Area Questions - 1

The three sides of a triangle are 7 cm, 8 cm and 9 cm respectively. What is the area of the triangle ?
  • 12.5 sq.cm
  • 10$\displaystyle \sqrt{5}$ sq.cm
  • 12 sq.cm
  • 12$\displaystyle \sqrt{5}$ sq.cm
  • 12$\displaystyle \sqrt{8}$ sq.cm
Explanation   
sides of a triangle

a, b, c are sides, h is its altitude.

Perimeter = a + b + c.

Area = $\displaystyle \frac{1}{2}$ x base x height.

OR

$\displaystyle \sqrt{\text{s}(\text{s} \ – \ \text{a})(\text{s} \ - \ \text{b})(\text{s} \ - \ \text{c})}$

s = semi perimeter

s = $\displaystyle \frac{\text{a} \ + \ \text{b} \ + \ \text{c}}{2}$ => s = $\displaystyle \frac{7 + 8 + 9}{2} = \frac{24}{2}$ = 12 cm.

Area = $\displaystyle \sqrt{12(12 – 7)(12 – 8)( 12 – 9)}$

= $\displaystyle \sqrt{12(5)(4)(3)} = \sqrt{720} = \sqrt{144 \times 5}$

= 12$\displaystyle \sqrt{5}$ sq.cm.

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The perimeter of an isosceles triangle is 26 cm, and the lateral side to the base is in the ratio 4:5. What is the area of the triangle ?
  • 26 sq.cm
  • 5$\displaystyle \sqrt{39}$ sq.cm
  • 32 sq.cm
  • 6$\displaystyle \sqrt{6}$ sq.cm
  • 5$\displaystyle \sqrt{26}$ sq.cm
Explanation   
isosceles triangle

a is the length of each equal sides, b = third side.

Perimeter = 2a + b.

Area = $\displaystyle \frac{\text{b}}{4} \sqrt{4\text{a}^2 - \text{b}^2}$

Let sides be 4a : 4a : 5a = 26 cm.

13a = 26 or a = 2.

The two equal sides = 4 x 2 = 8 cm, 4 x 2 = 8cm, third side = 5 x 2 = 10 cm.

Area = $\displaystyle \frac{10}{4} \sqrt{4 \times 8^2 - 10^2} = 2.5\sqrt{256 - 100} = 2.5\sqrt{156}$

=> $\displaystyle 2.5\sqrt{4 \times 39} => 5\sqrt{39}$ sq.cm.

Workspace
The area of a triangle is 48 sq.cm and its base is 16 cm. What is the length of the altitude ?
  • 8 cm
  • 3 cm
  • 6 cm
  • 12 cm
  • 16 cm
Explanation   

Area of a triangle = $\displaystyle \frac{1}{2} \times$ base x height.

=> $\displaystyle \frac{1}{2} \times$ 16 x h = 48 sq.cm.

=> 8h = 48 => h = $\displaystyle \frac{48}{8}$ = 6 cm.

Altitude = 6 cm.

Workspace
The ratio of the areas of two similar equilateral triangles is 25 : 16. What is the ratio of sides ?
  • 1 : 2
  • 25 : 16
  • 4 : 5
  • 5 : 4
  • 3 : 4
Explanation   

Ratio of sides = $\displaystyle \sqrt{25} : \sqrt{16}$ = 5 : 4.

Workspace
What is the ratio between the area of a square of side ‘s’ and an equilateral triangle, both having equal length of side ‘s’?
  • 4 : $\displaystyle \sqrt{3}$
  • 4 : 3
  • $\displaystyle \sqrt{8}$ : 3
  • $\displaystyle \sqrt{3}$ : 4
  • 3 : 4
Explanation   

Area of a square = side2 or S2.

Area of a equilateral triangle = $\displaystyle \frac{\sqrt{3}}{4} \times$ side2

Ratio = $\displaystyle \frac{\text{Area of the square}}{\text{Area of the equilateral triangle}}$

=> $\displaystyle \frac{\text{Side}^2}{\sqrt{3} \ \text{S}^2} = \frac{\text{S}^2 \times 4}{\sqrt{3} \times \text{S}^2}$

=> $\displaystyle \frac{4}{\sqrt{3}}$ or 4 : $\displaystyle \sqrt{3}$

Workspace
Find the area of an isosceles triangle with sides 12 cm, 12 cm and 8 cm ?
  • 32 $\displaystyle \sqrt{6}$
  • 16 $\displaystyle \sqrt{2}$
  • 32 $\displaystyle \sqrt{3}$
  • 32 $\displaystyle \sqrt{2}$
  • 24 $\displaystyle \sqrt{2}$
Explanation   

a is the length of each equal sides, b = third side.

Perimeter = 2a + b.

Area of isosceles triangle.

Area = $\displaystyle \frac{\text{b}}{4} \sqrt{4\text{a}^2 \ – \ \text{b}^2}$

a = 12 cm, a = 12 cm, b = 8 cm.

= $\displaystyle \frac{8}{4} \sqrt{4 \times 12^2 – 8^2} = 2 \sqrt{512}$

=> 2$\displaystyle \sqrt{256 \times 2} = 32 \sqrt{2 \ \text{sqcm}}$

Workspace

If the side of an equilateral triangle is increased by 50%, then by how much will the area increase ?

I. Increases 2.25 times

II. Increases by 125%

III. 225% of earlier

  • only I
  • only II
  • Both I and II
  • All I, II and III
  • None of the above
Explanation   

Area of equilateral triangle = $\displaystyle \frac{\sqrt{3}}{4} \text{a}^2$, here a = side.

If the side is increased by 50%, then side becomes 1.5a.

$\displaystyle \frac{\sqrt{3}}{4} \times$ (1.5a)2 = $\displaystyle \frac{\sqrt{3}}{4} \times$ (2.25a)2.

$\displaystyle \frac{\sqrt{3}}{4} \times$ a2 = 2.25.

=> 2.25 times of or 225% of, or increases by 225% - 100% = 125%.

Workspace
If the area of an equilateral triangle is 16$\displaystyle \sqrt{3}$ sq.cm. What is the perimeter of the triangle ?
  • 32$\displaystyle \sqrt{3}$ cm
  • 24 cm
  • 24$\displaystyle \sqrt{3}$ cm
  • 36 cm
  • 39 cm
Explanation   

Area of equilateral triangle = $\displaystyle \frac{\sqrt{3}}{4} \text{a}^2 = 16 \ \sqrt{3}$

$\displaystyle \sqrt{3} \ \text{a}^2 = 64\sqrt{3}$ sq.cm.

a2 = $\displaystyle \frac{64\sqrt{3}}{\sqrt{3}}$ = a2 = 64 = a = 8 cm or side = 8 cm.

Perimeter of equilateral triangle = 3a = 3 x 8 = 24 cm.

Workspace
What is the altitude of an equilateral triangle with sides 8 cm ?
  • 4 cm
  • 3 $\displaystyle \sqrt{3}$ cm
  • 3 cm
  • 4 $\displaystyle \sqrt{3}$ cm
  • 6 cm
Explanation   

Altitude of equilateral triangle = $\displaystyle \frac{\sqrt{3}}{2} \times \text{side}$

Altitude = $\displaystyle \frac{\sqrt{3}}{2} \times 8 = 4 \sqrt{3}$ cm.

Workspace
If the altitude of an equilateral triangle is 5 $\displaystyle \sqrt{3}$ cm, then its area is ?
  • 15 sq.cm
  • 20 $\displaystyle \sqrt{3}$ sq.cm
  • 25 $\displaystyle \sqrt{3}$ sq.cm
  • 25 sq.cm
  • 5 $\displaystyle \sqrt{15}$ sq.cm
Explanation   

Altitude = $\displaystyle \frac{\sqrt{3} \ \text{a}}{2} = 5 \sqrt{3}$ ;     a = side.

Side = $\displaystyle \frac{10 \sqrt{3}}{\sqrt{3}}$ = 10 cm,

Area = $\displaystyle \frac{\sqrt{3}}{4} \times 10^2 = 25 \sqrt{3}$ sq.cm.

Workspace

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