Mensuration: Area and Perimeter

Find the perimeter of a regular hexagon with each side measuring $6.5$ cm.

If the perimeter of a rectangle is $24$ cm and its length is $8$ cm, find its breadth.

A piece of wire is $44$ cm long. If it is bent to form a circle, what is its radius? (Take $\pi = \frac{22}{7}$)

An isosceles triangle has equal sides of $8$ cm each and the third side is $10$ cm. Find its perimeter.

A walking track is designed in the shape of a rectangle with semi-circular ends. The rectangular part of the track has a length of $70$ m, and the radius of the semi-circular ends is $14$ m. If a student walks around the edge of the track exactly three times, what is the total distance covered? (Take $\pi = \frac{22}{7}$)

Find the perimeter of a quarter-circle of radius $7$ cm. (Take $\pi = \frac{22}{7}$)

The length of a rectangular field is twice its breadth. If the perimeter is $120$ m, find the length of the field.

Two circles touch each other externally. The sum of their areas is $130\pi\text{ cm}^2$ and the distance between their centers is $14\text{ cm}$. Find the difference in their circumferences.

The perimeter of a sector of a circle of radius $5.6\text{ cm}$ is $27.2\text{ cm}$. Find the length of the arc of the sector.

An architect designs a courtyard in the shape of a large circle of radius $10\text{ m}$. Inside the courtyard, there is a square pond such that the vertices of the square lie on the circle. Find the total perimeter of the four segment-shaped grass areas between the circular boundary and the square pond. (Use $\pi = 3.14$ and $\sqrt{2} = 1.414$)

Calculate the area of a ring with outer radius $10$ cm and inner radius $7$ cm. (Use $\pi \approx \frac{22}{7}$)

A bicycle wheel has a diameter of $70$ cm. How many revolutions does it make to cover $220$ meters? (Use $\pi \approx \frac{22}{7}$)

If the area of a circle is numerically equal to its circumference, what is the radius of the circle?

The perimeter of a circle is $176$ cm. Find its radius using $\pi \approx \frac{22}{7}$.

The radii of two circles are $8$ cm and $6$ cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

The radii of two circles are $19$ cm and $9$ cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

A circular park is $40$ m in diameter. It has a $2$ m wide gravel path around it on the outside. Find the cost of gravelling the path at Rs $10$ per m$^2$. (Use $\pi \approx 3.14$)

A field is in the form of a circle. A fence is to be erected around it at the rate of Rs $24$ per meter. The cost of fencing is Rs $5280$. The field is then ploughed at the rate of Rs $0.50$ per m$^2$. Find the cost of ploughing the field (use $\pi \approx \frac{22}{7}$). Why does using $\pi \approx \frac{22}{7}$ lead to a terminating decimal for the cost in this specific problem?

Four equal circles, each of radius $5$ cm, touch each other as shown. Find the area of the shaded region between them (use $\pi \approx 3.14$). Why is this area necessarily a transcendental number times a rational if $\pi$ is not approximated?

The diameter of a cycle wheel is $70$ cm. How many revolutions will it make to cover $110$ km? Use $\pi \approx \frac{22}{7}$. If we used $\pi \approx 3.14$, would the number of revolutions increase or decrease?

A circular arc of a sector subtends an angle of $60^{\circ}$ at the center of a circle with radius $21$ cm. Find the length of the arc. (Take $\pi = \frac{22}{7}$)

Find the length of the arc of a circle of radius 14 cm which subtends an angle of $45^\circ$ at the center.

What is the area of a circle that can be inscribed in a square of side 10 cm?

If the angle of a sector is $p^\circ$ and radius is $R$, which of the following is the area of the sector?

If a wire is bent into the shape of a square, then the area of the square is 81 $cm^2$. When the same wire is bent into a semi-circular shape, find the radius of the semi-circle. (Take $\pi = \frac{22}{7}$)

The diameter of a circular pond is 17.5 m. It is surrounded by a path of width 2 m. Find the cost of paving the path at Rs 100 per square meter. (Use $\pi = 3.14$)

An arc of a circle is of length $5\pi$ cm and the sector it bounds has an area of $20\pi$ $cm^2$. Find the radius of the circle.

A circular clock's minute hand is $14$ cm long. How much distance does the tip of the minute hand travel between 8:00 AM and 8:20 AM? (Take $\pi = \frac{22}{7}$)

An equilateral triangle has a side of $12$ cm. With each vertex as center, arcs are drawn with radius $6$ cm. Find the sum of the lengths of the three arcs contained within the triangle.

A cow is tethered to one corner of a rectangular grassy field of dimensions $20$ m $\times$ $16$ m by a rope $14$ m long. What is the length of the arc along which the cow can graze within the field? (Take $\pi = \frac{22}{7}$)

Which shape has a larger area: a rectangle with sides $6 \times 4$ or a square with side $5$?

The base of a triangle is doubled while the height remains the same. How does the area change?

A square has a side of $7 \text{ cm}$. What is its perimeter?

Calculate the area of a right-angled triangle where the legs are $6 \text{ cm}$ and $8 \text{ cm}$.

If the base of a parallelogram is $x+3$ and its height is $x-3$, and its area is $16 \text{ units}^2$, find the value of $x$.

The length of a rectangle is $2 \text{ cm}$ more than its width. If the perimeter is $28 \text{ cm}$, find its area.

Find the area of an isosceles triangle with base $10 \text{ cm}$ and equal sides of $13 \text{ cm}$ each.

An isosceles triangle has a perimeter of $32$ cm and the ratio of the equal side to its base is $3:2$. Find the area of the triangle.

In a triangle $ABC$, the medians $AD$, $BE$, and $CF$ intersect at $G$. If the area of $\triangle ABC$ is $108$ cm$^2$, find the area of the quadrilateral $BDGF$.

The area of a triangle with base $24$ cm is equal to the area of a square with side $12$ cm. Find the altitude of the triangle.

A triangle has sides $10$ cm and $10$ cm and the third side is $12$ cm. Calculate the area.

Find the area of a triangle with sides $7$ cm, $24$ cm, and $25$ cm.

If $s$ is the semi-perimeter of a triangle with sides $a, b, c$, the formula for area is:

What is the area of a triangle with sides $12$ cm, $16$ cm, and $20$ cm?

A triangular side wall of a flyover has been used for advertisements. The sides of the wall are $122$ m, $22$ m and $120$ m. The advertisements yield an earning of Rs $5000$ per m$^2$ per year. A company hired one of its walls for $3$ months. How much rent did it pay?

Find the area of a triangle two sides of which are $18$ cm and $10$ cm and the perimeter is $42$ cm.

A kite in the shape of a square with a diagonal $32$ cm and an isosceles triangle of base $8$ cm and sides $6$ cm each is made of three different shades. How much paper of each shade has been used in it? (Find the area of the bottom isosceles triangle)

Find the area of a shaded region in a layout where a large triangular plot (sides $40\text{ m}, 70\text{ m}, 90\text{ m}$) contains a smaller triangular flower bed (sides $10\text{ m}, 17\text{ m}, 21\text{ m}$).

The perimeter of a triangle is $50\text{ cm}$. One side of the triangle is $4\text{ cm}$ longer than the smallest side and the third side is $6\text{ cm}$ less than twice the smallest side. Find the area of the triangle.

A quadrilateral $ABCD$ has $AB = 9\text{ m}, BC = 40\text{ m}, CD = 28\text{ m}, DA = 15\text{ m}$ and $\angle ABC = 90^\circ$. Find the total area of the quadrilateral.

The diameter of a wheel is $84\text{ cm}$. How many revolutions will it make to cover $792\text{ m}$?

In a circle of radius $10\text{ cm}$, an arc subtends an angle of $90^\circ$ at the centre. Find the area of the minor sector. (Use $\pi = 3.14$)

Find the area of a circle whose circumference is $44\text{ cm}$.

A wire is in the form of a circle of radius $28\text{ cm}$. If it is bent into a square, find the side of the square.

A circular park is surrounded by a road $7\text{ m}$ wide. If the radius of the park is $21\text{ m}$, find the area of the road.

The area of a circle is $220\text{ cm}^2$. Find the area of a square inscribed in this circle.

An equilateral triangle of side $6\text{ cm}$ is inscribed in a circle. Find the area of the circle. (Use $\pi = 3.14$)

Find the area of a sector of a circle whose radius is $r$ and the length of the arc is $l$.

A field is in the form of a circle. A fence is put around it at the rate of Rs $24$ per meter, costing Rs $5280$. The field is then ploughed at the rate of Rs $0.50$ per m$^2$. Find the total cost of ploughing the field.

If the perimeter of a semi-circular protractor is $72$ cm, calculate its diameter. (Use $\pi = \frac{22}{7}$)

Find the area of an equilateral triangle with side $2$ cm using Heron's formula.

A rectangle with sides $6$ cm and $8$ cm is always cyclic. Calculate its semi-perimeter $s$.

For a triangle with sides $a, b, c$, Heron's formula is $Area = \sqrt{s(s-a)(s-b)(s-c)}$. In the context of Brahmagupta's formula, which 'side' is effectively zero?

Which of the following conditions must be met to use Brahmagupta's formula $A = \sqrt{(s-a)(s-b)(s-c)(s-d)}$?

A cyclic quadrilateral has sides $a=3, b=3, c=4, d=4$. This is a rectangle. Use Brahmagupta's formula to find its area.

If the semi-perimeter $s$ of a cyclic quadrilateral is $10$ cm and its sides are $a, b, c, d$, what is the maximum possible value for $(s-a)(s-b)(s-c)(s-d)$ when all sides are equal?

A kite is inscribed in a circle (making it a right kite). If its distinct side lengths are $7$ cm and $24$ cm, find its area using Brahmagupta's formula.

Prove the connection: A cyclic quadrilateral has sides $a, b, c, d$. If $d \rightarrow 0$, the formula $\sqrt{(s-a)(s-b)(s-c)(s-d)}$ becomes Heron's formula. For a cyclic quadrilateral with sides $10, 17, 21, 0$, find the area using both perspectives.

A cyclic quadrilateral has sides $2\text{ cm}$, $3\text{ cm}$, $4\text{ cm}$, and $x\text{ cm}$. If the semi-perimeter is $6\text{ cm}$, find the value of $x$ and then calculate the area of the quadrilateral.

A decorative tile is a cyclic quadrilateral with sides $a=10\text{ cm}$, $b=10\text{ cm}$, $c=16\text{ cm}$, and $d=16\text{ cm}$. By observing the symmetry and using Brahmagupta's formula, determine its area.