Area, Perimeter, Circumference Word Problems

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

Hemi is creating a Maori-inspired artwork on a circular canvas. The outer edge of the canvas has a circumference of 188.4cm. He leaves a 5cm border around the edge unpainted. What is the area of the painted region?

Answer explanation

Radius of canvas

= Circumference ÷ (2π) = 188.4 ÷ (2π) = 30 cm

Area of canvas

= πr² = π × 30² ≈ 2827.43 cm²

Radius of painted area

= 30 - 5 = 25 cm

Area of painted region

= π × 25² ≈ 1963.5 cm²

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

Find the perimeter of the figure.

• 10 ft

• 14 ft

• 20 ft

• 18 ft

The radius of a cookie is 5 inches. What is the cookie’s

circumference?

• 10 in

• 2.5 in

• 15.7 in

• 31.4 in