Area and Perimeter Word Problems

A triangular wall needs to be painted. The triangle is an equilateral triangle with 35 metre sides and the height of the wall is 30 metres. Nina wants to make a gold border around the wall. What is the length of the gold border that is needed?

Answer explanation

An equilateral triangle means that all 3 sides are equal. This means you need to add up all 3 sides to get the length of the gold border.

The garden has two sections: a triangle section for vegetables and a circular section for flowers, with the rest grass. The vegetables section is a right-angle with sides of 3.45m and 2.1m. The flower section is 5.7m from one side to the other. What is the total area of each section of the garden? If grass seed costs $1.77/m², how much will it cost to buy enough seed for the grass area?

Answer explanation

Vegetable Section (Triangle)

Area of triangle = (base × height) ÷ 2

Area = (3.45 × 2.1) ÷ 2 = 3.6225 m²

Flower Section (Circle)

Area of circle = π × r²

Where r is the radius (half the diameter) - Radius = 5.7 ÷ 2 = 2.85m

Area = π × 2.85² ≈ 25.5173 m²

Grass Section

subtract vegetable and flower areas from the total garden area.

TOTAL GARDEN AREA: 52.46m² (8.6 x 6.1)

VEGETABLE & FLOWER GARDENS:

3.6225 m² + 25.5173 m² = 29.1398m²

GRASS AREA = 52.46 - 29.1398 = 23.32m²

Grass Seed Cost = Grass area × Price per m²

         = 23.32m² × $1.77/m²  = 41.2764

         = $41.30

For a school event, circular pizzas with a radius of 12cm are ordered. If each student should get 50cm² of pizza, how many students can one pizza feed?

Answer explanation

Area of pizza = πr² = π × 12² =  452.39 cm²

Number of students = 452.39  ÷ 50 ≈ 9.05

= 9 students

In Hobbiton, hobbit houses have circular windows. A hobbit needs to replace a window which has a circumference of 4.8 metres. What is the area of the window?

Answer explanation

We need to find the radius first using the circumference formula: 

C = 2πr

4.8 = 2πr

4.8 = 2 × 3.14 × r

 r = 4.8 ÷ (2 × 3.14) 

 r = 0.764 metres

Calculate the area

A = πr²

A  = 3.14 × (0.764)²

       = 3.14 × 0.584 

The school is painting a mural on a wall shaped like a trapezium. The parallel sides of the wall are 8 metres and 12 metres long, and the height between them is 5 metres. How much paint will they need if 1L covers 10 square metres?

Answer explanation

Area of trapezium

= ½(a+b)h

= ½(8+12) × 5

= 50m²

Paint needed

= 50 ÷ 10

A church is commissioning a circular stained glass window with a smaller circular design in its centre. The outer circle has a radius of 2.4 metres, while the inner circle has an area that is 1/9 of the total window area. What is the radius of the inner circle?

Answer explanation

Area of outer circle

= πr² = π(2.4)² = 18.1 m² (rounded to one dp)

Area of inner circle

=The inner circle's area is 1/9 of the total area

   =  18.1 ÷ 9 = 2.01 m²

use the area formula for a circle to find the radius of the inner circle:

   2.01 = πr²

   r² = 2.01 ÷ π

   r = √(2.01 ÷ π) ≈ 0.8 m

A farmer notices a mysterious crop circle in his field. It's composed of two circles: a larger outer circle and a smaller inner circle. The area between the two circles is 200 square metres. If the radius of the larger circle is 10 metres, what is the radius of the smaller circle?

Answer explanation

 Area between circles

= π(R² - r²) = 200

=  π(10² - r²) = 200

= 100π - πr² = 200

=  πr² = 100π - 200

= r² = 100 - 200/π ≈ 36.34

A school is renovating a classroom with an unusual shape. The room consists of a rectangle measuring 85m by 62m, with a semicircle attached to one of the longer sides. What is the total area of the floor that needs to be covered with new tiles?

Answer explanation

Area of rectangle

= 85 m × 62 m = 527 m²

Calculate the area of the semicircle:

 The diameter of the semicircle is equal to the length of the rectangle 

   - Radius of semicircle = 85 m ÷ 2 = 42.5 m

   - Area of full circle = πr² = π(42.5)² ≈ 567 m²

   - Area of semicircle = 567 m² ÷ 2 ≈ 284 m²

Total area = Area of rectangle + Area of semicircle

         = 527 m² + 284 m² 

A new skateboard park is being designed in the shape of a trapezium. The parallel sides are 58m and 45m long, and the non-parallel sides are both 50m long. Inside the park, a circular area with a radius of 18 metres will be set aside for beginners. What percentage of the total park area is the beginners' circle?

Answer explanation

  Area of trapezium = 2575m²

  Area of circle = πr² = π × 16² =  804.2477 = 804.25m²

  Percentage = (804.25 ÷ 2575) × 100 = 31.233%

An artist is creating an installation using a square metal sheet. She cuts out a circle from the centre of the square. The area of the remaining metal (the square minus the circle) is 144cm². If the side length of the square is 14cm, what is the radius of the circular cut-out?

Answer explanation

Area of square = 14² = 196 cm²

  Area of circle =

πr² = 196 - 144  

     πr² = 52

     r² = 52 ÷ π ≈ 16.55

  Radius = 4.07 cm