Sec 3 A-Math · Coordinate geometry

Area of a rectilinear figure questions with answers

Practice questions on area of a rectilinear figure, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Find the area of triangle $ABC$ with vertices $A(-2, -2)$ , $B(6, 2)$ and $C(0, 5)$ .

Step 1 of 3

List the vertices in order round the shape, and repeat the first at the end

\text{Area} = \dfrac{1}{2}\left| \begin{matrix} -2 & 6 & 0 & -2 \\[8pt] -2 & 2 & 5 & -2 \end{matrix} \right|

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Questions

Question 1Triangle
Find the area of triangle $ABC$ with vertices $A(2, 3)$ , $B(8, -1)$ and $C(6, 6)$ .
Show the worked solution
1. Step 1: List the vertices in order round the shape, and repeat the first at the end $\text{Area} = \dfrac{1}{2}\left| \begin{matrix} 2 & 8 & 6 & 2 \\[8pt] 3 & -1 & 6 & 3 \end{matrix} \right|$
2. Step 2: Multiply down to the right and add; multiply up to the right and add $(2)(-1) + 8(6) + 6(3)$ $= 64$ $(8)(3) + 6(-1) + 2(6)$ $= 30$
3. Step 3: Half the difference (taken as positive) $\text{Area} = \dfrac{1}{2}|64 - 30|$ $= 17$
Question 2Triangle
Find the area of triangle $ABC$ with vertices $A(-1, 0)$ , $B(4, -4)$ and $C(5, 3)$ .
Show the worked solution
1. Step 1: List the vertices in order round the shape, and repeat the first at the end $\text{Area} = \dfrac{1}{2}\left| \begin{matrix} -1 & 4 & 5 & -1 \\[8pt] 0 & -4 & 3 & 0 \end{matrix} \right|$
2. Step 2: Multiply down to the right and add; multiply up to the right and add $(-1)(-4) + 4(3) + 5(0)$ $= 16$ $(4)(0) + 5(-4) + (-1)(3)$ $= -23$
3. Step 3: Half the difference (taken as positive) $\text{Area} = \dfrac{1}{2}|16 - (-23)|$ $= 19.5$
Question 3Quadrilateral
Find the area of quadrilateral $ABCD$ with vertices $A(1, 1)$ , $B(5, -2)$ , $C(8, 3)$ and $D(3, 6)$ .
Show the worked solution
1. Step 1: List the vertices in order round the shape, and repeat the first at the end $\text{Area} = \dfrac{1}{2}\left| \begin{matrix} 1 & 5 & 8 & 3 & 1 \\[8pt] 1 & -2 & 3 & 6 & 1 \end{matrix} \right|$
2. Step 2: Multiply down to the right and add; multiply up to the right and add $(1)(-2) + 5(3) + 8(6) + 3(1)$ $= 64$ $(5)(1) + 8(-2) + 3(3) + 1(6)$ $= 4$
3. Step 3: Half the difference (taken as positive) $\text{Area} = \dfrac{1}{2}|64 - 4|$ $= 30$
Question 4Quadrilateral
Find the area of quadrilateral $ABCD$ with vertices $A(1, 3)$ , $B(6, 0)$ , $C(9, 4)$ and $D(4, 8)$ .
Show the worked solution
1. Step 1: List the vertices in order round the shape, and repeat the first at the end $\text{Area} = \dfrac{1}{2}\left| \begin{matrix} 1 & 6 & 9 & 4 & 1 \\[8pt] 3 & 0 & 4 & 8 & 3 \end{matrix} \right|$
2. Step 2: Multiply down to the right and add; multiply up to the right and add $(1)(0) + 6(4) + 9(8) + 4(3)$ $= 108$ $(6)(3) + 9(0) + 4(4) + 1(8)$ $= 42$
3. Step 3: Half the difference (taken as positive) $\text{Area} = \dfrac{1}{2}|108 - 42|$ $= 33$

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Worked example

Find the area of triangle

ABC

with vertices

A(-2, -2)

B(6, 2)

and

C(0, 5)

Step 1 of 3

List the vertices in order round the shape, and repeat the first at the end

\text{Area} = \dfrac{1}{2}\left| \begin{matrix} -2 & 6 & 0 & -2 \\[8pt] -2 & 2 & 5 & -2 \end{matrix} \right|

Questions

Question 1Triangle

Find the area of triangle $ABC$ with vertices $A(2, 3)$ , $B(8, -1)$ and $C(6, 6)$ .

Show the worked solution

Step 1: List the vertices in order round the shape, and repeat the first at the end $\text{Area} = \dfrac{1}{2}\left| \begin{matrix} 2 & 8 & 6 & 2 \\[8pt] 3 & -1 & 6 & 3 \end{matrix} \right|$
Step 2: Multiply down to the right and add; multiply up to the right and add $(2)(-1) + 8(6) + 6(3)$ $= 64$ $(8)(3) + 6(-1) + 2(6)$ $= 30$
Step 3: Half the difference (taken as positive) $\text{Area} = \dfrac{1}{2}|64 - 30|$ $= 17$

Question 2Triangle

Find the area of triangle $ABC$ with vertices $A(-1, 0)$ , $B(4, -4)$ and $C(5, 3)$ .

Show the worked solution

Step 1: List the vertices in order round the shape, and repeat the first at the end $\text{Area} = \dfrac{1}{2}\left| \begin{matrix} -1 & 4 & 5 & -1 \\[8pt] 0 & -4 & 3 & 0 \end{matrix} \right|$
Step 2: Multiply down to the right and add; multiply up to the right and add $(-1)(-4) + 4(3) + 5(0)$ $= 16$ $(4)(0) + 5(-4) + (-1)(3)$ $= -23$
Step 3: Half the difference (taken as positive) $\text{Area} = \dfrac{1}{2}|16 - (-23)|$ $= 19.5$

Question 3Quadrilateral

Find the area of quadrilateral $ABCD$ with vertices $A(1, 1)$ , $B(5, -2)$ , $C(8, 3)$ and $D(3, 6)$ .

Show the worked solution

Step 1: List the vertices in order round the shape, and repeat the first at the end $\text{Area} = \dfrac{1}{2}\left| \begin{matrix} 1 & 5 & 8 & 3 & 1 \\[8pt] 1 & -2 & 3 & 6 & 1 \end{matrix} \right|$
Step 2: Multiply down to the right and add; multiply up to the right and add $(1)(-2) + 5(3) + 8(6) + 3(1)$ $= 64$ $(5)(1) + 8(-2) + 3(3) + 1(6)$ $= 4$
Step 3: Half the difference (taken as positive) $\text{Area} = \dfrac{1}{2}|64 - 4|$ $= 30$

Question 4Quadrilateral

Find the area of quadrilateral $ABCD$ with vertices $A(1, 3)$ , $B(6, 0)$ , $C(9, 4)$ and $D(4, 8)$ .

Show the worked solution

Step 1: List the vertices in order round the shape, and repeat the first at the end $\text{Area} = \dfrac{1}{2}\left| \begin{matrix} 1 & 6 & 9 & 4 & 1 \\[8pt] 3 & 0 & 4 & 8 & 3 \end{matrix} \right|$
Step 2: Multiply down to the right and add; multiply up to the right and add $(1)(0) + 6(4) + 9(8) + 4(3)$ $= 108$ $(6)(3) + 9(0) + 4(4) + 1(8)$ $= 42$
Step 3: Half the difference (taken as positive) $\text{Area} = \dfrac{1}{2}|108 - 42|$ $= 33$