Composite shapes with circles
Find the area of composite shapes formed by combining or removing rectangles, triangles and circular regions.
Worked examples
Area of a rectangle combined with a semicircle
Straightforward
Problem
A shape is formed by attaching a semicircle to the top of a rectangle that is cm wide and cm tall. Find the total area correct to 2 decimal places.
1
Find the area of the rectangle.
cm²
2
Find the radius of the semicircle. The diameter equals the width of the rectangle.
cm
3
Find the area of the semicircle using .
cm²
4
Add both areas for the total.
cm²
Answer
cm²
Area of a rectangle with a circular region removed
Moderate
Problem
A rectangular piece of wood is cm long and cm wide. A circular hole of radius cm is cut from the centre. Find the remaining area correct to 2 decimal places.
1
Find the area of the rectangle.
cm²
2
Find the area of the circular hole using .
cm²
3
Subtract the circle area from the rectangle area.
cm²
Answer
cm²
Area of a circular border
Challenging
Problem
A circular pond has a radius of m. A garden path of uniform width m is built around the outside. Find the area of the path correct to 2 decimal places.
1
Find the radius of the outer circle (pond radius plus path width).
m
2
Find the area of the outer circle.
m²
3
Find the area of the inner circle (the pond).
m²
4
Subtract to find the area of the path.
m²
Answer
m²
Practise
Q1·Straightforward
A shape is formed by attaching a semicircle to one of the longer sides of a rectangle that is cm wide and cm tall. Find the total area correct to 2 decimal places.
Explanation
Rectangle: cm²\
Semicircle: cm²\
Total cm²
Semicircle: cm²\
Total cm²
Q2·Straightforward
A shape is formed by attaching a semicircle to one short end of a rectangle that is cm long and cm wide. Find the total area correct to 2 decimal places.
Explanation
Rectangle: cm²\
Semicircle: cm²\
Total cm²
Semicircle: cm²\
Total cm²
Q3·Straightforward
A shape consists of a rectangle cm wide and cm tall with a semicircle attached to the top (along the cm side). Find the total area correct to 2 decimal places.
Explanation
Rectangle: cm²\
Semicircle: cm²\
Total cm²
Semicircle: cm²\
Total cm²
Q4·Straightforward
A doorway is modelled as a rectangle cm wide and cm tall with a semicircle attached to the top. Find the total area correct to 2 decimal places.
Explanation
Rectangle: cm²\
Semicircle: cm²\
Total cm²
Semicircle: cm²\
Total cm²
Q5·Moderate
A rectangular piece of metal is cm long and cm wide. A circular hole of radius cm is cut from the centre. Find the remaining area correct to 2 decimal places.
Explanation
Rectangle: cm²\
Circle: cm²\
Remaining area cm²
Circle: cm²\
Remaining area cm²
Q6·Moderate
A square tile has side length cm. A semicircle is cut from one side of the tile (the semicircle has diameter cm). Find the remaining area correct to 2 decimal places.
Explanation
Square: cm²\
Semicircle: cm²\
Remaining area cm²
Semicircle: cm²\
Remaining area cm²
Q7·Moderate
A rectangular sign is cm long and cm tall. Two circular holes, each with radius cm, are cut from the sign. Find the remaining area correct to 2 decimal places.
Explanation
Rectangle: cm²\
Two circles: cm²\
Remaining area cm²
Two circles: cm²\
Remaining area cm²
Q8·Moderate
A rectangle is cm long and cm wide. A semicircle with diameter cm is removed from one short end. Find the remaining area correct to 2 decimal places.
Explanation
Rectangle: cm²\
Semicircle: cm²\
Remaining area cm²
Semicircle: cm²\
Remaining area cm²
Q9·Challenging
A circular garden has a radius of m. A path of uniform width m is built around the outside of the garden. Find the area of the path correct to 2 decimal places.
Explanation
Outer circle: m²\
Garden: m²\
Path area m²
Garden: m²\
Path area m²
Q10·Challenging
A square piece of cardboard has side length cm. A circle is inscribed in the square, touching all four sides. Find the area of the cardboard that is outside the circle, correct to 2 decimal places.
Explanation
Square: cm²\
Circle: cm²\
Shaded area cm²
Circle: cm²\
Shaded area cm²
Q11·Challenging
A swimming pool is shaped like a rectangle m long and m wide, with a semicircle of radius m attached to each short end. A square safety platform with side m sits in the centre of the pool. Find the area of the water surface correct to 2 decimal places.
Explanation
Rectangle: m²\
Two semicircles one full circle: m²\
Platform: m²\
Water surface m²
Two semicircles one full circle: m²\
Platform: m²\
Water surface m²
Q12·Challenging
A triangular garden has base m and perpendicular height m. A circular pond of radius m is dug in the garden. Find the remaining garden area correct to 2 decimal places.
Explanation
Triangle: m²\
Pond: m²\
Remaining area m²
Pond: m²\
Remaining area m²