Measuring Space: Perimeter and Area MCQs for UPSC Prelims
125 practice questions covering Measuring Space: Perimeter and Area, organised into 0 topics.
125 questions include a written explanation.
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Sample questions
Q1
easy
A farmer knows the four side lengths of a four-sided field and also the length of one diagonal. How should the area of the field be found?
AMultiply the two longest sides together
BAdd the four side lengths
CTake half the product of the two longest sides
DSplit the field along that diagonal into two triangles, find each area by Heron's formula and add them
Show answer and explanation
Correct answer: D - Split the field along that diagonal into two triangles, find each area by Heron's formula and add them
The diagonal cuts the field into two triangles, and each triangle then has all three of its sides known, which is exactly what Heron's formula needs; adding the two areas gives the whole field. Multiplying two sides works only for a rectangle. Adding the four sides gives the perimeter, a length, not an area. Half the product of two sides is not an area rule for a general quadrilateral.
Q2
easy
Two flat shapes are found to have exactly the same perimeter. What can be said about their areas?
AThey must be equal, because perimeter decides area
BThey must be different, because equal areas need different perimeters
CThey are equal only if both shapes are triangles
DThey may be equal or may be different; equal perimeters do not fix the area
Show answer and explanation
Correct answer: D - They may be equal or may be different; equal perimeters do not fix the area
Perimeter measures the boundary and area measures the surface inside, and knowing one does not fix the other. A $10\ \text{cm}$ by $2\ \text{cm}$ rectangle and a $6\ \text{cm}$ square both have perimeter $24\ \text{cm}$ but areas of $20\ \text{cm}^2$ and $36\ \text{cm}^2$. Two congruent shapes show that equal perimeters can go with equal areas too, so neither 'must be equal' nor 'must be different' holds, and the shape being a triangle changes nothing.
Q3
easy
Two radii are drawn in a circle, cutting it into two sectors. One of the sectors has a central angle of $110^\circ$. What is the central angle of the other sector?
A$180^\circ$
B$250^\circ$
C$290^\circ$
D$70^\circ$
Show answer and explanation
Correct answer: B - $250^\circ$
The two central angles together make one full turn, so the other is $360^\circ-110^\circ = 250^\circ$. Taking $180^\circ-110^\circ = 70^\circ$ treats the two angles as making a straight line instead of a full turn. An angle of $180^\circ$ would need the first sector to be a semicircle, and $290^\circ$ comes from subtracting $70^\circ$ from $360^\circ$.
Q4
medium
Two circles, each of radius $21\ \text{cm}$, are drawn so that each passes through the centre of the other. They cross at two points, and these points cut each circle into a minor arc of $120^\circ$ and a major arc of $240^\circ$. The outline of the whole figure is made of the two major arcs. Find the length of that outline. Take $\pi = \frac{22}{7}$.
A$264\ \text{cm}$
B$88\ \text{cm}$
C$132\ \text{cm}$
D$176\ \text{cm}$
Show answer and explanation
Correct answer: D - $176\ \text{cm}$
Each circle measures $2\pi r = 132\ \text{cm}$ round, so a $240^\circ$ arc is $132\times\frac{240}{360} = 88\ \text{cm}$. Two such arcs give $176\ \text{cm}$. The value $264\ \text{cm}$ adds both circles in full, which counts the parts hidden inside the other circle. $88\ \text{cm}$ is one major arc only, and $132\ \text{cm}$ is one whole circle.
Q5
medium
A rectangular sheet measuring $18\ \text{cm}$ by $8\ \text{cm}$ is cut up and rearranged into a square of exactly the same area. Find the perimeter of that square.
A$144\ \text{cm}$
B$12\ \text{cm}$
C$48\ \text{cm}$
D$52\ \text{cm}$
Show answer and explanation
Correct answer: C - $48\ \text{cm}$
The sheet covers $18\times 8 = 144\ \text{cm}^2$, so the square has side $12\ \text{cm}$ and perimeter $4\times 12 = 48\ \text{cm}$. The value $52\ \text{cm}$ is the rectangle's own perimeter, which does not survive the rearrangement. The value $12\ \text{cm}$ is the side of the square, not the distance round it, and $144\ \text{cm}$ is the area figure with a length unit.
Q6
easy
A straight line segment $AB$ is $40\ \text{cm}$ long. A semicircular arc is drawn with $AB$ as its diameter. How long is that arc? Take $\pi = 3.14$.
A$125.6\ \text{cm}$
B$31.4\ \text{cm}$
C$40\ \text{cm}$
D$62.8\ \text{cm}$
Show answer and explanation
Correct answer: D - $62.8\ \text{cm}$
The radius is $20\ \text{cm}$, so the semicircular arc is $\pi r = 3.14\times 20 = 62.8\ \text{cm}$. The value $125.6\ \text{cm}$ is the whole circle, twice as long as needed. Halving again gives $31.4\ \text{cm}$, which would belong to a quarter circle. The value $40\ \text{cm}$ is the straight distance from $A$ to $B$, not the curved path.
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