If S represents the area of a square inscribed in a circle and T represents the area of an equilateral triangle inscribed in the same circle, then which of the following is correct ?
- (1)3√3 T = 8 S
- (2)3√3 T = 4 S
- (3)4 T = 3√3 S
- (4)8 T = 3√3 S
Answer
Why
Correct — option (4), 8 T = 3√3 S.
Let the circle's radius be r.
Step 1 — the inscribed square.
Its diagonal is a diameter: diagonal = 2r
Side = 2r ÷ √2 = √2 r
S = (√2 r)² = 2r²
Step 2 — the inscribed equilateral triangle.
Its centre is the circle's centre, two-thirds of the way along each median from the vertex.
r = (2/3) × height = (2/3) × (√3/2) × side = side ÷ √3
Side = √3 r
T = (√3/4) × (√3 r)² = (√3/4) × 3r² = 3√3 r²/4
Step 3 — divide T by S.
T ÷ S = (3√3 r²/4) ÷ 2r² = 3√3/8
Step 4 — cross-multiply.
8 T = 3√3 S
Size check: 3√3 ≈ 5.196, so T ÷ S ≈ 0.65. T ≈ 1.30 r² is smaller than S = 2r², as the relation requires.
The idea to remember: in a circle of radius r, the inscribed square has side √2 r and the inscribed equilateral triangle has side √3 r.
Why the others are wrong
- (1)3√3 T = 8 S — Option (1) turns the correct fraction upside down: it gives T ÷ S = 8/(3√3) ≈ 1.54, which would make the triangle larger than the square.
The working gives T = 3√3 r²/4 ≈ 1.30 r² and S = 2r², so T ÷ S = 3√3/8 ≈ 0.65.
- (2)3√3 T = 4 S — Option (2) gives T ÷ S = 4/(3√3) ≈ 0.77, above the correct 3√3/8 ≈ 0.65.
Test it with the areas: 3√3 T = 3√3 × 3√3 r²/4 = 27r²/4 = 6.75r², while 4 S = 8r². The two sides are not equal.
- (3)4 T = 3√3 S — Option (3) says T ÷ S = 3√3/4 ≈ 1.30. That is T divided by r², not by the square's area, since T = 3√3 r²/4.
The square's area is 2r², not r², so the ratio halves to 3√3/8 and the relation becomes 8 T = 3√3 S.
Concept
A polygon is inscribed in a circle when all its vertices lie on the circle; the circle's radius is then the polygon's circumradius.
Square: the diagonal is a diameter, so side = √2 r and area = 2r².
Equilateral triangle: the centre lies two-thirds of the way along each median from the vertex, so r = side ÷ √3. Side = √3 r and area = 3√3 r²/4.
In one circle the areas stand in the order triangle < square < circle: about 1.30 r², 2 r² and 3.14 r².
RPSC's 2023 syllabus lists "Perimeter and Area of Plane figures" under Basic Numeracy in Reasoning & Mental Ability.
Inscribed figures rest on circle theorems. An inscribed right angle, with its vertex and the ends of both arms on the circle, stands on a diameter. Each corner of an inscribed square is such a right angle, so the diagonal joining the two neighbouring corners is a diameter.
A regular hexagon inscribed in a circle has side equal to the radius, because it splits into six equilateral triangles; its area, 3√3 r²/2, is twice that of the inscribed equilateral triangle.
Key facts
- Square inscribed in a circle of radius r: side √2 r, area 2r².
- Equilateral triangle inscribed in a circle of radius r: side √3 r, area 3√3 r²/4.
- Area of an equilateral triangle of side a = (√3/4) a².
- T ÷ S = 3√3/8 ≈ 0.65, so 8 T = 3√3 S.
- Regular hexagon inscribed in a circle of radius r: side r, area 3√3 r²/2.
Every vertex of the square and the triangle lies on the circle.
Study next
Common traps
- Taking the square's side as r. The radius is half the diagonal, not the side, so the side is √2 r and the area 2r².
- Taking the triangle's side as 2r or r. An equilateral triangle in a circle of radius r has side √3 r.
- Putting the 8 on the wrong side when cross-multiplying. T ÷ S = 3√3/8 becomes 8 T = 3√3 S.
A question can inscribe two figures in one circle and ask for the relation between their areas, as this one does, or ask for one figure's area from the radius.
A question can also draw a circle inside a figure, or a figure around a circle, and compare the areas.
Related PYQs
The sides of a triangle are 5, 12 and 13 units. A rectangle is constructed, which is equal in area to the triangle. It has a width of 10 units, then the perimeter of this rectangle is :
- (1) 30 units
- (2) 26 units
- (3) 13 units
- (4) 40 units
Answer(2)
Same skill of finding and comparing areas of plane figures. There a rectangle of width 10 equals in area a triangle with sides 5, 12 and 13, and its perimeter is asked (RPSC's key: option (2), 26 units); here the areas of a square and an equilateral triangle inscribed in one circle are related.
Eight equidistant points lie on a circle. Using these points as vertices, right angled triangles are drawn such that one side of each triangle is diameter of the circle. The number of such possible right angled triangles is :
- (1) 8
- (2) 16
- (3) 20
- (4) 24
Answer(4)
Same geometry of figures inscribed in a circle. That question counts right-angled triangles on eight equidistant points of a circle with one side a diameter (RPSC's key: option (4), 24); it rests on a diameter making a right angle at the circle, the fact that also makes an inscribed square's diagonal a diameter.
Practice
- practice — not a real PYQ
A square is inscribed in a circle of radius 7 cm. What is the area of the square ?
- (a)49 cm²
- (b)98 cm²
- (c)196 cm²
- (d)154 cm²
Answer(2) — The diagonal is the diameter, 14 cm, and a square's area is diagonal² ÷ 2 = 196 ÷ 2 = 98 cm². Option (1) is r², option (3) is the area of the square of side 14 cm drawn around the circle, and option (4) is the circle's area, (22/7) × 49. - practice — not a real PYQ
What is the ratio of the area of an equilateral triangle inscribed in a circle to the area of an equilateral triangle circumscribed about the same circle ?
- (a)1 : 2
- (b)1 : 3
- (c)1 : 4
- (d)1 : √3
Answer(3) — The inscribed triangle has circumradius r, so side √3 r. The circumscribed triangle has inradius r; an equilateral triangle's inradius is half its circumradius, so its circumradius is 2r and its side 2√3 r. Sides in the ratio 1 : 2 give areas in the ratio 1 : 4.Option (1) is the ratio of the sides; options (2) and (4) match neither ratio.