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Area, Perimeter and Volume with Algebra

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 13 marks

A triangle has base (x+5) cm and perpendicular height 2x cm.

A square has sides of length x cm.

The area of the triangle is three times the area of the square.

Work out the value of x.

Hint

Area of a triangle =\frac12\times base \times height; set up an equation and remember x cannot be 0.

Worked solution
  1. Area of triangle =\frac12(x+5)(2x)=x(x+5)=x^2+5x
  2. Area of square =x^2
  3. x^2+5x=3x^2
  4. 2x^2-5x=0
  5. x(2x-5)=0
  6. x=0 is impossible for a length, so 2x-5=0
  7. x=2.5

Question 22 marks

A pyramid has a square base with sides of length 2x cm and perpendicular height h cm.

A cube has edges of length x cm.

The pyramid and the cube have the same volume.

Volume of a pyramid =\frac13\times area of base \times h

Work out h in terms of x.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

The area of the base is (2x)^2, not 2x^2.

Worked solution
  1. Area of base =(2x)^2=4x^2
  2. Volume of pyramid =\frac13\times 4x^2\times h=\frac{4x^2h}{3}
  3. Volume of cube =x^3
  4. \frac{4x^2h}{3}=x^3
  5. 4x^2h=3x^3
  6. h=\frac{3x^3}{4x^2}=\frac{3x}{4}

Question 3Challenge6 marks

Two circles have the same centre.

The smaller circle has radius x cm.

The larger circle has radius (x+3) cm.

The area of the region between the two circles is equal to the area of the smaller circle.

(a)

Work out the value of x.

Give your answer in the form a+b\sqrt2, where a and b are integers.

4 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Work out the area of the larger circle.

Give your answer in the form (p+q\sqrt2)\pi cm^2, where p and q are integers.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Write the area of the ring as the larger circle's area minus the smaller circle's area, then divide through by \pi.

Worked solution

Part (a)

  1. Area between the circles =\pi(x+3)^2-\pi x^2
  2. Set equal to the smaller circle: \pi(x+3)^2-\pi x^2=\pi x^2
  3. Divide by \pi: x^2+6x+9-x^2=x^2
  4. x^2-6x-9=0
  5. x=\dfrac{6\pm\sqrt{36+36}}{2}=\dfrac{6\pm 6\sqrt2}{2}
  6. x=3\pm3\sqrt2
  7. 3-3\sqrt2 is negative, so reject it
  8. x=3+3\sqrt2

Part (b)

  1. Radius of larger circle =x+3=6+3\sqrt2
  2. (6+3\sqrt2)^2=36+36\sqrt2+18=54+36\sqrt2
  3. Area =(54+36\sqrt2)\pi cm^2
  4. (Check: the larger circle is twice the smaller one, 2\pi(3+3\sqrt2)^2=2\pi(27+18\sqrt2) ✓)

More on this topic: Area, Perimeter and Volume with Algebra worksheet with full solutions

All AQA Level 2 Further Maths practice questions

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