π ≈ 3.14)

Answer: 78.5 cm²

Workings:

Formula for Area: A = πr²

Given radius (r) = 5 cm, π ≈ 3.14

A = 3.14 × 5²

A = 3.14 × 25

A = 78.5 cm²

Square Perimeter Example: A square has a side length of 7 meters. What is its perimeter?

Answer: 28 meters

Workings:

Formula for Perimeter: P = 4s

Given side (s) = 7 meters

P = 4 × 7

P = 28 meters

Square Area Example: A square has a side length of 7 meters. What is its area?

Answer: 49 m²

Workings:

Formula for Area: A = s²

Given side (s) = 7 meters

A = 7²

A = 49 m²

Rectangle Perimeter Example: A rectangle has a length of 10 cm and a width of 4 cm. What is its perimeter?

Answer: 28 cm

Workings:

Formula for Perimeter: P = 2(l + w)

Given length (l) = 10 cm, width (w) = 4 cm

P = 2 × (10 + 4)

P = 2 × 14

P = 28 cm

Rectangle Area Example: A rectangle has a length of 10 cm and a width of 4 cm. What is its area?

Answer: 40 cm²

Workings:

Formula for Area: A = l × w

Given length (l) = 10 cm, width (w) = 4 cm

A = 10 × 4

A = 40 cm²

Triangle Perimeter Example: A triangle has side lengths of 6 cm, 8 cm, and 10 cm. What is its perimeter?

Answer: 24 cm

Workings:

Formula for Perimeter: P = a + b + c

Given side lengths: 6 cm, 8 cm, 10 cm

P = 6 + 8 + 10

P = 24 cm

Triangle Area Example: A triangle has a base of 12 cm and a height of 5 cm. What is its area?

Answer: 30 cm²

Workings:

Formula for Area: A = ½ × base × height

Given base = 12 cm, height = 5 cm

A = ½ × 12 × 5

A = 6 × 5

A = 30 cm²

Parallelogram Perimeter Example: A parallelogram has adjacent sides of 9 meters and 6 meters. What is its perimeter?

Answer: 30 meters

Workings:

Formula for Perimeter: P = 2(a + b)

Given adjacent sides: 9 meters and 6 meters

P = 2 × (9 + 6)

P = 2 × 15

P = 30 meters

Parallelogram Area Example: A parallelogram has a base of 15 cm and a height of 8 cm. What is its area?

Answer: 120 cm²

Workings:

Formula for Area: A = base × height

Given base = 15 cm, height = 8 cm

A = 15 × 8

A = 120 cm²

Trapezium Perimeter Example: A trapezium has parallel sides of 7 cm and 11 cm, and non-parallel sides of 5 cm and 6 cm. What is its perimeter?

Answer: 29 cm

Workings:

Formula for Perimeter: P = a + b + c + d

Given side lengths: 7 cm, 11 cm, 5 cm, 6 cm

P = 7 + 11 + 5 + 6

P = 29 cm

Trapezium Area Example: A trapezium has parallel sides of 8 meters and 12 meters, and a height of 5 meters. What is its area?

Answer: 50 m²

Workings:

Formula for Area: A = ½ × (8 + 12) × 5

Given parallel sides (a) = 8 m, (b) = 12 m, height (h) = 5 m

A = ½ × 20 × 5

A = 10 × 5

A = 50 m²

3D Shapes

3D shapes have three dimensions: length, width, and height. Key measurements include surface area (total area of all faces) and volume (space occupied by the shape).

Sphere

A perfectly round 3D object where all points on the surface are equidistant from the center.

Properties:

1 curved surface.
No edges or vertices.

Formulas (π ≈ 3.14):

Volume: V = (4/3)πr³
Surface Area: SA = 4πr²

Cone

A 3D shape with a circular base and a single vertex (apex).

Properties:

1 circular base.
1 curved surface.
1 vertex (apex).
1 circular edge.

Formulas (π ≈ 3.14):

Volume: V = (1/3)πr²h (where h is perpendicular height)
Surface Area: SA = πrl + πr² (where l is slant height)
(Slant height l = √(r² + h²) using Pythagoras)

Cylinder

A 3D shape with two identical circular bases and one curved surface.

Properties:

2 circular faces (bases).
1 curved surface.
2 circular edges.
No vertices.

Formulas (π ≈ 3.14):

Volume: V = πr²h
Surface Area: SA = 2πrh + 2πr²

Prism

A 3D shape with two identical polygonal bases and flat rectangular sides connecting corresponding edges of the bases. The name of the prism is based on the shape of its base (e.g., triangular prism, rectangular prism/cuboid).

Properties (General):

2 identical polygonal bases.
Rectangular side faces (for right prisms).
Number of faces, edges, vertices depends on the base shape.

Formulas (General for a right prism):

Volume: V = Area of base × height (or length)
Surface Area: SA = (2 × Area of base) + (Perimeter of base × height)

3D Shape Examples: Volume and Surface Area

Sphere Volume Example: Calculate the volume of a sphere with a radius of 4 cm. (Use π ≈ 3.14)

Answer: 267.95 cm³

Workings:

Formula for Volume: V = (4/3)πr³

Given radius (r) = 4 cm, π ≈ 3.14

V = (4/3) × 3.14 × 4³

V = (4/3) × 3.14 × 64

V = 1.333... × 3.14 × 64

V ≈ 267.95 cm³

Sphere Surface Area Example: Calculate the surface area of a sphere with a radius of 4 cm. (Use π ≈ 3.14)

Answer: 200.96 cm²

Workings:

Formula for Surface Area: SA = 4πr²

Given radius (r) = 4 cm, π ≈ 3.14

SA = 4 × 3.14 × 4²

SA = 4 × 3.14 × 16

SA = 200.96 cm²

Cone Volume Example: Calculate the volume of a cone with a radius of 3 cm and a perpendicular height of 7 cm. (Use π ≈ 3.14)

Answer: 65.94 cm³

Workings:

Formula for Volume: V = (1/3)πr²h

Given radius (r) = 3 cm, height (h) = 7 cm, π ≈ 3.14

V = (1/3) × 3.14 × 3² × 7

V = (1/3) × 3.14 × 9 × 7

V = 3.14 × 3 × 7

V = 3.14 × 21

V = 65.94 cm³

Cone Surface Area Example: Calculate the surface area of a cone with a radius of 3 cm and a perpendicular height of 4 cm. (Use π ≈ 3.14)

Answer: 75.36 cm²

Workings:

1. Calculate slant height (l) using Pythagoras: l = √(r² + h²)

l = √(3² + 4²) = √(9 + 16) = √25 = 5 cm

2. Formula for Surface Area: SA = πrl + πr²

Given radius (r) = 3 cm, slant height (l) = 5 cm, π ≈ 3.14

SA = (3.14 × 3 × 5) + (3.14 × 3²)

SA = (3.14 × 15) + (3.14 × 9)

SA = 47.1 + 28.26

SA = 75.36 cm²

Cylinder Volume Example: Calculate the volume of a cylinder with a radius of 2 meters and a height of 5 meters. (Use π ≈ 3.14)

Answer: 62.8 m³

Workings:

Formula for Volume: V = πr²h

Given radius (r) = 2 m, height (h) = 5 m, π ≈ 3.14

V = 3.14 × 2² × 5

V = 3.14 × 4 × 5

V = 3.14 × 20

V = 62.8 m³

Cylinder Surface Area Example: Calculate the surface area of a cylinder with a radius of 2 meters and a height of 5 meters. (Use π ≈ 3.14)

Answer: 87.92 m²

Workings:

Formula for Surface Area: SA = 2πrh + 2πr²

Given radius (r) = 2 m, height (h) = 5 m, π ≈ 3.14

SA = (2 × 3.14 × 2 × 5) + (2 × 3.14 × 2²)

SA = (2 × 3.14 × 10) + (2 × 3.14 × 4)

SA = 62.8 + 25.12

SA = 87.92 m²

Cuboid Volume Example: Calculate the volume of a cuboid with length 10 cm, width 4 cm, and height 5 cm.

Answer: 200 cm³

Workings:

Formula for Volume: V = length × width × height

Given length = 10 cm, width = 4 cm, height = 5 cm

V = 10 × 4 × 5

V = 40 × 5

V = 200 cm³

Cuboid Surface Area Example: Calculate the surface area of a cuboid with length 10 cm, width 4 cm, and height 5 cm.

Answer: 220 cm²

Workings:

Formula for Surface Area: SA = 2(lw + lh + wh)

Given length (l) = 10 cm, width (w) = 4 cm, height (h) = 5 cm

SA = 2((10 × 4) + (10 × 5) + (4 × 5))

SA = 2(40 + 50 + 20)

SA = 2(110)

SA = 220 cm²

Quick Quiz on Shapes

1. A 2D shape has 4 equal sides and 4 right angles. What is it?

Answer: Square

2. What is the formula for the area of a circle with radius 'r'? (Use π)

Answer: A = πr²

3. A 3D shape has one circular base and one vertex. What is it?

Answer: Cone

4. What is the general formula for the volume of a prism?

Answer: Volume = Area of base × height (or length)

5. What is the key difference between a rectangle and a square?

Answer: A square must have four equal sides, while a rectangle only needs to have equal opposite sides.

6. What is the formula for the volume of a cylinder with radius 'r' and height 'h'?

Answer: V = πr²h

7. How many flat faces does a cuboid (a rectangular prism) have?

Answer: 6 faces.

8. What is the formula for the surface area of a sphere with radius 'r'?

Answer: SA = 4πr²

9. A 2D shape has opposite sides that are parallel, but its angles are not necessarily right angles. What is it?

Answer: A parallelogram.

10. What is the perimeter of a triangle with side lengths of 5cm, 6cm, and 7cm?

Answer: 18cm

Workings:

Perimeter is the sum of all side lengths.
5cm + 6cm + 7cm = 18cm.

Exam Style Questions

1. A rectangular garden measures 15 meters by 8 meters. A circular pond with a radius of 3 meters is built in the center. Calculate the area of the garden *not* covered by the pond. (Use π ≈ 3.14)

Answer: 91.74 m²

Workings:

1. Calculate the area of the rectangular garden:

Area_garden = length × width = 15 m × 8 m = 120 m²

2. Calculate the area of the circular pond:

Area_pond = πr² = 3.14 × 3² = 3.14 × 9 = 28.26 m²

3. Subtract the pond area from the garden area:

Area_not_covered = Area_garden - Area_pond = 120 m² - 28.26 m² = 91.74 m²

2. A triangular piece of land has a base of 20 meters and a height of 15 meters. If fencing costs £12 per meter, how much will it cost to fence the entire perimeter of the land if the other two sides are 25 meters each?

Answer: £840

Workings:

1. Calculate the perimeter of the triangular land:

Perimeter = side1 + side2 + side3 = 20 m + 25 m + 25 m = 70 m

2. Calculate the total cost of fencing:

Cost = Perimeter × Cost per meter = 70 m × £12/m = £840

(Note: The height of 15m is not needed for perimeter calculation but would be for area.)

3. A rectangular swimming pool has a length of 25 meters and a width of 10 meters. The depth of the pool is uniformly 2 meters. How much water (in cubic meters) is needed to fill the pool?

Answer: 500 m³

Workings:

The pool is a cuboid (a rectangular prism).

Formula for Volume of a cuboid: V = length × width × height

Given length = 25 m, width = 10 m, height (depth) = 2 m

V = 25 × 10 × 2

V = 250 × 2

V = 500 m³

4. A school playground is in the shape of a parallelogram with a base of 30 meters and a perpendicular height of 18 meters. The school wants to cover the playground with a special safety surface that costs £25 per square meter. What is the total cost of covering the playground?

Answer: £13,500

Workings:

1. Calculate the area of the parallelogram playground:

Area = base × height = 30 m × 18 m = 540 m²

2. Calculate the total cost of covering the playground:

Cost = Area × Cost per square meter = 540 m² × £25/m² = £13,500

5. A cylindrical water tank has a radius of 2 meters and a height of 5 meters. What is the volume of the tank? (Use π ≈ 3.14)

Answer: 62.8 m³

Workings:

Formula for Volume of a cylinder: V = πr²h

Given radius (r) = 2 m, height (h) = 5 m, π ≈ 3.14

V = 3.14 × 2² × 5

V = 3.14 × 4 × 5

V = 3.14 × 20

V = 62.8 m³

Key Shape Facts to Remember

Perimeter is the distance around a 2D shape.
Area is the space inside a 2D shape.
Volume is the space occupied by a 3D shape.
Surface Area is the total area of all surfaces of a 3D shape.
Know the specific formulas for common shapes like circles, rectangles, triangles, cylinders, and cuboids.
For prisms, the volume is always (Area of Base) × Height.