Geometry - Solved Questions and Answers

Last Updated : 23 Jul, 2025

Geometry is the branch of mathematics that deals with the properties, measurements, and relationships of points, lines, angles, surfaces, and solids in space.

Geometry questions and answers are provided below for you to learn and practice.

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.