SME
Grade 7 Math · Problem type
A circle in a square
Use this for “A circle of radius 5 is inside a square. What area is left in the corners?”
Diagram
s = 10 cm (side of the square)
Q = 100 cm² (area of the square)
L = 21.4602 cm² (area left in the corners)
This is an example. Type your own values to start a new problem.
r
Radius
example
s
Side of the square
calculated
Q
Area of the square
worked out (not typed)
A
Area of the circle
worked out (not typed)
L
Area left in the corners
worked out (not typed)
Formulas
s
=
2
×
r
(side
of
the
square
=
2
×
radius)
10 = 2 × 5
Q
=
s
2
(area
of
the
square
=
side
of
the
square²)
100
=
10
2
A
=
π
×
r
2
(area
of
the
circle
=
π
×
radius²)
25π
=
π
×
5
2
L
=
Q
−
A
(area
left
in
the
corners
=
area
of
the
square
−
area
of
the
circle)
21.4602 = 100 − 25π
Assumptions
•
The circle touches all four sides, so the square’s side is the diameter: 2r.
•
The area left in the corners is the square’s area minus the circle’s.
•
The corners are always about 21% of the square, whatever its size.
Step-by-step
WE KNOW
r = 5 cm (radius)
FIND
side of the square (s), area of the square (Q), area of the circle (A), area left in the corners (L)
Step 1 · Find side of the square (s)
Use s = 2 × r (side of the square = 2 × radius)
The circle touches both sides: the side is a diameter.
s
=
2
×
5
s
=
10
cm
Step 2 · Find area of the square (Q)
Use Q = s² (area of the square = side of the square²)
A
square’
s
area
is
its
side
times
itself.
Q
=
10
2
Q
=
100
cm²
Step 3 · Find area of the circle (A)
Use A = π × r² (area of the circle = π × radius²)
Square the radius, then multiply by π.
A
=
π
×
5
2
A
=
25
×
π
A
=
25π
cm²
(≈
78.54
cm²)
Step 4 · Find area left in the corners (L)
Use L = Q − A (area left in the corners = area of the square − area of the circle)
Take
the
circle’
s
area
from
the
square’
s
.
L
=
100
−
25π
L
≈
21.4602
cm²
Check
Put the numbers back in. Both sides should match.
10 = 2 × 5 ✓
100
=
10
2
✓
25π
=
π
×
5
2
✓
21.4602 ≈ 100 − 25π ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Related lessons
Circumference and area of circles
Main lesson
Circumference and wheels
Use this for “A 28-inch wheel turns 25 times. How far does the bike go?”
Circumference is proportional to diameter
Use this for “A circle has diameter 4. What is its circumference, and where is it on the graph?”
Circumference or area?
Use this for “Is it circumference or area?” before choosing the formula.