If the side of a square is decreased by 40%, find the percentage change in its area.

decrease of 64%. Let the original side = S, so original area = S^2. The side is decreased by 40 percent, so new side = S x (1 - 40/100). New area = (new side)^2 = S^2 x (1 - 40/100)^2, since area scales with the square of the side. Squaring the multiplier gives new area = 36% of the original area. So the change is a decrease of 64 percent.

If the side of a square is increased by 10%, find the percentage change in its area.

increase of 21%. Let the original side = S, so original area = S^2. The side is increased by 10 percent, so new side = S x (1 + 10/100). New area = (new side)^2 = S^2 x (1 + 10/100)^2, since area scales with the square of the side. Squaring the multiplier gives new area = 121% of the original area. So the change is a increase of 21 percent.

Find the area of a triangle in which two sides measure 60 cm and 32 cm, with an included angle of 30 degrees.

480 sq cm. Let the two sides be a = 60 cm and b = 32 cm, with included angle theta = 30 degrees. Area of a triangle = (1/2) x a x b x sin(theta). sin(30 deg) = 1/2. Area = (1/2) x 60 x 32 x 1/2 = 480 sq cm.

Find the area of a triangle in which two sides measure 34 cm and 24 cm, with an included angle of 90 degrees.

408 sq cm. Let the two sides be a = 34 cm and b = 24 cm, with included angle theta = 90 degrees. Area of a triangle = (1/2) x a x b x sin(theta). sin(90 deg) = 1. Area = (1/2) x 34 x 24 x 1 = 408 sq cm.

A rectangular field is 53 m long and 38 m wide. Find the cost of fencing it at Rs 12 per metre.

Rs 2184. Let length = L = 53 m, breadth = B = 38 m, rate = Rs 12 per metre. Perimeter to be fenced = 2 x (L + B) = 2 x (53+38) = 182 m. Cost of fencing = perimeter x rate = 182 x 12 = Rs 2184.

A rectangular room is 28 m long and 26 m wide. Find the cost of flooring it at Rs 100 per sq m.

Rs 72800. Let length = L = 28 m, breadth = B = 26 m, rate = Rs 100 per sq m. Area of the room = L x B = 28 x 26 = 728 sq m. Cost of flooring = area x rate = 728 x 100 = Rs 72800.

A room is 28 m long, 22 m wide, and 12 m high. Find the area of its four walls.

1200 sq m. Let length = L = 28 m, breadth = B = 22 m, height = h = 12 m. Area of the four walls = perimeter of the floor x height = 2 x (L + B) x h. Perimeter = 2 x (28+22) = 100 m. Area of walls = 100 x 12 = 1200 sq m.

Area: Formulas, Shortcuts and Solved Examples

Area questions ask how much surface a shape covers, or what it costs to cover or fence it.

This lesson gives each formula and the reason it works. It then solves three examples and lists the traps that cost marks.

The idea in brief

Rectangle: l × b.
Triangle: ½ × base × height.
Trapezium: ½ × (a + b) × h.
Four walls: 2(l + b) × h.
Trap: fencing is perimeter, flooring is area.

What Area Questions Test

What is area?

Area is the space a flat shape covers. It is measured in square units, such as sq cm or sq m.

Perimeter is different. It is the length around the shape, in plain units like m.

When do you need area, and when perimeter?

Use area for flooring, painting, tiling and covering a surface.

Use perimeter for fencing, a border or a boundary rope.

Check 1

Fencing a field needs the area of the field.

Core Formulas and Why They Work

What are the rectangle and square formulas?

Rectangle: area = l × b. Perimeter = 2(l + b).Square: area = a². Perimeter = 4a.

Why it works: l × b counts the unit squares in l rows of b squares each. A square is a rectangle with l = b.

Rectangle, triangle and trapezium. The dashed line is the height, always at right angles to the base.

How do you find a triangle’s area?

Area = ½ × base × height.Area = ½ × a × b × sin C. Use this when two sides and the angle between them are known.

Why it works: a triangle is half of a rectangle or parallelogram with the same base and height. With sin 30° = ½ and sin 90° = 1, the numbers get easy.

What is the trapezium formula?

Area = ½ × (sum of parallel sides) × height.

Why it works: split it into a rectangle and two right triangles. Or join two copies into a parallelogram with base (a + b).

Sources: NCERT Class 8 Mathematics, Mensuration.

How do you find the area of four walls?

Four walls = 2(l + b) × h.

Why it works: unroll the four walls into one long strip. Its length is the perimeter of the floor, and its height is h.

What happens to area when sides change by a percentage?

If a side changes by x%, the area of a square changes by x + x + x²/100 percent.

Why it works: the area is side times side. Both factors change, so the changes combine.

The four walls opened out into one strip. Area = strip length × height = 2(l + b) × h.

What are the circle, parallelogram and rhombus formulas?

Circle: area = πr². Circumference = 2πr.Parallelogram: area = base × height.Rhombus: area = ½ × d₁ × d₂. Here d₁ and d₂ are the two diagonals.

Why they work: a parallelogram can be cut and moved into a rectangle of the same base and height. A rhombus is four equal right triangles. Each has legs d₁/2 and d₂/2.

How do you find the area of a path or border?

Path area = outer area − inner area.

Why it works: the path is whatever is left when you remove the inner shape from the outer shape.

For a path of width w around an l by b field, the outer shape is (l + 2w) by (b + 2w).

Check 2

The area of four walls is 2(l + b) × h.

Three Solved Examples

Easy: a field is 53 m by 38 m. What does fencing cost at Rs 12 per metre?

Fencing needs the perimeter. Perimeter = 2(53 + 38) = 182 m.

Cost = 182 × 12 = Rs 2184.

Medium: find the area of a triangle with sides 60 cm and 32 cm and an angle of 30° between them.

Use ½ × a × b × sin C.

That is ½ × 60 × 32 × ½ = 480.

So the area is 480 sq cm.

Exam level: the side of a square is cut by 40%. What is the change in area?

Let the old side be 100. The old area is 10,000.

The new side is 60. The new area is 3,600.

The area falls by 6,400, which is 64% of the old area.

Step by step: a park is 40 m by 30 m. A path 2 m wide runs around the outside. What is the path area?

Step 1: inner area = 40 × 30 = 1200 sq m.

Step 2: the path adds 2 m on each side. Outer size = (40 + 4) by (30 + 4) = 44 by 34.

Step 3: outer area = 44 × 34 = 1496 sq m.

Step 4: path area = 1496 − 1200 = 296 sq m.

Step by step: a room is 8 m by 6 m and 3 m high. Find the cost of painting four walls at Rs 20 per sq m.

Step 1: four walls = 2(l + b) × h = 2(8 + 6) × 3.

Step 2: that is 28 × 3 = 84 sq m.

Step 3: cost = 84 × 20 = Rs 1680. The floor and ceiling are not painted, so they are left out.

Check 3

A 10% increase in a square’s side raises its area by 10%.

Common Traps and Shortcuts

Which traps cost marks?

Mixing up area and perimeter. Fencing is perimeter. Flooring is area.

Mixing units. Convert cm to m before you multiply.

Using a slant side as the height in a triangle or trapezium.

Applying the same percentage to area as to the side.

Which shortcuts save time?

For a right-angled triangle, the area is half the product of the two shorter sides.

For flooring cost, multiply l × b first and the rate last.

For the change in a square’s area, square the multiplier: 0.6² = 0.36.

Check that your answer has the right unit before you pick an option.

Check 4

A right-angled triangle with legs 34 and 24 has area 408.

0/0 understanding checks correct

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