Geometry - Solved Questions and Answers

Last Updated : 26 Jun, 2025

Question 1: In a circle with a radius of 10 cm, find the length of a chord that is 6 cm from the center.

Solution:

Let the radius r = 10cm and the distance from the center to the chord d = 6.
Use the formula for the chord length:
L = 2\sqrt{r^2 - d^2}
L = 2\sqrt{10^2 - 6^2}
= 2\sqrt{100 - 36}
= 2√64
= 2 × 8
= 16 cm

Question 2: Find the sum of the interior angles of a polygon with 8 sides.

Solution:

Sum of interior angles of an n-sided polygon = (n−2) × 180^∘
Sum = (8 − 2) × 180 = 6 × 180 = 1080^∘

Question 3: Find the measure of an angle if five times its complement is 10° less than twice its supplement.

Solution:

According to the problem:
5 × (90^∘− x) = 2 × (180^∘− x) − 10^∘
Now, solve the equation:
450^∘− 5x = 360^∘− 2x − 10^∘
450^∘− 350^∘= −2x + 5x
100^∘ = 3x
x = 33.33^∘

Question 4: In a triangle ΔXYZ, if 3∠X = 4∠Y = 5∠Z3, then find the value of ∠X.

Solution:

Let: 3∠X = 4∠Y = 5∠Z = k
From this, we can express the angles as:
∠X = k/3
∠Y = k/4
∠Z = k/5
The sum of the angles in a triangle is 180^∘:
∠X + ∠Y+ ∠Z = 180^∘
Substituting the expressions for ∠X, ∠Y, and ∠Z:
k/3 + k/4 + k5 = 180^∘
20k + 15k + 12k/60 = 180^∘
47k = 10800
k = 229.79^∘
∠X = k/3 = 229.79/3 = 76.6.

Question 5: What is the formula for calculating the volume of a cylinder, and how do you apply it to a cylinder with a radius of 7 cm and a height of 10 cm?

Solution:

To calculate the volume of a cylinder, you can use the formula:
Volume = πr²h
r = 7
h =10
volume = π(7 × 7) × 10
490π cm³

Question 6: A triangle has sides of lengths 12 cm, 12 cm, and 9 cm. Calculate the area of the triangle.

Solution:

A triangle has a base b=10 and two equal sides of 12 cm each.
we can use Heron's formula.
The formula for the area of a triangle with sides of lengths a, b, and c is:
A=√s(s−a)(s−b)(s−c)
Where:
s = a + b + c/2

12+12+9/2=33/2=16.5cm
A=√16.5(16.5−12)(16.5−12)(16.5−9)
A = √16.5 × 4.5 × 4.5 × 7.5
16.5 × 4.5 = 74.25
74.25 × 4.5 = 333.125
333.125 × 7.5 = 2498.4375
Now, take the square root:
A = √2498.4375 ≈ 49.98cm2
The area of the triangle is approximately 50 cm².

Question 7: The distance between the centers of two circles with radii 8 cm and 5 cm is 20 cm. What is the length of the traverse common tangent to the circles?

Solution:

Length of traverse common tangent = √[(Distance between their centres)²-(r₁ + r₂)²]
= √[(20)²- (8 + 5)²]
= √(400 - 169)
= √ (231)
= 15.2 cm

Question 8: If each interior angle of a regular polygon is 120∘120^\circ120∘, what is the number of sides of the polygon?

Solution:

Interior angle = 120^∘
Exterior angle = 180^∘− 120^∘= 60^∘
Number of sides of polygon = 360^∘/exterior angle = 360^∘/60^∘= 6
The number of sides of the polygon is 6.

To learn about Geometric Shapes, you can refer to Geometric Shapes in Maths.

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Geometry - Solved Questions and Answers

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